Jan

01/05/21
Gleb Smirnov  ETH Zurich
Infinitely many nonisotopic real symplectic forms on the quadric surface
AbstractA real symplectic manifold is a symplectic manifold endowed with an involution which is antisymplectic. Given a real symplectic manifold, we may ask: are there any antiinvariant symplectic forms which are cohomologous but not isotopic within antiinvariant forms? In this talk, we will show that such disconnectivity indeed appears for certain real quadric surfaces.

01/05/21
 UC San Diego
Organizational meeting

01/05/21
Emily Zhu  Ph.D. student  UC San Diego
Ramsey Theory: Old and New
AbstractWe introduce some classical problems in graph Ramsey Theory. We also discuss new results on a lessclassical problem, namely, multicolor Ramsey numbers for some triple system paths of length three. The latter is joint work with Tom Bohman. Come for the tikzpictures, stay for the Ramsey Theory (or vice versa)!

01/05/21
Danna Zhang  UC San Diego
High dimensional testing for nonGaussian data
AbstractHigh dimensional nonGaussian data are increasingly encountered in a wide range of applications. It poses new challenges to traditional statistical tools. In this talk, we will present some recent development on methodologies and theories for the analysis of fattailed data as well as some high dimensional estimation and inference problems.

01/07/21
Jose Perea  Michigan State University
Learning functions on the space of persistence diagrams
AbstractThe persistence diagram is an increasingly useful shape descriptor from Topological Data Analysis, but its use alongside typical machine learning techniques requires mathematical finesse. We will describe in this talk a mathematical framework for featurization of said descriptors, and we show how it addresses the problem of approximating continuous functions on compact subsets of the space of persistence diagrams. We will also show how these techniques can be applied to problems in semisupervised learning where these descriptors are relevant.

01/07/21
Joshua Lam  Ph.D. student  Harvard University
CalabiYau varieties and Shimura varieties
AbstractI will discuss the Attractor Conjecture for CalabiYau
varieties, which was formulated by Moore in the nineties, highlighting
the difference between CalabiYau varieties with and without Shimura
moduli. In the Shimura case, I show that the conjecture holds and gives
rise to an explicit parametrization of CM points on certain Shimura
varieties; in the case without Shimura moduli, I'll present
counterexamples to the conjecture using unlikely intersection theory.
\\
\\
Part of this is joint work with Arnav Tripathy. 
01/12/21
Minju Lee  Graduate student  Yale University
Invariant measures for horospherical actions and Anosov groups
AbstractLet $\Gamma$ be an Anosov subgroup of a connected semisimple real linear Lie group $G$. For a maximal horospherical subgroup $N$ of $G$, we show that the space of all nontrivial $NM$invariant ergodic and $A$quasiinvariant Radon measures on $\Gamma \backslash G$, up to proportionality, is homeomorphic to $\mathbb{R}^{\mathrm{rank} (G)1}$, where $A$ is a maximal real split torus and $M$ is a maximal compact subgroup which normalizes $N$.
\\
\\
This is joint work with Hee Oh. 
01/12/21
Jianfeng Lin  UC San Diego
Isotopy of the Dehn twist on $K3\#K3$ after a single stabilization
AbstractKronheimerMrowka recently proved that the Dehn twist along a 3sphere in the neck of $K3\#K3$ is not smoothly isotopic to the identity. This provides a new example of selfdiffeomorphisms on 4manifolds that are isotopic to the identity in the topological category but not smoothly so. (The first such examples were given by Ruberman.)
\\
\\
In this talk, we study the BauerFuruta invariant as an element in the Pin(2)equivariant stable homotopy group of spheres. We use it to show that this Dehn twist is not smoothly isotopic to the identity even after a single stabilization (connected summing with the identity map on S2 cross S2). This gives the first example of exotic phenomena on simplyconnected smooth 4manifolds that do not disappear after a single stabilization. In particular, it implies that one stabilization is not enough in the diffeomorphism isotopy problem for 4manifolds. It gives an interesting comparison with AucklyKimMelvinRubermanSchwartz's theorem that one stabilization is enough in the surface isotopy problem. 
01/12/21
Stephanie Wang  Department of Computer Science and Engineering, UC San Diego
Capturing Surfaces with Differential Forms

01/12/21
Sam Spiro  Ph.D. Candidate  UC San Diego
Some Selected Picks from Extremal Combinatorics
AbstractIn this talk things will get extreme, but not with number fields or schemes. Instead we'll focus on combinatorics, though we won't discuss clever tricks. We speak only of glorious theorems, as well as some notorious problems. And for all those that attend, there is a surprise at the end!

01/12/21
David Stapleton  UC San Diego
Projective space, hypersurfaces, and algebraic geometry
AbstractWe give a quick and friendly introduction to projective space and then introduce and explore some of the most elementary and fundamental examples in algebraic geometry: hypersurfaces in projective space (especially cubic hypersurfaces).

01/13/21
Riccardo Tione  EPFL
Anisotropic energies: examples, rectifiability and regularity
AbstractAnisotropic energies are functionals defined by integrating over a generalized surface (such as a current or a varifold) an integrand depending on the tangent plane to the surface. In the case of a constant positive integrand, one obtains the area functional, and hence one can see anisotropic energies as a generalization of it. A long standing question in geometric measure theory is to establish regularity properties of critical points to such functionals. In this talk, I will discuss some recent developments on this theory, addressing in particular the question of rectifiability of stationary points and regularity of stationary Lipschitz graphs.
\\
\\
The talk is based on joint work with Antonio De Rosa. 
01/13/21
Li Wang  University of Texas at Arlington
A SelfConsistentField Iteration for Orthogonal Canonical Correlation Analysis

01/14/21
Dan Daniel ErdmannPham  UC Berkeley
Hydrodynamics of the inhomogeneous lTASEP and its Application to Protein Synthesis
AbstractThe inhomogeneous lTASEP is an interacting particle process wherein particles stochastically enter, unidirectionally traverse, and finally exit a onedimensional lattice segment at rates that may depend on a particle's location within the lattice. Its homogeneous version is known to exhibit various phase transitions in macroscopic observables like particle density and current, with fluctuations governed by what is known as the KPZ equation. In this talk, we begin to extend such results to the inhomogeneous setting by developing the socalled hydrodynamic limit, which governs the system dynamics on an LLNtype scale. If time permits, we apply our results to elucidate the key determinants of protein synthesis, which motivated the introduction of TASEP fifty years ago.
\\
\\
This is based on joint work with Khanh Dao Duc and Yun S. Song. 
01/14/21

01/14/21
Philippe Robert  INRIA, Paris
Stochastic Models of Neural Synaptic Plasticity
AbstractIn neuroscience, learning and memory are usually associated to longterm changes of neuronal connectivity. Synaptic plasticity refers to the set of mechanisms driving the dynamics of neuronal connections, called synapses and represented by a scalar value, the synaptic weight. SpikeTiming Dependent Plasticity (STDP) is a biologicallybased model representing the time evolution of the synaptic weight as a functional of the past spiking activity of adjacent neurons.
\\
\\
In this talk we present a new, general, mathematical framework to study synaptic plasticity associated to different STDP rules. The system composed of two neurons connected by a single synapse is investigated and a stochastic process describing its dynamical behavior is presented and analyzed. We show that a large number of STDP rules from neuroscience and physics can be represented by this formalism. Several aspects of these models are discussed and compared to canonical models of computational neuroscience. An important subclass of plasticity kernels with a Markovian formulation is also defined and investigated via averaging principles.
\\
\\
Joint work with Gaetan Vignoud 
01/14/21
Aranya Lahiri  Indiana University
Resolutions of locally analytic principal series representations of $GL_2(F)$
AbstractLocally analytic representations of $p$adic analytic groups
have played a crucial role in many areas of arithmetic and
representation theory (including in $p$adic local Langlands program)
since their introduction by Schneider and Teitelbaum. In this talk we
will briefly review some aspects of the theory of locally analytic
representations. Then, for a locally analytic representation $V$ of
$GL_2(F)$ we will construct a coefficient system attached to the
BruhatTits tree of $Gl_2(F)$. Finally we will use this coefficient
system to construct a resolution for locally analytic principal series
of $GL_2(F)$. 
01/14/21
Sam Babichenko  Undergraduate student  UC San Diego
``Mean Field Games and Interacting Particle Systems'' following David Lacker.

01/15/21
Roberto Svaldi  EPFL
Applications of birational geometry to holomorphic foliations, part 2
AbstractThis will be the continuation to Calum's talk. The plan,
building on what Calum explained, is to discuss some recent work
building towards the birational classification of holomorphic foliations
on projective varieties (particularly 3folds) in the spirit of the
Minimal Model program.
\\
\\
We will explain some applications of these ideas to the study of the
dynamics and geometry of foliations and foliation singularities.
\\
\\
Features works of C. Spicer and P. Cascini and joint work with C. Spicer. 
01/19/21
Felix Weilacher  Graduate student  Carnegie Mellon University
Marked groups with isomorphic Cayley graphs but different Descriptive combinatorics
AbstractWe discuss the relationship between the Borel measurable / Baire measurable combinatorics of the action of a finitely generated group on its Bernoulli shift and the discrete combinatorics of the multiplication action of that group on itself. Our focus is on various chromatic numbers of graphs generated by these actions. We show that marked groups with isomorphic Cayley graphs can have Borel measurable / Baire measurable chromatic numbers which differ by arbitrarily much. In the Borel twoended, Baire measurable, and measurable hyperfinite settings, we show our constructions are nearly best possible (up to only a single additional color), and we discuss prospects for improving our constructions in the general Borel setting. Along the way, we will get tightness of some bounds of Conley and Miller on Baire measurable and measurable chromatic numbers of locally finite Borel graphs.

01/19/21
Foling Zou  University of Michigan
Equivariant nonabelian Poincar\'e duality and equivariant factorization homology of Thom spectra
AbstractThis is joint work with Asaf Horev and Inbar Klang. Factorization homology theories are invariants of $n$manifolds with coefficients in suitable $E_n$algebras. Let $G$ be a finite group and $V$ be a finite dimensional $G$representation. The equivariant factorization homology for $V$framed $G$manifolds have $E_V$algebra as coefficients. We show that when coefficient algebra $A$ is the Thom spectrum of an $E_{V+W}$map for a large enough representation $W$, the factorization homology of $A$ can be computed by a certain Thom spectrum. With nonabelian Poincar\'e duality theorem, we are able to simplify the result in some cases. In particular, we compute $\mathrm{THR}(\mathrm{H}\mathbb{F}_{2})$, $\mathrm{THR}(\mathrm{H}\mathbb{Z}_{(2)})$, $\mathrm{THH}_{C_2}(\mathrm{H}\mathbb{F}_2)$.

01/19/21
Xiaochuan Tian  UC San Diego
Numerical methods for nonlocal models: asymptotically compatible schemes and multiscale modeling
AbstractNonlocal continuum models are in general integrodifferential equations in place of the conventional partial differential equations. While nonlocal models show their effectiveness in modeling a number of anomalous and singular processes in physics and material sciences, for example, the peridynamics model of fracture mechanics, they also come with increased difficulty in computation with nonlocality involved. In this talk, we will give a review of the asymptotically compatible schemes for nonlocal models with a parameter dependence. Such numerical schemes are robust under the change of the nonlocal length parameter and are suitable for multiscale simulations where nonlocal and local models are coupled. Some open questions will also be discussed.

01/19/21
Cameron Cinel  Ph.D. student  UC San Diego
Polynomial Identity Algebras and the Kurosh Problem
AbstractThe Kurosh problem can be seen as an analogue of the Burnside problem for algebras. It asks whether or not a finitely generated algebra over a field has finite dimension. While the answer is negative in general, you don't have to go all the way to \textbf{Chinatown} to find a class of algebras for which the answer is affirmative.
\\
\\
In this talk, we will show that for algebras satisfying a polynomial identity (PI algebras), the Kurosh problem is true. Along the way, we will have a \textbf{(The) Conversation} about the basics of the theory of PI algebras, discussing their properties, constructing specific identities for classes of algebras, and looking at their structure. Time permitting, before we say our \textbf{(Long) Goodbyes} we will also look at another type of \textbf{Nice (Guys)} identities: central polynomials. Additionally, we will use them to prove a not so \textbf{Small (Town Crime)} result, Rowan's theorem.
\\
\\
Hopefully, after this talk, as \textbf{Twilight} turns into \textbf{Night (Moves)}, you will go to \textbf{(The Big) Sleep} dreaming about PI algebras. 
01/19/21
Zhouli Xu  UC San Diego
Generalized Poincare Conjecture, Homotopy Groups of Spheres, and the Motivic Adams spectral sequence
AbstractI will introduce and discuss some recent development of a fundamental problem in topology  the classification of continuous maps between spheres up to homotopy. These mathematical objects are called homotopy groups of spheres. I will start with some geometric background  its connection to the Generalized Poincare Conjecture for example. I will then introduce some classical and new methods of doing such computations, using certain spectral sequences. If time permits, I will discuss some recent development using motivic homotopy theory, a theory that was designed to use algebraic topology to study algebraic geometry, but has now been applied successfully in the reverse direction. Old and new open problems will be mentioned along the discussion.

01/20/21
Yiming Zhao  MIT
Mass transport problem on the unit sphere via Gauss map
AbstractIn this talk, I will discuss when two probability measures on the unit sphere can be transported to one another using the Gauss map of a convex body. Here, a convex body is a compact convex subset of the Euclidean nspace with nonempty interior. Notice that the boundary of a convex body might not be smoothin general, it can even contain a fractal structure. This problem can be viewed as the problem of reconstructing a convex body using partial data regarding its Gauss map. When smoothness is assumed, it reduces to a MongeAmpere type equation on the sphere. However, in this talk, we will work with generic convex bodies and talk about how variational argument can work in this setting.
\\
\\
This is joint work with K\'aroly B"{o}r"{o}czky, Erwin Lutwak, Deane Yang, and Gaoyong Zhang. 
01/21/21
Daniel Kane  UC San Diego
Point Location and Active Learning
AbstractIn the point location problem one is given a hyperplane arrangement and an unknown point. By making linear queries about that point one wants to determine which cell of the hyperplane arrangement it lies in. This problem has an unexpected connection to the problem in machine learning of actively learning a halfspace. We discuss these problems and their relationship and provide a new and nearly optimal algorithm for solving them.

01/21/21
Gwen McKinley  UC San Diego
Counting integer partitions with the method of maximum entropy
AbstractWe give an asymptotic formula for the number of partitions of an integer n where the sums of the kth powers of the parts are also fixed, for some collection of values k. To obtain this result, we reframe the counting problem as an optimization problem, and find the probability distribution on the set of all integer partitions with maximum entropy among those that satisfy our restrictions in expectation (in essence, this is an application of Jaynes' principle of maximum entropy). This approach leads to an approximate version of our formula as the solution to a relatively straightforward optimization problem over realvalued functions. To establish more precise asymptotics, we prove a local central limit theorem using an equidistribution result of Green and Tao.
\\
\\
A large portion of the talk will be devoted to outlining how our method can be used to rederive a classical result of Hardy and Ramanujan, with an emphasis on the intuitions behind the method, and limited technical detail. This is joint work with Marcus Michelen and Will Perkins. 
01/21/21
Kwun Angus Chung  University of Michigan
$v$adic convergence for exp and log in function fields and applications to $v$adic $L$values
AbstractClassically over the rational numbers, the exponential and
logarithm series converge $p$adically within some open disc of
$\mathbb{C}_p$. For function fields, exponential and logarithm series
arise naturally from Drinfeld modules, which are objects constructed by
Drinfeld in his thesis to prove the Langlands conjecture for
$\mathrm{GL}_2$ over function fields. For a ``finite place'' $v$ on such a
curve, one can ask if the exp and log possess similar $v$adic
convergence properties. For the most basic case, namely that of the
Carlitz module over $\mathbb{F}_q[T]$, this question has been long
understood. In this talk, we will show the $v$adic convergence for
Drinfeld(Hayes) modules on elliptic curves and a certain class of
hyperelliptic curves. As an application, we are then able to obtain a
formula for the $v$adic $L$value $L_v(1,\Psi)$ for characters in these
cases, analogous to Leopoldt's formula in the number field case. 
01/21/21
Amber Puha  Cal State San Marcos
From Queueing Theory to Modern Stochastic Networks: A Mathematical Perspective
AbstractWaiting in a queue (or a line) for some type of service is a commonplace experience. People do this at a grocery store, bank, amusement park or DMV, to give just a few examples. These customer service systems feature inherent randomness, which impacts performance. Modern computer and communications systems, manufacturing processes, transportation systems and even biological networks experience similar stochastic effects. Stochastic network theory is the study of the performance and optimal control of such systems. At the beginning of the talk, I will show a few simple examples of where mathematics plays an integral role in illuminating system behavior. Following that I will discuss some of the mathematical challenges associated with analyzing the performance of modern networks.
\\
\\
Finally, I will end by discussing workin progress related to modern call centers that is joint with Amy Ward (U. Chicago Booth) and Yueyang Zhong (U. Chicago Booth). 
01/22/21
Yalong Cao  IPMU
GopakumarVafa type invariants for CalabiYau 4folds
AbstractGopakumarVafa type invariants on CalabiYau 4folds (which
are nontrivial only for genus zero and one) are defined by
KlemmPandharipande from GromovWitten theory, and their integrality
is conjectured. In this talk, I will explain how to give a sheaf
theoretical interpretation of them using counting invariants on moduli
spaces of one dimensional stable sheaves.
\\
\\
Based on joint works with D. Maulik and Y. Toda. 
01/26/21
Evangelos Nikitopoulos  UC San Diego
Noncommutative $C^k$ Functions, Multiple Operator Integrals, and Derivatives of Operator Functions
AbstractLet $A$ be a $C^*$algebra, $f \colon \mathbb{R} \to \mathbb{C}$ be a continuous function, and $\tilde{f} \colon A_{\text{sa}} \to A$ be the functional calculus map $A_{\text{sa}} \ni a \mapsto f(a) \in A$. It is elementary to show that $\tilde{f}$ is continuous, so it is natural to wonder how the differentiability properties of $f$ relate/transfer to those of $\tilde{f}$. This turns out to be a delicate, complicated problem. In this talk, I introduce a rich class $NC^k(\mathbb{R}) \subseteq C^k(\mathbb{R})$ of noncommutative $C^k$ functions $f$ such that $\tilde{f}$ is $k$times differentiable. I shall also discuss the interesting objects, called multiple operator integrals, used to express the derivatives of $\tilde{f}$.

01/26/21
Suhan Zhong  UC San Diego
A Lagrange multiplier expression method for bilevel polynomial optimization
AbstractBilevel optimization problem is a twolevel optimization problem, where a subset of its variables is constrained in the optimizer set of another optimization problem parameterized by the remaining variables. In this talk, we introduce a Lagrange multiplier expression method for bilevel polynomial optimization based on polynomial optimization relaxations. Each relaxation is obtained from the KurashKuhnTucker (KKT) conditions for the lower level optimization and the exchange technique for semiinfinite programming. The global convergence of the method is proved under some general assumptions. And some numerical examples will be given to show the efficiency of the method.

01/26/21
Tom Bachmann  LMU Munich
Cellular motivic invariants of Z[1/2]
AbstractReport on work in progress, joint with Paul Arne Oestvaer.
\\
\\
A cellular motivic invariant is a special type of functor from the
category of commutative rings (or the opposite of schemes, say) to
spectra. Examples include algebraic Ktheory, motivic cohomology, \'e{}tale
cohomology and algebraic cobordism. DwyerFriedlander observed that for
2adic \'e{}tale Ktheory and certain related invariants, the value on
Z[1/2] can be described in terms of a fiber square involving the values
on the real numbers, the complex numbers, and the field with three elements.
\\
\\
I will explain a generalization of this result to arbitrary 2adic
cellular motivic invariants. As an application, we show that $\pi_0$ of the
motivic sphere spectrum over Z[1/2] is given by the GrothendieckWitt
ring of Z[1/2], up to odd torsion. 
01/26/21
Jason O'Neill  Ph.D. Candidate  UC San Diego
A note on $k$wise oddtown problems
AbstractConsider a town of $n$ people and suppose this town wants to impose the following rules on its clubs (formally subsets of the towns $n$ residents). First, each club must have an odd number of members. Second, each distinct pair of clubs in the town must have an even number of members in common. What is the maximum number of clubs this town can have while adhering to the rules?

01/26/21
Ioana Dumitriu  UC San Diego
Random matrices, random graphs, and applications to machine learning
AbstractThe last decade has seen tremendous progress in applying random matrix methods to adjacency matrices or Laplacians of random graphs, in order to understand their spectra and be able to apply the new results to algorithms in machine learning, coding theory, data science, etc. Nevertheless, many problems remain. I will present some of the most interesting tools and new results and mention some (still) open problems.

01/27/21
Yi Lai  UC Berkeley
A family of 3d steady gradient solitons that are flying wings
AbstractWe find a family of 3d steady gradient Ricci solitons that are flying wings. This verifies a conjecture by Hamilton. For a 3d flying wing, we show that the scalar curvature does not vanish at infinity. The 3d flying wings are collapsed. For dimension $n \geq 4$, we find a family of $Z2$ $\times$ $O(n  1)$symmetric but nonrotationally symmetric ndimensional steady gradient solitons with positive curvature operator. We show that these solitons are noncollapsed.

01/29/21
Rahul Pandharipande  ETH
K3 surfaces: curves, sheaves, and moduli
AbstractI will talk about some results and open questions related
to the moduli of maps of curves to K3 surfaces, sheaves
on K3 surfaces, and moduli of K3 surfaces themselves.
Feb

02/02/21
Matthieu Joseph  Graduate student  Ecole Normale Superieure de Lyon
Rigidity and flexibility phenomenons in isometric orbit equivalence
AbstractIn an ongoing work, we introduce the notion of isometric orbit equivalence for probability measure preserving actions of marked groups. This notion asks the Schreier graphings defined by the actions of the marked groups to be isomorphic. In the first part of the talk, we will prove that pmp actions of a marked group whose Cayley graph has a discrete automorphisms group are rigid up to isometric orbit equivalence. In a second time, we will explain how to construct pmp actions of the free group that are isometric orbit equivalent but not conjugate.

02/02/21
Adi Tcaciuc  MacEwan University
The Invariant Subspace Problem for rankone perturbations
AbstractThe Invariant Subspace Problem is one of the most famous problems in Operator Theory, and is concerned with the search of nontrivial, closed, invariant subspaces for bounded operators acting on a separable Banach space. Considerable success has been achieved over the years both for the existence of such subspaces for many classes of operators, as well as for nonexistence of invariant subspaces for particular examples of operators. However, for the most important case of a separable Hilbert space, the problem is still open.
\\
\\
A natural, related question deals with the existence of invariant subspaces for perturbations of bounded operators. These types of problems have been studied for a long time, mostly in the Hilbert space setting. In this talk I will present a new approach to these ``perturbation'' questions, in the more general setting of a separable Banach space. I will focus on the recent history, presenting several new results that were obtained along the way with this new approach, and examining their connection and relevance to the Invariant Subspace Problem. 
02/02/21
Valentin Duruisseaux  Graduate Student  UC San Diego
A Variational Approach to Accelerated Optimization
AbstractEfficient optimization has become one of the major concerns in data analysis. There has been a lot of focus on firstorder optimization algorithms because of their low cost per iteration. In 1983, Nesterov's Accelerated Gradient method (NAG) was shown to converge in $O(1/k^2)$ to the minimum of the convex objective function $f(x)$, improving on the $O(1/k)$ convergence rate exhibited by the standard gradient descent methods, which is the phenomenon referred to as acceleration. It was shown that NAG limits to a second order ODE, as the time step goes to 0, and that the objective function $f(x(t)$) converges to its optimal value at a rate of $O(1/t^2)$ along the trajectories of this ODE. In this talk, we will discuss how the convergence of $f(x(t))$ can be accelerated in continuous time to an arbitrary convergence rate $O(1/t^p)$ in normed spaces, by considering flow maps generated by a family of timedependent Bregman Lagrangian and Hamiltonian systems which is closed under time resca
ling. We will then discuss how this variational framework can be exploited together with the timeinvariance property of the family of Bregman Lagrangians using adaptive geometric integrators to design efficient explicit algorithms for symplectic accelerated optimization. Finally, we will discuss briefly the generalization from normed spaces to Riemannian manifolds. 
02/02/21
Mingcong Zeng  Utrecht
Real bordism and its friends
AbstractIn the 1960's, Kervaire and Milnor boiled down the problem of counting smooth structure on spheres of dimension greater than 4 to the computation of stable homotopy groups of spheres and the Kervaire invariant one problem. In the following decades, the elements of Kervaire invariant one whose dimension are less or equals to 62 are shown to exist, and finally, Hill, Hopkins and Ravenel in their 2016 paper show that the Kervaire invariant one elements doesn't exist for dimension larger or equals to 254, leaving the 126dimensional case open.
\\
\\
The $C_2$equivariant Real bordism spectrum and its norms are crucial in HHR's solution, and computation of them is a central topic in equivariant stable homotopy theory. In this talk, I will explore two aspects of Real bordism and its norms:
\\
\\
1. How computation in Real bordism helps us to understand LubinTate Etheories at p = 2. In particular, we can understand almost all actions of finite subgroups of the Morava stablizer groups on Etheories in homotopy.
\\
\\
2. The relation between Real bordism and the Segal conjecture. This relation allows us to bring new tools and perspective into this equivariant computation, and we will show how a spectral sequence based on (Real) topological Hochschild homology can help in understanding Real bordism and its norms.
\\
\\
This talk is based on joint work with Beaudry, Hill, Lawson, Meier and Shi. 
02/02/21
Itai Maimon  Graduate student  UC San Diego
Topological Quantum Computation: The Toric Code
AbstractOne of the largest problems in Quantum Computation is how you deal with errors. Alexei Kitaev invented a method whereby we can use the discretization of a manifold to encode logical information in a subspace of the Hilbert space that corresponds to the homology of the Manifold itself. This has been vastly generalized, but we will restrict to looking at his original example of the toric code. We will go through how it can be used as an errorcorrecting code and several methods on how to actually compute on such a code. If time permits we can discuss recent results where expander graphs (and in general combinatorial methods) are used to construct a very good code which is then used to construct a manifold that solves a certain problem in differential topology.

02/02/21
Xiaochuan Tian  UC San Diego
An invitation to non local models
AbstractThere has been a growing interest in the study of nonlocal models as more general and sometimes more realistic alternatives to the conventional PDE models. We will give an introduction to nonlocal models in this talk. In particular, we will focus on the nonlocal models with a finite range of nonlocal interactions, which serve as bridges connecting the classical PDEs, nonlocal discrete models and the fractional differential equations. This talk will cover topics including nonlocal modeling, nonlocal calculus and numerical analysis for the nonlocal models.

02/03/21
Chester Holtz  UC San Diego
Adversarial Examples & Provable Robustness
AbstractModern machine learning methods (i.e. neural networks) have been very successful in tasks such as image classification and speech recognition, but have been shown to be extremely brittle to small, adversariallychosen perturbations of their inputs. This is a critical issue in many deep learning applications (e.g. object detection, robotic perception, ranking and recommendation, etc.). In this talk, I will provide an overview of the problem of adversarial robustness, formally introduce some general principles (what we know and what we don't know about this phenomenon), and discuss heuristic solutions (methods that appear to work in practice) and recent certification techniques (how do we provably  and efficiently  guarantee robustness?).

02/03/21
Jonathan Zhu  Princeton University
Explicit Lojasiewicz inequalities for mean curvature flow shrinkers
AbstractLojasiewicz inequalities are a popular tool for studying the stability of geometric structures. For mean curvature flow, Schulze used Simon's reduction to the classical Lojasiewicz inequality to study compact tangent flows. For round cylinders, Colding and Minicozzi instead used a direct method to prove Lojasiewicz inequalities. We'll discuss similarly explicit Lojasiewicz inequalities and applications for other shrinking cylinders and Clifford shrinkers.

02/04/21
Tino Ullrich  TU Chemnitz
A New Subsampling Technique for Random Points and Optimal Least Squares Approximation of HighDimensional Functions
AbstractWe provide a new general upper bound for the minimal L2worstcase recovery error in the framework of RKHS, where only n function samples are allowed. This quantity can be bounded in terms of the singular numbers of the compact embedding into the space of square integrable functions. It turns out that in many relevant situations this quantity is asymptotically only worse by square root of log(n) compared to the singular numbers. The algorithm which realizes this behavior is a weighted least squares algorithm based on a specific set of sampling nodes which works for the whole class of functions simultaneously. These points are constructed out of a random draw with respect to distribution tailored to the spectral properties of the reproducing kernel (importance sampling) in combination with a subsampling procedure coming from the celebrated proof of Weaver's conjecture, which was shown to be equivalent to the KadisonSinger problem. For the above multivariate setting, it is still a fundamental open problem whether sampling algorithms are as powerful as algorithms allowing general linear information like Fourier or wavelet coefficients. However, the gap is now rather small. As a consequence, we may study wellknown scenarios where it was widely believed that sparse grid sampling recovery methods perform optimally. It turns out that this is not the case for dimensions d greater than 2.
\\
\\
This is joint work with N. Nagel and M. Schaefer from TU Chemnitz. 
02/04/21
Naomi Sweeting  Harvard University
Kolyvagin's conjecture and higher congruences of modular forms
AbstractGiven an elliptic curve E, Kolyvagin used CM points on
modular curves to construct a system of classes valued in the Galois
cohomology of the torsion points of E. Under the conjecture that not
all of these classes vanish, he gave a description for the Selmer group
of E. This talk will report on recent work proving new cases of
Kolyvagin's conjecture. The methods follow in the footsteps of Wei
Zhang, who used congruences between modular forms to prove Kolyvagin's
conjecture under some technical hypotheses. We remove many of these
hypotheses by considering congruences modulo higher powers of p. The
talk will explain the difficulties associated with higher congruences of
modular forms and how they can be overcome. I will also provide an
introduction to the conjecture and its consequences, including a `converse theorem': algebraic rank one implies analytic rank one. 
02/04/21
Cristina Costantini  Universite di ChietiPescara, Italy
Obliquely reflecting diffusions in nonsmooth domains: some new existence and uniqueness results

02/04/21
Ronghui Lily Xu  UC San Diego
Learning survival from electronic medical/health records (EMR/EHR) data using high dimensional claims codes
AbstractOur work was motivated by the analysis projects using the linked US SEERMedicare database to study mortality in men of age 65 years or older who were diagnosed with prostate cancer. Such data sets contain up to 100,000 human subjects and over 20,000 claim codes. For studying mortality the number of deaths are the ``effective'' sample size, so here we are in the situation of p is greater than n which is referred to as having highdimensional predictors. In addition, a patient might die of cancer, or of other causes such as heart disease etc. These are referred to as competing risks. How to best perform prediction which inevitably involves variable selection for this type of complex survival data had not been previously investigated. Interest may also lie in comparing treatments such as radical prostatectomy versus conservative treatment. In this case the data were obviously not randomized with regard to the treatment assignments, and confounding most likely exists, possibly even beyond the commonly captured clinical variables in the SEER database. We will showcase research work done by our former PhD students from the UCSD Math Dept to account for such unobserved confounding, as well as efforts to make use of the high dimensional claims codes which have been shown to contain rich information about the patients survival.

02/05/21
Kisun Lee  UC San Diego
Finding and certifying numerical roots of systems of equations
AbstractNumerical algebraic geometry studies methods to approach problems in algebraic geometry numerically. Especially, finding roots of systems of equations using theory in algebraic geometry involves symbolic algorithm which requires expensive computations. However, numerical techniques often provides faster methods to tackle these problems. We establish numerical techniques to approximate roots of systems of equations and ways to certify its correctness.
\\
\\
As techniques for approximating roots of systems of equations, homotopy continuation method will be introduced. Since numerical approaches rely on heuristic method, we study how to certify numerical roots of systems of equations. Krawczyk method from interval arithmetic and Smale's alpha theory will be used as main paradigms for certification. Furthermore, as an approach for multiple roots, we establish the local separation bound of a multiple root. For a regular quadratic multiple zero, we give their local separation bound and study how to certify an approximation of such multiple roots. 
02/07/21
Richard Bamler  University of California, Berkeley
Structure theory of noncollapsed limits of Ricci flows, Part 1
AbstractThis talk, the first on the third paper https://arxiv.org/abs/2009.03243 of a series, is partly a continuation of talks given in the fall.
See: http://www.math.ucsd.edu/\~{}benchow/ccseminar.html
\\
\\
Some review will be given to make the talks more selfcontained. 
02/09/21
Tamara Kucherenko  City College of New York
Flexibility of the Pressure Function
AbstractOur settings are onedimensional compact symbolic systems. We discuss the flexibility of the pressure function of a continuous potential (observable) with respect to a parameter regarded as the inverse temperature. The points of nondifferentiability of this function are of particular interest in statistical physics since they correspond to qualitative changes of the characteristics of a dynamical system referred to as phase transitions. It is well known that the pressure function is convex, Lipschitz, and has an asymptote at infinity. We show that these are the only restrictions. We present a method to explicitly construct a continuous potential whose pressure function coincides with any prescribed convex Lipschitz asymptotically linear function starting at a given positive value of the parameter.
\\
\\
This is based on joint work with Anthony Quas. 
02/09/21
Tim de Laat  Universit"{a}t M"{u}nster
Gelfand pairs, spherical functions and (exotic) group $C^*$algebras
AbstractFor a nonamenable group $G$, there may be many (exotic) group $C^*$algebras that lie naturally between the universal and the reduced $C^*$algebra of $G$. Let $G$ be a simple Lie group or an appropriate locally compact group acting on a tree. I will explain how the $L^p$integrability properties of different spherical functions on $G$ (relative to a maximal compact subgroup) can be used to distinguish between different (exotic) group $C^*$algebras. This recovers results of Samei and Wiersma. Additionally, I will explain that under certain natural assumptions, the aforementioned exotic group $C^*$algebras are the only ones coming from $G$invariant ideals in the FourierStieltjes algebra of $G$.
\\
\\
This is based on joint work with Dennis Heinig and Timo Siebenand. 
02/09/21
Robert Burklund  MIT
Classification of manifolds and the Adams spectral sequence
AbstractA classical question in differential topology is the following: Classify all simplyconnected, closed, smooth (2n)manifolds whose only nontrivial homology groups are $H_0, H_n$ and $H_{2n}$.
In this talk I will survey the history of the high dimensional side of this question and how its resolution requires a surprisingly deep understanding of the Adams spectral sequence computing the stable homotopy groups of spheres. Time permitting, I will then discuss how the situation changes as we relax our topological restrictions on the manifold (for example allowing $H_{ne}$, $H_{ne+1}$, ... $H_{n+e}$ to be nontrivial for a small number e).
\\
\\
This talk represents joint work with Jeremy Hahn and Andy Senger. 
02/09/21
Alexander Guldemond  UC San Diego
Secondderivative SQP methods for largescale nonconvex nonlinear optimization
AbstractInterior Point methods and Sequential Quadratic Programming (SQP) methods have become two of the most crucial methods for solving largescale nonlinear optimization problems. The two methods take very different approaches to solving the same problem. SQP methods find approximate solutions to a sequence of linearly constrained quadratic subproblems in which a quadratic model of the Lagrangian is minimized subject to a linear model of the constraints. Typically, the QP subproblems are solved using an activeset method, giving the problem a majorminor iteration pattern in which each iteration of the activeset method solves an indefinite system. In contrast, interior point methods follow a continuous path towards the optimal solution by perturbing the firstorder optimality conditions of the problem. In this talk, we discuss a shifted primal dual interior point method and its potential applicability in solving the QP subproblem of an SQP method. We also discuss some of
the potential issues with this approach that we hope to overcome. 
02/09/21
Jack Jerry Garzella  Ph.D. Student  UC San Diego
Type Theory: It's the New Set Theory
AbstractSet theory has long been considered the foundation of all mathematical thought. However, people who prove theorems on computers for a living don't use set theory anymoreand now some suggest that mathematicians should do the same. We'll discuss the problems people have with Set Theory and its main alternative, Type Theory.

02/09/21
Aaron Pollack  UC San Diego
Modular forms and sums of four squares
AbstractHow many ways can a positive integer be written as the sum of four squares? There is a simple formula for the number of ways, which goes back to Jacobi. I'll introduce modular forms and sketch how they provide an answer to this question.

02/10/21
Jiyoung Choi  Graduate Student  UC San Diego
Nash equilibrium problems

02/11/21
Massimo Fornasier  Technische Universit"{a}t M"{u}nchen
Consensusbased Optimization on the Sphere
AbstractI present new stochastic multiparticle models for global optimization of nonconvex functions on the sphere. These models belong to the class of ConsensusBased Optimization methods. In fact, particles move over the manifold driven by a drift towards an instantaneous consensus point, computed as a combination of the particle locations weighted by the cost function according to Laplace's principle. The consensus point represents an approximation to a global minimizer. The dynamics is further perturbed by a random vector field to favor exploration, whose variance is a function of the distance of the particles to the consensus point. In particular, as soon as the consensus is reached, then the stochastic component vanishes. In the first part of the talk, I present the wellposedness of the model on the sphere and we derive rigorously its meanfield approximation for large particle limit.
\\
\\
In the second part I address the proof of convergence of numerical schemes to global minimizers provided conditions of wellpreparation of the initial datum. The proof combines the meanfield limit with a novel asymptotic analysis, and classical convergence results of numerical methods for SDE. We present several numerical experiments, which show that the proposed algorithm scales well with the dimension and is extremely versatile. To quantify the performances of the new approach, we show that the algorithm is able to perform essentially as good as ad hoc state of the art methods in challenging problems in signal processing and machine learning, namely the phase retrieval problem and the robust subspace detection.
\\
\\
Joint work with H. Huang, L. Pareschi, and P. S"{u}nnen 
02/11/21
Allechar Serrano Lopez  University of Utah
Counting elliptic curves with prescribed torsion over imaginary quadratic fields
AbstractA generalization of Mazur's theorem states that there are 26
possibilities for the torsion subgroup of an elliptic curve over a
quadratic extension of $\mathbb{Q}$. If $G$ is one of these groups, we
count the number of elliptic curves of bounded naive height whose
torsion subgroup is isomorphic to $G$ in the case of imaginary quadratic
fields. 
02/12/21
Ljudmila Kamenova  Stony Brook University
Algebraic nonhyperbolicity of hyperkahler manifolds
AbstractA projective manifold is algebraically hyperbolic if the
degree of any curve is bounded from above by its genus times a
constant, which is independent from the curve. This is a property which
follows from Kobayashi hyperbolicity. We prove that hyperkahler
manifolds are not algebraically hyperbolic when the Picard rank is at
least 3, or if the Picard rank is 2 and the SYZ conjecture on existence
of Lagrangian fibrations is true. We also prove that if the automorphism
group of a hyperkahler manifold is infinite, then it is algebraically
nonhyperbolic.
\\
\\
These results are a joint work with Misha Verbitsky. 
02/16/21
Nishant Chandgotia  Tata Institute of Fundamental Research
About Borel and almost Borel embeddings for Z\^{}D actions
AbstractKriegerâ€™s generator theorem says that all free ergodic measure preserving actions (under natural entropy constraints) can be modelled by a full shift. Recently, in a sequence of two papers Mike Hochman noticed that this theorem can be strengthened: He showed that all free homeomorphisms of a Polish space (under entropy constraints) can be Borel embedded into the full shift. In this talk we will discuss some results along this line from a recent paper with Tom Meyerovitch and ongoing work with Spencer Unger.
\\
\\
With Meyerovitch, we established a condition called flexibility under which a large class of systems are almost Borel universal, meaning that such systems can model any free Z\^{}d action on a Polish space up to a null set. The condition of flexibility covered a large class of examples including those of domino tilings and the space of proper 3colourings and answered questions by Robinson and Sahin. However extending the embedding to include the null set is a daunting task and there are many partial results towards this. Using tools developed by Gao, Jackson, Krohne and Seward, along with Spencer Unger we were able to get Borel embedding of symbolic systems (as opposed to all Borel systems) under some very similar assumptions which still covered all the examples that we were interested in. This answered questions by Gao and Jackson and recovered results announced by Gao, Jackson, Krohne and Seward. 
02/16/21
Andrew Lobb  Durham University
Foursided pegs fitting round holes fit all smooth holes
AbstractGiven a smooth Jordan curve and a cyclic quadrilateral (a cyclic quadrilateral is a quadrilateral that can be inscribed in a circle) we show that there exist four points on the Jordan curve forming the vertices of a quadrilateral similar to the one given. The smoothness condition cannot be dropped (since not all cyclic quadrilaterals can be inscribed in all triangles), while the cyclicity is necessary (since the circle is itself a smooth Jordan curve). The proof involves some results in symplectic topology. No prior knowledge assumed.
\\
\\
Joint work with Josh Greene. 
02/16/21
Ben Hayes  University of Virginia
A random matrix approach to absorption in free products
AbstractI'll discuss joint with JekelNelsonSinclair. We give the first free entropy proof of Popa's famous result that the generator MASA in a free group factor is maximal amenable, and we partially recover Houdayer's results on amenable absorption and Gamma stability. Moreover, we give a unified approach to all these results using 1bounded entropy. The main techniques are concentration of measure on unitary groups as well as Voiculescu's asymptotic freeness theorem.

02/16/21
Adu Vengal  UC San Diego
Sp00ky Groups and the General Burnside Problem
AbstractConsider the following statement: If $G$ is a finitely generated group, and all elements of $G$ have finite order, then $G$ is a finite group. Is it true? Nope. We'll construct a counterexample (the Grigorchuk group), and then talk a little about the properties and representations of any such counterexample.

02/16/21
Brandon Seward  UC San Diego
Bernoulli shifts and entropy theory
AbstractIn ergodic theory, one often studies measurepreserving actions of countable groups on probability spaces. Bernoulli shifts are a class of such actions that are particularly simple to define, but despite several decades of study some elementary questions about them still remain open, such as how they are classified up to isomorphism. Progress in understanding Bernoulli shifts has historically gone handinhand with the development of a tool known as entropy. In this talk, I will review classical concepts and results, which apply in the case where the acting group is amenable, and then I will discuss recent developments that are beginning to illuminate the case of nonamenable groups.

02/17/21
Mat Langford  University of Tennessee, Knoxville
Ancient solutions out of polytopes
AbstractI will show how to construct a very large family of new examples of convex ancient and translating solutions to mean curvature flow in all dimensions. At $t=\infty$, these examples resemble a family of standard Grim hyperplanes of certain prescribed orientations. The existence of such examples has been suggested by Hamilton and Huiskenâ€”Sinestrari. Our examples include solutions with symmetry group $D\times \mathbb Z_2$, where $D$ is the symmetry group of any given regular polytope, and, surprisingly, many examples which admit only a single reflection symmetry. We also exhibit a family of eternal solutions which do not evolve by translation, settling a conjecture of Brian White in the negative. Time permitting, I will present further structure and partial classification results for this class of solutions, as well as some open questions and conjectures.
\\
\\
Joint with T. Bourni and G. Tinaglia. 
02/18/21
Mikhail Belkin  UC San Diego
A theory of optimization and transition to linearity in deep learning
AbstractThe success of deep learning is due, to a large extent, to the remarkable effectiveness of gradientbased optimization methods applied to large neural networks. In this talk I will discuss some general mathematical principles allowing for efficient optimization in overparameterized nonlinear systems, a setting that includes deep neural networks. Remarkably, it seems that optimization of such systems is "easy". In particular, optimization problems corresponding to these systems are not convex, even locally, but instead satisfy locally the PolyakLojasiewicz (PL) condition allowing for efficient optimization by gradient descent or SGD. We connect the PL condition of these systems to the condition number associated to the tangent kernel and develop a nonlinear theory parallel to classical analyses of overparameterized linear equations.
\\
\\
In a related but conceptually separate development, I will discuss a new perspective on the remarkable recently discovered phenomenon of transition to linearity (constancy of NTK) in certain classes of large neural networks. I will show how this transition to linearity results from the scaling of the Hessian with the size of the network.
\\
\\
Joint work with Chaoyue Liu and Libin Zhu. 
02/18/21
Zuhair Mullath  University of Massachusetts, Amherst
Unobstructed Galois deformation problems associated to GSp(4)
AbstractTo a cuspidal automorphic representation of GSp(4) over
$\mathbb Q$, one can associate a compatible system of Galois
representations $\{\rho_p\}_{p \; \mathrm{prime}}$. For $p$ sufficiently
large, the deformation theory of the mod$p$ reduction $\overline
\rho_p$ is expected to be unobstructed  meaning the universal
deformation ring is a power series ring. The global obstructions to
deforming $\overline \rho_p$ is controlled by certain adjoint BlochKato
Selmer groups, which are expected to be trivial for $p$ large enough.
\\
\\
I will talk about some recent results showing that there are no local
obstructions to the deformation theory of $\overline \rho_p$ for almost
all $p$.
\\
\\
This is joint work with M. Broshi, C. Sorensen, and T. Weston. 
02/18/21
Alina Bucur  UC San Diego
AWM Colloquium

02/19/21
Ruijie Yang  Stonybrook University
Decomposition theorem for semisimple local systems
AbstractIn complex algebraic geometry, the decomposition
theorem asserts that semisimple geometric objects remain semisimple
after taking direct images under proper algebraic maps. This was
conjectured by Kashiwara and is proved by Mochizuki and Sabbah in a
series of long papers via harmonic analysis and Dmodules.
In this talk, I would like to explain a simpler proof in the case of
semisimple local systems using a more geometric approach. As a
byproduct, we recover a weak form of Saito's decomposition theorem for
variations of Hodge structures.
\\
\\
Joint work in progress with Chuanhao Wei. 
02/21/21
Richard Bamler  University of California, Berkeley
Structure theory of noncollapsed limits of Ricci flows, Part 3
AbstractThis talk, the first on the third paper https://arxiv.org/abs/2009.03243 of a series, is partly a continuation of talks given in the fall. See: http://www.math.ucsd.edu/\~{}benchow/ccseminar 20.html

02/22/21
Jonathan Novak  UC San Diego
HCIZ, BGW, and other capital letters
AbstractThis talk will be about a pair of related matrix integrals, the HarishChandra/ItzyksonZuber integral and the BrezinGrossWitten integral, which play an important role in random matrix theory, representation theory, and mathematical physics. While these integrals cannot be exactly evaluated, an old conjecture says that they admit asymptotic expansions whose coefficients are themselves generating functions for some unspecified combinatorial invariants of compact Riemann surfaces (or smooth projective curves).

02/23/21
Gianluca Basso  Postdoctoral Scholar  Universit\'{e} Claude Bernard Lyon 1
Topological dynamics beyond Polish groups
AbstractWhen $G$ is a Polish group, one way of knowing that it has nice
dynamics is to show that $M(G)$, the universal minimal flow of $G$, is
metrizable. For nonPolish groups, this is not the relevant dividing
line: the universal minimal flow of the symmetric group of a set of
cardinality $\kappa$ is the space of linear orders on $\kappa$not
a metrizable space, but still nice, for example.
In this talk, we present a set of equivalent properties of topological
groups which characterize having nice dynamics. We show that the class
of groups satisfying such properties is closed under some topological
operations and use this to compute the universal minimal flows of some
concrete groups, like $\mathrm{Homeo}(\omega_{1})$.
\\
\\
This is joint work with Andy Zucker. 
02/23/21
Mehrdad Kalantar  University of Houston
Noncommutative boundary maps and C*algebras of quasiregular representations
AbstractWe investigate some structural properties of C*algebras generated by quasiregular representations of stabilizers of boundary actions of discrete groups G. Our main tool is the notion of (noncommutative) boundary maps, namely Gequivariant unital positive maps from GC*algebras to C(B), where B is the Furstenberg boundary of G. We completely describe the tracial structure and characterize the simplicity of these C*algebras. As an application, we show that the C*algebra generated by the quasiregular representation associated to Thompson's groups $F < T$ does not admit traces and is simple.
\\
\\
This is joint work with Eduardo Scarparo. 
02/23/21
Vyacheslav Kungurtsev  Department of Computer Science, Czech Technical University, Prague
LevenbergMarquardt Algorithms for Nonlinear Inverse Least Squares
AbstractLevenbergMarquardt (LM) algorithms are a class of methods that add a regularization term to a GaussNewton method to promote better convergence properties. This talk presents three works on this class of methods. The first discusses a new method that simultaneously achieves all types of state of the art convergence guarantees for unconstrained problems. Stochastic LM is discussed next, which is an algorithm to handle noisy data. An example is presented on data assimilation. Finally, a LM method is presented to handle equality constraints, with examples from inverse problems in PDEs.

02/23/21
Juanita Pinzon Caicedo  University of Notre Dame
Toroidal integer homology spheres have irreducible SU(2)representations
AbstractThe fundamental group is one of the most powerful invariants to distinguish closed threemanifolds. One measure of the nontriviality of a threemanifold is the existence of nontrivial SU(2)representations. In this talk I will show that if an integer homology threesphere contains an embedded incompressible torus, then its fundamental group admits irreducible SU(2)representations.
\\
\\
This is joint work with Tye Lidman and Raphael Zentner. 
02/23/21
Scotty Tilton  Ph.D. Student  UC San Diego
Stay at (Co)Home: Links, Blowups, and Grass, man
AbstractReal Grassmannians' uses in geometry are manifold, but in general, their integral (co)homology groups were unknown. Until now. I won't Stiefel myself any longer, and I will (co)change your views on this class of manifolds. At the end of this talk, you should be able to differentiate Grassmannian manifolds and feel right at (co)home with them, K? This talk will Blow Up your mind, Link some ideas you may not have heard of, and you'll take an ExitPath out with your mind (po)set straight.

02/24/21
Robin Neumayer  Northwestern
$d_p$ Convergence and $\epsilon$regularity theorems for entropy and scalar curvature lower bounds
AbstractIn this talk, we consider Riemannian manifolds with almost nonnegative scalar curvature and Perelman entropy. We establish an $\epsilon$regularity theorem showing that such a space must be close to Euclidean space in a suitable sense. Interestingly, such a result is false with respect to the GromovHausdorff and Intrinsic Flat distances, and more generally the metric space structure is not controlled under entropy and scalar lower bounds. Instead, we introduce the notion of the $d_p$ distance between (in particular) Riemannian manifolds, which measures the distance between $W^{1,p}$ Sobolev spaces, and it is with respect to this distance that the $\epsilon$ regularity theorem holds. We will discuss various applications to manifolds with scalar curvature and entropy lower bounds, including a compactness and limit structure theorem for sequences, a uniform $L^\infty$ Sobolev embedding, and a priori $L^p$ scalar curvature bounds for $p<1$.
\\
\\
This is joint work with ManChun Lee and Aaron Naber. 
02/24/21
Xiaoou Pan  Ph.D. Student  UC San Diego
Implicit regularization in overparameterized models
AbstractWe study the benign overfitting phenomenon induced by simple optimization algorithms in deep learning.
Oftentimes the neural network is overparameterized in the sense that the number of parameters exceeds the
training data size, but the obtained solution generalizes well to unseen data. The generalization stems from
an implicit regularization of the optimization algorithm. We present the recent theoretical development of
overparameterization for linear/nonlinear models, together with some numerical experiments. 
02/25/21
Hao Shen  University of WisconsinMadison
Stochastic quantization and YangMills
AbstractWe briefly overview the current developments of rigorous constructions in "stochastic quantizationâ€  an active field linking quantum field theory with stochastic PDE.
We then focus on stochastic quantization of the YangMills model in 2 and 3 space dimensions.
This includes constructing the Langevin dynamic for the formal YangMills measure, defining the state space of gauge orbits, proving gauge equivariance of the dynamic, and making sense of Wilson loop observables in this context. We will also discuss some future directions.
\\
\\
The talk is based on several works mostly joint with A.Chandra, I.Chevyrev, and M.Hairer. 
02/25/21
Sergey Kryazhimskiy  UC San Diego  Department of Biology
Emergence and propagation of epistasis in metabolic networks
AbstractEpistasis is a situation when the effect of one mutation changes as other mutations are introduced into the genome. Epistasis is used in genetics to probe functional relationships between genes, and it also plays an important role in evolution. However, there is no theory to understand how functional relationships at the molecular level translate into epistasis at the level of wholeorganism phenotypes, such as fitness. I will present a simple model of a hierarchical metabolic network with firstorder kinetics which helps us gain some intuition in this problem. I will derive two rules for how epistasis between mutations with small effects propagates from lower to higherlevel phenotypes and how such epistasis depends on the topologyof the network. Most importantly, weak epistasis at a lower level may be distorted as it propagates to higher levels. These results suggest that pairwise intergene epistasis should be common and it should generically depend on the genetic background and environment. Furthermore, the epistasis coefficients measured for highlevel phenotypes may not be sufficient to fully infer the underlying functional relationships.

02/25/21
Tim Trudgian  UNSW Canberra at ADFA
Verifying the Riemann hypothesis to a new height
AbstractSadly, I won't have time to prove the Riemann hypothesis in
this talk. However, I do hope to outline recent work in a record
partialverification of RH.
\\
\\
This is joint work with Dave Platt, in
Bristol, UK. 
02/25/21
Eva Loeser  Graduate Student  UC San Diego
On Heavy Traffic Limit for a Processor Sharing Queue with Soft Deadlines, following Gromoll and Kruk (2007).

02/25/21
Nathan Kuncel  University of Minnesota
Improving Graduate School Admissions
AbstractThe effectiveness of a graduate school admissions ultimately
rests on the quality of the information collected and the decision
making process that is used to arrive at a decision. Admissions relies
on faculty judgment combining a variety of tools including grades, test
scores, letters of recommendation, interviews, and student essays to
identify the best candidates. Unfortunately, current practice often
falls short of well established best practices leading to lower quality
decisions and the possible introduction of bias. In this talk, I will
make the case that improvement is urgently needed and then lay out both
a short and long term place for modernizing graduate school admissions. 
02/26/21
Mircea Musta\c{t}\u{a}  University of Michigan
The minimal exponent of hypersurface singularities
AbstractI will introduce and discuss an invariant of
hypersurface singularities, Saito's minimal exponent (a.k.a. Arnold
exponent in the case of isolated singularities). This can be considered
as a refinement of the log canonical threshold, which is interesting in
the case of rational singularities. I will focus on recent work on this
invariant and remaining open problems. 
02/26/21
Eric Lybrand  Graduate Student  UC San Diego
Doing it For the Bit: Applications of Quantization in Data Science and Signal Processing
Mar

03/01/21
Yuxing Deng  Beijing Institute of Technology
Steady gradient Ricci solitons, Part 1

03/02/21
Sam Mellick  Postdoctoral Scholar  \'{E}cole normale sup\'{e}rieure de Lyon
TBA

03/02/21
Daniel P. Robinson  Department of Industrial and Systems Engineering, Lehigh University, Bethlehem
A Fast Secondorder Method for GroupSparse Optimization
AbstractI discuss an optimization framework for solving problems with group sparsity inducing regularization. Such regularizers include Lasso (L1), group Lasso, and latent group Lasso. The framework computes iterates by optimizing over small dimensional subspaces, thus keeping the cost per iteration relatively low. Theoretical convergence results and numerical tests on various learning problems will be presented.

03/02/21
Sven Raum  Stockholm University
Right angled Hecke operator algebras and representation theory
AbstractWith every Coxeter system one can associate a family of algebras considered as deformation of its group algebra. These are socalled Hecke algebras, which are classical objects of study in combinatorics and representation theory. Complex Hecke algebras admit a natural *structure and a *representation on Hilbert space. Taking the norm and SOTclosure in such representation, one obtains Hecke operator algebras, which have recently seen increased attention.
\\
\\
In this talk, I will introduce Hecke operator algebras from scratch, focusing on the case of rightangled Coxeter groups. This case is particularly interesting from an operator algebraic perspective, thanks to its description by iterated amalgamated free products. I will survey known results on the structure of Hecke operator algebras, before I describe recent work that clarified the factor decomposition of Hecke von Neumann algebras. Two applications to representation theory will be presented. I will finish with some results on the scope and limits of Ktheoretic classification of rightangled Hecke C*algebras.
\\
\\
This is joint work with Adam Skalski. 
03/02/21
Evangelos ``Vaki" Nikitopoulos  UC San Diego
Partial Seussification of a Proof of Liouville's Theorem using Brownian Motion

03/02/21
Luca Spolaor  UC San Diego
Regularity of the freeboundary for the Obstacle Problem

03/03/21
Kaizheng Wang  Columbia University
Clustering via uncoupled regression
AbstractIn this talk we consider a canonical clustering problem where one receives unlabeled samples drawn from a balanced mixture of two elliptical distributions and aims for a classifier to estimate the labels. Many popular methods including PCA and kmeans require individual components of the mixture to be somewhat spherical, and perform poorly when they are stretched. To overcome this issue, we propose a nonconvex program seeking for an affine transform to turn the data into a onedimensional point cloud concentrating around 1 and 1, after which clustering becomes easy. Our theoretical contributions are twofold: (1) we show that the nonconvex loss function exhibits desirable geometric properties when the sample size exceeds some constant multiple of the dimension, and (2) we leverage this to prove that an efficient firstorder algorithm achieves nearoptimal statistical precision without good initialization. We also propose a general methodology for clustering with flexible choices of feature transforms and loss objectives.

03/03/21
Roberto Rubalcaba  Associate Professor  San Diego City College
Pathways to invite and retain underrepresented minorities and women in STEM at UCSD
AbstractIn this talk I will describe pathways to invite and retain underrepresented minorities and women local to San Diego and Tijuana in mathematics and sciences at UCSD. I will describe Math Jams, which I pioneered at San Diego City College, and how I can lead Math Jams at Living Learning Communities (LLCs) such as the African Black Diaspora LLC at Sixth College and Raza LLC at Eleanor Roosevelt College. I will discuss the Hesabu Circle, a safe space for black students of all ages preK to postdoc, and how math circles can be created for underrepresented minorities and women at UCSD supporting undergraduate and graduate students. I will discuss student success statistics with my Umoja and Puente students at San Diego City College and Upward Bound students.

03/03/21
Yuxing Deng  Beijing Institute of Technology
Steady gradient Ricci solitons, Part 2

03/04/21
Ronald DeVore  Texas A\&M University
Deep Learning and Neural Networks: The Mathematical View
AbstractDeep Learning is much publicized and has had great empirical success on challenging problems in learning. Yet there is no quantifiable proof of performance and certified guarantees for these methods. This talk will give an overview of Deep Learning from the viewpoint of mathematics and numerical computation.

03/04/21
Soumya Sankar  The Ohio State University
Counting elliptic curves with a rational $N$isogeny
AbstractThe classical problem of counting elliptic curves with a
rational Nisogeny can be phrased in terms of counting rational points
on certain moduli stacks of elliptic curves. Counting points on stacks
poses various challenges, and I will discuss these along with a few ways
to overcome them. I will also talk about the theory of heights on stacks
developed in recent work of Ellenberg, Satriano and ZureickBrown and
use it to count elliptic curves with an $N$isogeny for certain $N$. The
talk assumes no prior knowledge of stacks and is based on joint work
with Brandon Boggess. 
03/04/21
Varun Khurana  UC San Diego
On deep hedging following Buehler et al.

03/05/21
Antonella Grassi  Universit\'{a} di Bologna and University of Pennsylvania
Kodaira's birational classification of singular elliptic fibers (and threefolds with Qfactorial and non Qfactorial terminal singularities)
AbstractKodaira classified the singular elliptic fibers
occurring on relatively minimal elliptic surfaces (over C). I will
explain a birational Kodaira's classifications for higher dimensional
elliptic fibrations. (Based on work in collaboration with T. Weigand) 
03/05/21
Sam Spiro  UC San Diego
Theorems Featuring Facts of FFT Tables of Full Frequency
AbstractThe FFT problem, which was inspired by work of Guldemond, can be stated as follows: how can you fill a 3x3 grid with F's and T's such that it contains as many copies of the word "FFT" as possible? For example, the following two grids each contain 5 copies of the word FFT (we allow the word to be written forwards or backwards, and to appear in rows, columns, or diagonals):
\\
\\
\[\begin{matrix} F F T\\ F F T\\ F F T\end{matrix}\hspace{30pt} \begin{matrix} F T F\\ T F F\\ F F T\end{matrix}\]
\\
\\
Grubb claimed that there exists a grid containing 6 copies of FFT. Eight minutes later he claimed that actually, the best you could do is 5. He offered no proof of either claim. In this talk we consider a generalization of the FFT problem. Namely, given a word $w$ of length $n$ and a grid $G$ of letters, let $f(w,G)$ be the number of times $w$ appears in $G$, and let $f(w)=\max_G f(w,G)$. We determine $f(w)$ for a number of words, and in particular we determine $f(FFT)$, solving the FFT problem. I, Sam Spiro, will be the only person talking for the entire hour that the talk is given. Absolutely nothing out of the ordinary will happen during the talk. 
03/08/21
PakYeung Chan  UC San Diego
Curvature estimates for steady Ricci solitons

03/09/21
Pengyu Yang  Postdoctoral Scholar  ETH Zurich
Equidistribution of expanding translates of lines in $\mathrm{SL}_3(\mathbb{R})/\mathrm{SL}_3(\mathbb{Z})$
AbstractLet $X=\mathrm{SL}_3(\mathbb{R})/\mathrm{SL}_3(\mathbb{Z})$ and $a(t)=\mathrm{diag}(t^2,t^{1},t^{1})$. The expanding horospherical group $U^+$ is isomorphic to $\mathbb{R}^2$. A result of Shah tells us that the $a(t)$translates of a nondegenerate realanalytic curve in a $(U^+)$orbit get equidistributed in $X$. It remains to study degenerate curves, i.e. planar lines $y=ax+b$. In this talk, we give a Diophantine condition on the parameter $(a,b)$ which serves as a necessary and sufficient condition for equidistribution.
\\
\\
Joint work with Kleinbock, SaxcÃ© and Shah. If time permits, I will also talk about generalisations to $\mathrm{SL}_n(\mathbb{R})/\mathrm{SL}_n(\mathbb{Z})$. Joint work with Shah. 
03/09/21
Joshua Wang  Harvard University
Floer and Khovanov homologies of band sums
AbstractGiven a nontrivial band sum of two knots, we may add full twists to the band to obtain a family of knots indexed by the integers. In this talk, I'll show that the knots in this family have the same knot Floer homology, the same instanton homology, but distinct Khovanov homology, generalizing a result of M. Hedden and L. Watson. A key component of the argument is a proof that each of the three knot homologies detects the trivial band. The main application is a verification of the generalized cosmetic crossing conjecture for split links.

03/09/21
Nathaniel ``Tanny'' Libman  Ph.D. Student  UC San Diego
Quiver Representations, Gabriel's Theorem, and Morita Equivalence
AbstractA quiver is defined as a directed graph with an attitude towards representation theory. In this talk, I will introduce quiver representations and discuss a fundamental classification result due to Gabriel. If time permits, I will also discuss one possible answer to the question, ``Why are quivers?'' There are no prerequisites, and there will be many examples.

03/09/21
Denise Rava  Graduate Student  UC San Diego
Additive Hazards Model: Explained Variation and a Neural Network extension
AbstractPrognostic models in survival analysis are aimed at understanding the relationship between patients' covariates and the distribution of survival time. Traditionally, semiparametric models, such as the Cox model and the Additive Hazards model, have been assumed. In this talk I will derive a measure of explained variation under the Additive Hazards model showing its properties. Moreover I will describe the development of a new flexible method for survival prediction: DeepHazard, a neural network for timevarying risks. I will show its performance on popular real datasets.

03/09/21
Amir Mohammadi  UC San Diego
Dynamics on homogeneous spaces and applications

03/10/21
Valentino Tosatti  McGill University
Smooth asymptotics for collapsing Ricciflat metrics
AbstractI will discuss the problem of understanding the collapsing behavior of Ricciflat Kahler metrics on a CalabiYau manifold that admits a holomorphic fibration structure, when the Kahler class degenerates to the pullback of a Kahler class from the base. I will present new work with HansJoachim Hein where we obtain a priori estimates of all orders for the Ricciflat metrics away from the singular fibers, as a corollary of a complete asymptotic expansion.

03/10/21
Anila Yadavalli  MathCEP Assistant Professor  University of Minnesota
Investigating the Experiences of Asian American Students in an Accelerated Mathematics Program
AbstractThe University of Minnesota Talented Youth Mathematics Program
(UMTYMP) is a selective, fiveyear accelerated mathematics program for
students in grades 612. During the course of the program, students take
advanced mathematics courses on University of Minnesota campuses,
starting with algebra and continuing through logic and proofs, linear
algebra, and multivariable calculus. The majority of UMTYMP students come
from one of three demographic groups: White/Caucasian, East Asian/East
Asian American, and South Asian/South Asian American. We use the term Asian/Asian American to describe students in the latter two demographic groups.
\\
\\
The model minority stereotype (MMS) is the classification of Asian/Asian
American students as gifted, ``academic whizzes'' who outperform their
peers (Choi \& Lahey, 2006). In 2020, we initiated an IRBapproved study
to understand the impact of MMS on Asian/Asian American students who are
labelled as ``gifted" and/or ``talented." In this talk, I will
discuss the process and results of the study, propose best practices for
instructors who interact with students navigating MMS, and suggest ideas
for followup studies on this topic. 
03/11/21
Majid Noroozi  William Chauvenet Postdoctoral Lecturer  Washington University in St. Louis
Estimation and Clustering in Popularity Adjusted Block Model
AbstractWe consider the Popularity Adjusted Block model (PABM) introduced by Sengupta and Chen (2018).
We argue that the main appeal of the PABM is the flexibility of the spectral properties of the graph
which makes the PABM an attractive choice for modeling networks that appear in biological sciences.
We expand the theory of PABM to the case of an arbitrary number of communities which possibly
grows with a number of nodes in the network and is not assumed to be known. We produce estimators
of the probability matrix and the community structure and provide nonasymptotic upper bounds for the
estimation and the clustering errors. We use the Sparse Subspace Clustering (SSC) approach for
partitioning the network into communities, the approach that, to the best of our knowledge, has not been
used for clustering network data. The theory is supplemented by a simulation study. In addition, we
show advantages of the PABM for modeling a butterfly similarity network and a human brain functional
network. 
03/11/21
Elham Izadi  UC San Diego
AWM Colloquium

03/12/21
Daniel Huybrechts  University of Bonn
Brilliant families of K3 surfaces
AbstractWe explain how Hodge theory unifies three a priori very different
types of deformations of K3 surfaces: twistor spaces, Brauer (or TateShafarevich)
families and Dwork families. All three share the property of transporting
complex multiplication from one fibre in the NoetherLefschetz locus to
any other. This phenomenon is at the moment observed in all three cases but
geometrically only explained for Brauer families. The motivation comes
from the Hodge conjecture for squares of K3 surfaces which is still open. 
03/15/21
Zilu Ma  UC San Diego
Tangent flows at infinity of 4dimensional steady Ricci soliton singularity models
AbstractWe will discuss joint work with R. Bamler, B. Chow, Y. Deng, and Y. Zhang on 4dimensional steady Ricci soliton singularity models with 3cylindrical tangent flows at infinity, as well as mention the somewhat parallel work with Y. Zhang on the existence of asymptotic shrinkers on steady solitons with $Ric \geq 0$.

03/17/21
Tucker McElroy  US Census Bureau
Polyspectral Factorization
AbstractThis presentation contributes to the theoretical background for a new quadratic prediction method for time series. We develop a theory of polyspectral factorization, providing new mathematical results for polyspectral densities. New bijections between a restricted space of higherdimensional cepstral coefficients (where the restrictions are induced by the symmetries of the polyspectra) and the autocumulants are derived. Applications to modeling are developed; in particular, it is shown that semiparametric nonlinear time series modeling can be accomplished by approximation of the cepstral representation of polyspectra.

03/18/21
Roberto Imbuzeiro Oliveira  IMPA, Rio de Janeiro
Sample average approximation with heavier tails
AbstractConsider an ``ideal" optimization problem where constraints and objective function are defined in terms of expectations over some distribution P. The sample average approximation (SAA)  a fundamental idea in stochastic optimization  consists of replacing the expectations by an average over a sample from P. A key question is how much the solutions of the SAA differ from those of the original problem. Results by Shapiro from many years ago consider what happens asymptotically when the sample size diverges, especially when the solution of the ideal problem lies on the boundary of the feasible set. In joint work with Philip Thompson (Purdue), we consider what happens with finite samples. As we will see, our results improve upon the nonasymptotic state of the art in various ways: we allow for heavier tails, unbounded feasible sets, and obtain bounds that (in favorable cases) only depend on the geometry of the feasible set in a small neighborhood of the optimal solution. Our results combine ``localization" and ``fixedpoint" type arguments inpired by the work of Mendelson with chainingtype inequalities. One of our contributions is showing what can be said when the SAA constraints are random.

03/24/21
Yi Lai  UC Berkeley
A family of 3dimensional steady gradient Ricci solitons that are flying wings
AbstractWe find a family of 3d steady gradient Ricci solitons that are flying wings. This verifies a conjecture by Hamilton. For a 3d flying wing, we show that the scalar curvature does not vanish at infinity. The 3d flying wings are collapsed. For dimension $n \geq 4$, we find a family of $\mathbb{Z}_2 \times O(n âˆ’ 1)$symmetric but nonrotationally symmetric ndimensional steady gradient solitons with positive curvature operator. We show that these solitons are noncollapsed.

03/25/21
Rachel Ward  University of Texas at Austin
Function Approximation via Sparse Random Features
AbstractRandom feature methods have been successful in various machine learning tasks, are easy to compute, and come with theoretical accuracy bounds. They serve as an alternative approach to standard neural networks since they can represent similar function spaces without a costly training phase. However, for accuracy, random feature methods require more measurements than trainable parameters, limiting their use for datascarce applications or problems in scientific machine learning. This paper introduces the sparse random feature method that learns parsimonious random feature models utilizing techniques from compressive sensing. We provide uniform bounds on the approximation error for functions in a reproducing kernel Hilbert space depending on the number of samples and the distribution of features. The error bounds improve with additional structural conditions, such as coordinate sparsity, compact clusters of the spectrum, or rapid spectral decay. We show that the sparse random feature method outperforms shallow networks for wellstructured functions and applications to scientific machine learning tasks.

03/30/21
Amir Sagiv  Columbia University
Local and optimal transport perspectives on uncertainty propagation
AbstractIn many scientific areas, a deterministic model (e.g., a differential equation) is equipped with parameters. In practice, these parameters might be uncertain or noisy, and so an honest model should provide a statistical description of the quantity of interest. Underlying this computational question is a fundamental one  If two ``similar" functions pushforward the same measure, are the new resulting measures close, and if so, in what sense? I will first show how the probability density function (PDF) can be approximated, using spectral and local methods, and present applications to nonlinear optics. We will then discuss the limitations of PDF approximation, and present an alternative Wassersteindistance formulation of this problem, which yields a much simpler theory.

03/30/21
Dilian Yang  University of Windsor
Selfsimilar kgraph C*algebras
AbstractA selfsimilar kgraph is a pair consisting of a discrete group and a kgraph, such that the group acts on the kgraph selfsimilarly. For such a pair, one can associate it a universal C*algebra, called the selfsimilar kgraph C*algebra. This class of C*algebras embraces many important and interesting C*algebras, such as the higher rank graph C*algebras of KumjianPask, the Katsura algebra, the Nekrashevych algebra, and the ExelPardo algebra. In this talk, I will present some results about those C*algebras, which are based on joint work with Hui Li.

03/30/21
Sam Spiro  UC San Diego
Introduction to Spectral Graph Theory
AbstractGiven a graph $G$, one can compute the eigenvalues of its adjacency matrix $A_G$. Remarkably, these eigenvalues can tell us quite a bit about the structure $G$. More generally, spectral graph theory consists of taking a graph $G$, associating to it a matrix $M_G$, and then using algebraic properties of $M_G$ to recover combinatorial information about $G$. In this talk we discuss some of the more common applications of spectral graph theory, as well as a very simple proof of the sensitivity conjecture due to Huang.

03/31/21
Zilu Ma  UC San Diego
Tangent flows at infinity of 4dimensional steady Ricci soliton singularity models
AbstractAccording to Perelman's work on Ricci flow with surgeries in dimension 3, we know that it is important to understand at least qualitative behaviors of singularity formation in order to perform surgeries. The situation in dimension 4 is much more complicated as some new types of singularity models may arise and the classification of the singularity models is far from complete. We expect that the singularity models should be solitons, selfsimilar solutions to the Ricci flow, and we expect that most of singularity models are shrinking gradient solitons with possible singularities by the recent work of Richard Bamler. Steady gradient Ricci solitons may also arise as singularity models and they are related to shrinking solitons with quadratic curvature growth. In a recent joint work with R. Bamler, B. Chow, Y. Deng, and Y. Zhang, we managed to classify tangent flows at infinity which can be viewed as a blowdown of 4 dimensional steady gradient Ricci soliton singularity models. When the tangent flow at infinity is 3cylindrical, we can give very good qualitative characterization of such steady solitons. We will also mention the somewhat parallel work with Y. Zhang on the existence of asymptotic shrinkers on steady solitons with nonnegative Ricci curvature.
Apr

04/01/21
Peter Koymans  MPIM
Malle's conjecture for nonic Heisenberg extensions
AbstractIn 2002 Malle conjectured an asymptotic formula for the number
of $G$extensions of a number field $K$ with discriminant bounded by
$X$. In this talk I will discuss recent joint work with Etienne Fouvry
on this conjecture. Our main result proves Malle's conjecture in the
special case of nonic Heisenberg extensions. 
04/01/21
Yi Ma  UC Berkeley
Deep Networks from First Principles
AbstractIn this talk, we offer an entirely ``white box'' interpretation of deep (convolution) networks from the perspective of data compression (and group invariance). In particular, we show how modern deep layered architectures, linear (convolution) operators and nonlinear activations, and even all parameters can be derived from the principle of maximizing rate reduction (with group invariance). All layers, operators, and parameters of the network are explicitly constructed via forward propagation, instead of learned via back propagation. All components of soobtained network, called ReduNet, have precise optimization, geometric, and statistical interpretation. There are also several nice surprises from this principled approach: it reveals a fundamental tradeoff between invariance and sparsity for class separability; it reveals a fundamental connection between deep networks and Fourier transform for group invariance â€“ the computational advantage in the spectral domain (why spiking neurons?); this approach also clarifies the mathematical role of forward propagation (optimization) and backward propagation (variation). In particular, the soobtained ReduNet is amenable to finetuning via both forward and backward (stochastic) propagation, both for optimizing the same objective.
\\
\\
This is joint work with students Yaodong Yu, Ryan Chan, Haozhi Qi of Berkeley, Dr. Chong You now at Google Research, and Professor John Wright of Columbia University. 
04/01/21
Elham Izadi  UC San Diego
Some fun facts about cubics
AbstractCubic hypersurfaces are the zero sets of homogeneous polynomials of degree 3. They have been, still are, and probably will be for quite some time, the subject of a lot of research. I will survey a few wellknown and fun facts about cubic hypersurfaces and will also mention some open problems.

04/06/21
Jenna Zomback  University of Illinois UrbanaChampaign
A backward ergodic theorem and its forward implications
AbstractIn the classical pointwise ergodic theorem for a probability measure preserving (pmp) transformation T, one takes averages of a given integrable function over the intervals {x, T(x), T2(x),..., Tn(x)} in front of the point x. We prove a "backward" ergodic theorem for a countabletoone pmp T, where the averages are taken over subtrees of the graph of T that are rooted at x and lie behind x (in the direction of T1). Surprisingly, this theorem yields forward ergodic theorems for countable groups, in particular, one for pmp actions of free groups of finite rank, where the averages are taken along subtrees of the standard Cayley graph rooted at the identity. This strengthens Bufetov's theorem from 2000, which was the most general result in this vein.
\\
\\
This is joint work with Anush Tserunyan. 
04/06/21
Li Gao  Technische Universit\"{a}t M\""{u}nchen"
On Complete Logarithmic Sobolev Inequalities
AbstractLogarithmic Sobolev inequalities (LSI) were first introduced by
Gross in the 70's, and later found rich connections to geometry,
probability, graph theory, optimal transport as well as information
theory. In recent years, logarithmic Sobolev inequalities for quantum
Markov semigroups have attracted a lot of attention and found
applications in quantum information theory and quantum manybody
system. For classical Markov semigroup on a probability space, an
important advantage of logSobolev inequalities is the tensorization
property that if two semigroups satisfies LSI, so does their tensor
product semigroup. Nevertheless, tensoraization property fails for LSI
in the quantum cases. In this talk, I'll present some recent progress
on tensor stable logSobolev inequalities for quantum Markov
semigroups.
\\
\\
This talk is based on joint works with Michael Brannan,
Marius Junge, Nicholas LaRacuente, Haojian Li and Cambyse Rouze. 
04/06/21
Lisa Piccirillo  MIT
Knot concordance and exotica
AbstractOne wellknown strategy for distinguishing smooth structures on closed 4manifolds is to produce a knot $K$ in $S^3$ which is (smoothly) slice in one smooth filling $W$ of $S^3$ but not slice in some homeomorphic smooth filling $Wâ€™$. There are many techniques for distinguishing smooth structures on complicated closed 4manifolds, but this strategy stands out for itâ€™s potential to work for 4manifolds $W$ with very little algebraic topology. However, this strategy had never actually been used in practice, even for complicated $W$. Iâ€™ll discuss joint work with Manolescu and Marengon which gives the first application of this strategy. Iâ€™ll also discuss joint work with Manolescu which gives a systematic approach towards using this strategy to produce exotic definite closed 4manifolds.

04/06/21
Jack Jerry Garzella  UC San Diego
Toric Varieites
AbstractAlgebraic Geometry is known for its abstract nonsense and its towers of abstraction (one might even say skyscraper sheaves of abstraction). But for this talk, we'll forget about all of that  we'll explore toric varieties, an extremely concrete way of constructing examples of algebraic varieties. We'll even see examples that are *gasp* not complex manifolds.

04/07/21
Andrew Zimmer  University of Wisconsin Madison
Analytic problems on domains with good intrinsic geometry
AbstractIn this talk, I will describe a new class of domains in complex Euclidean space which is defined in terms of the existence of Kaehler metrics with good geometric properties. By definition, this class is invariant under biholomorphism. It also includes many wellstudied classes of domains such as strongly pseudoconvex domains, finite type domains in dimension two, convex domains, homogeneous domains, and embeddings of Teichmuller space. Analytic problems are also tractable for this class, in particular we show that compactness of the dbarNeumann operator on (0,q)forms is equivalent to a growth condition of the Bergman metric. This generalizes an old result of FuStraube for convex domains.

04/07/21
Zehua Lai  University of Chicago
RechtRe Noncommutative ArithmeticGeometric Mean Conjecture is False
AbstractStochastic optimization algorithms have become indispensable in modern machine learning. An important question in this area is the difference between withreplacement sampling and withoutreplacement sampling  does the latter have superior convergence rate compared to the former? A paper of Recht and Re reduces the problem to a noncommutative analogue of the arithmeticgeometric mean inequality where n positive numbers are replaced by n positive definite matrices. If this inequality holds for all n, then withoutreplacement sampling (also known as random reshuffling) indeed outperforms withreplacement sampling in some important optimization problems. In this talk, We will explain basic ideas and techniques in polynomial optimization and the theory of noncommutative Positivstellensatz, which allows us to reduce the conjectured inequality to a semidefinite program and the validity of the conjecture to certain bounds for the optimum values. Finally, we show that RechtRe conjecture is false as soon as $n = 5$.
\\
\\
This is a joint work with LekHeng Lim. 
04/08/21
Mahesh Kakde  IISc, Bangalore
On the BrumerStark conjecture and applications to Hilbert's 12th problem
AbstractI will report on my joint work with Samit Dasgupta on the
BrumerStark conjecture proving existence of the BrumerStark units and
on a conjecture of Dasgupta giving a padic analytic formula for these
units. I will present a sketch of our proof of the BrumerStark
conjecture and also mention applications to Hilbert's 12th problem, or
explicit class field theory. 
04/08/21
Mahdi Soltanolkotabi  University of Southern California
Precise Tradeoffs for Adversarial Training
AbstractDespite breakthrough performance, modern learning models are known to be highly vulnerable to small adversarial perturbations in their inputs. While a wide variety of recent adversarial training methods have been effective at improving robustness to perturbed inputs (robust accuracy), often this benefit is accompanied by a decrease in accuracy on benign inputs (standard accuracy), leading to a tradeoff between often competing objectives. Complicating matters further, recent empirical evidence suggests that a variety of other factors (size and quality of training data, model size, etc.) affect this tradeoff in somewhat surprising ways. In this talk we will provide a precise and comprehensive understanding of the role of adversarial training in the context of linear regression with Gaussian features and binary classification in a mixture model. We precisely characterize the standard/robust accuracy and the corresponding tradeoff achieved by a contemporary minimax adversarial training approach in a highdimensional regime where the number of data points and the parameters of the model grow in proportion to each other. Our theory for adversarial training algorithms also facilitates the rigorous study of how a variety of factors (size and quality of training data, model overparametrization etc.) affect the tradeoff between these two competing accuracies.

04/09/21
Arkadij Bojko  University of Oxford
Wallcrossing for Hilbert schemes on fourfolds and Quotschemes on surfaces
AbstractVirtual counting of coherent sheaves has seen recently a large
development in complex dimension four, where it was defined for
CalabiYau fourfolds by BorisovJoyce and OhThomas. I will focus
on invariants for Hilbert schemes of points as they have not been well
understood before. The only known result expressed integrals of top
Chern classes of tautological vector bundles associated to smooth
divisors in terms of the MacMahon function and CaoKool conjectured
this holds for any line bundle. To address these questions I discuss
the conjectural wallcrossing formulae of Joyce and discuss how to
relate them to the conjectures on Hilbert schemes. On the other hand,
ArbesfeldJohnsonLimOpreaPandharipande studied Quotschemes on
surfaces and their virtual integrals giving explicit expressions for
their generating series. Interestingly, these satisfy similar
wallcrossing formulae as Hilbert schemes in the fourfold case when
the curve class is zero. As a consequence their general invariants
share a large similarity. Computing explicitly virtual fundamental
classes and integrals on both, we can firstly recover the results in
the five author paper from a small piece of data. Moreover, we obtain
a universal transformation comparing integrals on Hilbert schemes on
fourfolds and elliptic surfaces. 
04/09/21
Alejandro Morales  University of Massachusetts, Amherst
Juggling, refinements and symmetries for volumes of flow polytopes
AbstractFlow polytopes are an important class of polytopes in combinatorics whose lattice points and volumes have interesting properties and relations to other parts of geometric and algebraic combinatorics. These polytopes were recently related to (multiplex) juggling sequences of Butler, Graham and Chung. The ChanRobbinsYuen (CRY) polytope is a flow polytope with normalized volume equal to the product of consecutive Catalan numbers. Zeilberger proved this by evaluating the Morris constant term identity, but no combinatorial proof is known. There is a refinement of this formula that splits the largest Catalan number into Narayana numbers, which MÃ©szÃ¡ros gave an interpretation as the volume of a collection of flow polytopes. In this talk we will talk about the connection between juggling and flow polytopes and introduce a new refinement of the Morris identity with combinatorial interpretations both in terms of lattice points and volumes of flow polytopes.
\\
\\
The first part of the talk is based on joint work with Benedetti, Hanusa, Harris and Simpson and the second part is based on joint work with William Shi. 
04/12/21
Wenshuai Jiang  Zhejiang University
Quantitative estimates of singular set and hessian estimate of harmonic functions on Einstein manifolds, Part 1
AbstractIn these two talks, we will first discuss Cheeger and Naber's quantitative estimates of singular sets on manifolds with lower Ricci curvature and also review some recent developments. As consequences, we will discuss the hessian estimates of harmonic functions on Einstein manifolds.

04/13/21
Brian Tran  UC San Diego
Multisymplectic Hamiltonian Variational Integrators
AbstractThe multisymplectic structure of Lagrangian and Hamiltonian PDEs is a covariant generalization of the fieldtheoretic symplectic structure and encodes many important physical conservation laws. Multisymplectic integrators are a class of numerical methods which, at the discrete level, preserve the multisymplectic structure of a Lagrangian or Hamiltonian field theory. By preserving this structure at the discrete level, a multisymplectic integrator admits discrete analogs of the conservation laws encoded by multisymplecticity. Such methods have been used, for example, for structurepreserving modeling of nonlinear wave phenomena and for stable discretizations of plasma physics problems. There have also been recent investigations into the application of discrete multisymplectic structures for lattice quantum field theory.
\\
\\
Traditionally, such multisymplectic integrators have been constructed from the variational perspective in the Lagrangian framework, or from directly discretizing the equations of motion in the Hamiltonian framework and subsequently determining the conditions for the discretization method to be multisymplectic in an ad hoc manner. In this talk, after discussing the necessary background material, I will discuss a systematic framework for constructing multisymplectic Hamiltonian integrators variationally utilizing the notion of a Type II generating functional. This framework only requires a choice of finitedimensional function space and quadrature, so it is applicable to unstructured meshes, whereas traditional Hamiltonian multisymplectic integrators require rectangular meshes. As an application of this framework, I will derive the class of multisymplectic partitioned RungeKutta methods and show that, in this framework, discretizing via a tensor product partitioned RungeKutta expansion in spacetime is welldefined if and only if the partitioned RungeKutta methods are symplectic in space and time. This is joint work with Prof. Melvin Leok.
\\
\\
Time permitting, I will discuss future research directions, such as applications to lattice quantum field theory. 
04/13/21
Jason Crann  Carleton University
Local lifting and approximation properties for operator modules
AbstractWe introduce notions of finite presentation which serve as analogues of finitedimensionality for operator modules over completely contractive Banach algebras. With these notions we then introduce analogues of the local lifting property, nuclearity, and semidiscreteness. For a large class of operator modules, we show that the local lifting property is equivalent to flatness, generalizing the operator space result of Kye and Ruan. We pursue applications to abstract harmonic analysis, where, for a locally compact group G, we show that A(G)nuclearity of the inclusion $C*_r(G) \to C^*_r(G)**$ and $A(G)$semidiscreteness of $VN(G)$ are both equivalent to amenability of $G$. We also present the equivalence between $A(G)$injectivity of the crossed product $G\bar{\ltimes}M$, $A(G)$semidiscreteness of $G\bar{\ltimes} M$, and amenability of W*dynamical systems $(M,G,\alpha)$ with $M$ injective.

04/13/21
Artem Kotelskiy  Indiana University
Khovanov homology via Floer theory of the 4punctured sphere
AbstractConsider a Conway twosphere S intersecting a knot K in 4 points, and thus decomposing the knot into two 4ended tangles, T and Tâ€™. We will first interpret Khovanov homology Kh(K) as Lagrangian Floer homology of a pair of specifically constructed immersed curves C(T) and C'(Tâ€™) on the dividing 4punctured sphere S. Next, motivated by several tanglereplacement questions in knot theory, we will describe a recently obtained structural result concerning the curve invariant C(T), which severely restricts the types of curves that may appear as tangle invariants. The proof relies on the matrix factorization framework of KhovanovRozansky, as well as the homological mirror symmetry statement for the 3punctured sphere.
\\
\\
This is joint work with Liam Watson and Claudius Zibrowius. 
04/13/21
Junekey Jeon  UC San Diego
PointFree Measure Theory
AbstractPointfree measure theory is an approach to measure theory in a more abstract viewpoint. Specifically, it forgets the notion of ``points" and tries to recover the whole measure theory only with the notion of measurable sets. We will briefly see why this viewpoint has a potential to liberate us from the agony that we feel all the time when an uncountable collection pops up in measure theory. We will also talk about how to define ``measurable functions" in the pointfree way.

04/14/21
Zihui Zhao  University of Chicago
Boundary unique continuation of Dini domains
AbstractLet $u$ be a harmonic function in $\Omega \subset \mathbb{R}^d.$ It is known that in the interior, the singular set $\mathcal{S}(u) = \{u=\nabla u=0 \}$ is $(d2)$dimensional, and moreover $\mathcal{S}(u)$ is $(d2)$rectifiable and its Minkowski content is bounded (depending on the frequency of $u$). We prove the analogue near the boundary for $C^1$Dini domains: If the harmonic function $u$ vanishes on an open subset $E$ of the boundary, then near $E$ the singular set $\mathcal{S}(u) \cap \overline{\Omega}$ is $(d2)$rectifiable and has bounded Minkowski content. Dini domain is the optimal domain for which $\nabla$ u is continuous towards the boundary, and in particular every $C^{1,\alpha}$ domain is Dini. The main difficulty is the lack of monotonicity formula near the boundary of a Dini domain.
\\
\\
This is joint work with Carlos Kenig. 
04/14/21
Mengdi Wang  Princeton University
Regret bounds of modelbased reinforcement learning
AbstractWe discuss some recent results on modelbased methods for online reinforcement learning (RL). The goal of online RL is to adaptively explore an unknown environment and learn to act with provable regret bounds. In particular, we focus on finitehorizon episodic RL where the unknown transition law belongs to a generic family of models. We propose a model based `valuetargeted regression' RL algorithm that is based on optimism principle: In each episode, the set of models that are `consistent' with the data collected is constructed. The criterion of consistency is based on the total squared error of that the model incurs on the task of predicting values as determined by the last value estimate along the transitions. The next value function is then chosen by solving the optimistic planning problem with the constructed set of models. We derive a bound on the regret, for arbitrary family of transition models, using the notion of the socalled Eluder dimension proposed by Russo \& Van Roy (2014).

04/14/21
Wenshuai Jiang  Zhejiang University
Quantitative estimates of singular set and hessian estimate of harmonic functions on Einstein manifolds, Part 2
AbstractIn these two talks, we will first discuss Cheeger and Naber's quantitative estimates of singular sets on manifolds with lower Ricci curvature and also review some recent developments. As consequences, we will discuss the hessian estimates of harmonic functions on Einstein manifolds.

04/15/21
Subhabrata Sen  Harvard University
Large deviations for dense random graphs: beyond meanfield
AbstractIn a seminal paper, Chatterjee and Varadhan derived an LDP for the dense Erd\H{o}sR\'{e}nyi random graph, viewed as a random graphon. This directly provides LDPs for continuous functionals such as subgraph counts, spectral norms, etc. In contrast, very little is understood about this problem if the underlying random graph is \emph{inhomogeneous} or \emph{constrained}.
\\
\\
In this talk, we will explore large deviations for dense random graphs, beyond the ``meanfield'' setting. In particular, we will study large deviations for uniform random graphs with given degrees, and a family of dense block model random graphs. We will establish the LDP in each case, and identify the rate function. In the block model setting, we will use this LDP to study the upper tail problem for homomorphism densities of regular subgraphs. Our results establish that this problem exhibits a symmetry/symmetrybreaking transition, similar to one observed for Erd\H{o}sR\'{e}nyi random graphs.
\\
\\
Based on joint works with Christian Borgs, Jennifer Chayes, Souvik Dhara, Julia Gaudio and Samantha Petti. 
04/15/21
Michael Wakin  Colorado School of Mines
Spectral Properties of Timelimited Toeplitz Operators and Applications in Signal Processing
AbstractToeplitz operators are fundamental and ubiquitous in signal processing and information theory as models for convolutional (filtering) systems. Due to the fact that any practical system can access only signals of finite duration, however, timelimited restrictions of Toeplitz operators are also of interest. In the discretetime case, timelimited Toeplitz operators are simply Toeplitz matrices. In this talk we survey existing and present new bounds on the eigenvalues (spectra) of timelimited Toeplitz operators, and we discuss applications of these results in various signal processing contexts. As a special case, we discuss timefrequency limiting operators, which alternatingly limit a signal in the time and frequency domains. Slepian functions arise as eigenfunctions of these operators, and we describe applications of Slepian functions in spectral analysis of multiband signals, superresolution SAR imaging, and blind beamforming in antenna arrays.
\\
\\
This talk draws from joint work with numerous collaborators including Zhihui Zhu from the University of Denver. 
04/15/21
Lance Miller  University of Arkansas
Finiteness of quasicanonical lifts of elliptic curves
AbstractFix a prime integer $p$. Set $R$ the completed valuation ring
of the maximal unramified extension of $\mathbb{Q}_p$. For $X :=
X_1(N)$ the modular curve with $N$ at least 4 and coprime to $p$,
BuiumPoonen in 2009 showed that the locus of canonical lifts enjoys
finite intersection with preimages of finite rank subgroups of $E(R)$
when $E$ is an elliptic curve with a surjection from $X$. This is done
using Buium's theory of arithmetic ODEs, in particular interesting
homomorphisms $E(R) \to R$ which are arithmetic analogues of Manin maps.
\\
\\
In this talk, I will review the general idea behind this result and
other applications of arithmetic jet spaces to Diophantine geometry and
discuss a recent analogous result for quasicanonical lifts, i.e., those
curves with SerreTate parameter a root of unity. Here the ODE Manin
maps are insufficient, so we introduce a PDE version of Buium's theory
to provide the needed maps. All of this is joint work with A. Buium. 
04/16/21
Evgeny Shinder  University of Sheffield
Factorization centers, Cremona groups and the Grothendieck ring of varieties
AbstractI will state the question of uniqueness for centers of
blow ups and blow downs of birational maps, explain what is currently
known and give two applications. The first is to the structure of
Cremona groups, namely their nongeneration by involutions in dimension
$>$= 3. The second application is for the Grothendieck ring of
varieties, of dimension $<$= 2, over perfect fields.
\\
\\
Based on joint work with H.Y. Lin, and with H.Y. Lin and S. Zimmermann. 
04/20/21
Rachel Greenfeld  UCLA
Translational tilings in lattices
AbstractLet $F$ be a finite subset of $\mathbb{Z}^d$. We say that $F$ is a translational tile of $\mathbb{Z}^d$ if it is possible to cover $\mathbb{Z}^d$ by translates of $F$ without any overlaps. The periodic tiling conjecture, which is perhaps the most wellknown conjecture in the area, suggests that any translational tile admits at least one periodic tiling. In the talk, we will motivate and discuss the study of this conjecture. We will also present some new results, joint with Terence Tao, on the structure of translational tilings in lattices and introduce some applications.

04/20/21
Pieter Spaas  UCLA
Stable decompositions and rigidity for product equivalence relations
AbstractAfter discussing the motivation behind the talk and some necessary preliminaries, we will consider the ``stabilization" of a countable ergodic p.m.p. equivalence relation which is not Schmidt, i.e. admits no central sequences in its full group. Using a new local characterization of the Schmidt property, we show that this always gives rise to a socalled stable equivalence relation with a unique stable decomposition, providing the first nonstrongly ergodic such examples. We will also discuss some new structural results for product equivalence relations, which we will obtain using von Neumann algebraic techniques.

04/20/21
Xindong Tang  UC San Diego
Generalized Nash Equilibrium Problems of Polynomials
AbstractThe Generalized Nash Equilibrium Problem (GNEP) is a kind of games to find strategies for a group of players such that each playerâ€™s objective function is optimized, given other playersâ€™ strategies. If all the objective and constraining functions involved are polynomials, we call the problem a Generalized Nash Equilibrium Problem of Polynomials (GNEPP). When the constraining functions of each player are independent of other playerâ€™s strategies, the GNEP is called a (standard) Nash Equilibrium Problem (NEP). The GNEP is said to be convex if each playerâ€™s optimization is a convex optimization problem, given other playersâ€™ strategies.
\\
\\
For nonconvex Nash equilibrium problems that are given by polynomial functions, we formulate efficient polynomial optimization problems for computing Nash equilibria. We show that under generic assumptions, the method can find one or even all Nash equilibria if they exist, or detect nonexistence of Nash equilibria. For convex GNEPPs, we introduce rational and parametric expressions for Lagrange multipliers to formulate polynomial optimization for computing Generalized Nash Equilibria (GNEs). We prove that under some specific assumptions, the method can find a GNE if there exists one, or detect nonexistence of GNEs. Numerical experiments are presented to show the efficiency of the methods. The MomentSOS hierarchy of semidefinite relaxations are used to solve the polynomial optimization.
\\
\\
Moreover, we study the GaussSeidel method for solving the nonconvex GNEPPs. We give a certificate for a class of GNEPPs such that the GaussSeidel method is guaranteed to converge, and the numerical experiments show that the GaussSeidel method can solve many GNEPPs efficiently. 
04/20/21
Irving Dai  MIT
Equivariant knots via knot Floer homology
AbstractWe discuss how knot Floer homology can be used to study equivariant knots. We introduce some largesurgery correction terms that obstruct equivariant sliceness (and more generally, bound equivariant genus, following work of Juhasz and Zemke). We describe some crossingchange inequalities for these invariants. We also describe an amusing application to distinguishing (up to isotopy rel boundary) pairs of slice disks related by symmetries of a knot.
\\
\\
This is work in progress with Abhishek Mallick and Matthew Stoffregen. 
04/20/21
Iordan Ganev  Weizmann Institute of Science
The QR decomposition for radial neural networks
AbstractWe present a perspective on neural networks stemming from quiver representation theory. This point of view emphasizes the symmetries inherent in neural networks, interacts nicely with gradient descent, and has the potential to improve training algorithms. As an application, we prove an analogue of the QR decomposition for radial neural networks, which leads to a dimensional reduction result. We assume a basic machine learning background, while explaining all necessary representation theory concepts from first principles.
\\
\\
The talk is based on joint workinprogress with Robin Walters. 
04/20/21
Bryan Hu  UC San Diego
Potpourri of Number Theory
AbstractFermat's Last Theorem that $x^n + y^n = z^n$ has no positive integer solutions for n$>$2 was first written down by Fermat himself in the early 17th century and resisted proof until Andrew Wiles' monumental 1994 paper. During the 300 years in between, many others tried their hand and along the way developed a lot of interesting number theory. We will discuss more classical topics in algebraic number theory  cyclotomic fields, higher reciprocity laws, class field theory, etc.  in the context of historical attempts to prove the theorem. We will be able to verify (the first case of) Fermat's Last Theorem for pretty high prime exponents (p $<$= 156,442,236,847,241,729).

04/21/21
Yangyang Li  Princeton University
Generic Regularity of Minimal Hypersurfaces in Dimension 8
AbstractThe wellknown Simons's cone suggests that minimal hypersurfaces could be possibly singular in a Riemannian manifold with dimension greater than 7, unlike the lower dimensional case. Nevertheless, it was conjectured that one could perturb away these singularities generically. In this talk, I will discuss how to perturb them away to obtain a smooth minimal hypersurface in an 8dimension closed manifold, by induction on the ``capacity" of singular sets. This result generalizes the previous works by N. Smale and by ChodoshLiokumovichSpolaor to any 8dimensional closed manifold.
\\
\\
This talk is based on joint work with Zhihan Wang. 
04/21/21
Massimiliano Di Ventra  Department of Physics, UC San Diego
Digital MemComputing: from logic to dynamics to topology
AbstractMemComputing [1, 2] is a novel physicsbased approach to computation that employs time nonlocality (memory) to both process and store information on the same physical location. Its digital version [3, 4] is designed to solve combinatorial optimization problems. A practical realization of digital memcomputing machines (DMMs) can be accomplished via circuits of nonlinear dynamical systems with memory engineered so that periodic orbits and chaos can be avoided. A given logic problem is first mapped into this type of dynamical system whose point attractors represent the solutions of the original problem. A DMM then finds the solution via a succession of elementary instantons whose role is to eliminate solitonic configurations of logical inconsistency (``logical defects") from the circuit [5, 6]. I will discuss the physics behind memcomputing and show many examples of its applicability to various combinatorial optimization and Machine Learning problems demonstrating its advantages over traditional approaches [7, 8]. Work supported by DARPA, DOE, NSF, CMRR, and MemComputing, Inc.
\\
\\
{[1]} M. Di Ventra and Y.V. Pershin, Computing: the Parallel Approach, Nature Physics 9, 200 (2013).
\\
{[2]} F. L. Traversa and M. Di Ventra, Universal Memcomputing Machines, IEEE Transactions on Neural Networks and Learning Systems 26, 2702 (2015).
\\
{[3]} M. Di Ventra and F.L. Traversa, Memcomputing: leveraging memory and physics to compute efficiently, J. Appl. Phys. 123, 180901 (2018).
\\
{[4]} F. L. Traversa and M. Di Ventra, Polynomialtime solution of prime factorization and NPcomplete problems with digital memcomputing machines, Chaos: An Interdisciplinary Journal of Nonlinear Science 27, 023107 (2017).
\\
{[5]} M. Di Ventra, F. L. Traversa and I.V. Ovchinnikov, Topological field theory and computing with instantons, Annalen der Physik 529,1700123 (2017).
\\
{[6]} M. Di Ventra and I.V. Ovchinnikov, Digital memcomputing: from logic to dynamics to topology, Annals of Physics 409, 167935 (2019).
\\
{[7]} F. L. Traversa, P. Cicotti, F. Sheldon, and M. Di Ventra, Evidence of an exponential speedup in the solution of hard optimization problems, Complexity 2018, 7982851 (2018).
\\
{[8]} F. Sheldon, F.L. Traversa, and M. Di Ventra, Taming a nonconvex landscape with dynamical longrange order: memcomputing Ising benchmarks, Phys. Rev. E 100, 053311 (2019). 
04/22/21
Owen Barrett  University of Chicago
The derived category of the abelian category of constructible sheaves
AbstractNori proved in 2002 that given a complex algebraic variety
$X$, the bounded derived category of the abelian category of constructible sheaves on $X$ is
equivalent to the usual triangulated category $D(X)$ of bounded constructible complexes on $X$.
He moreover showed that given any constructible sheaf $\mathcal F$ on
$\mathbb{A}^n$, there is an injection $\mathcal F\hookrightarrow\mathcal G$ with
$\mathcal G$ constructible and $H^i(\mathbb{A}^n,\mathcal G)=0$ for $i>0$.
\\
\\
In this talk, I'll discuss how to extend Nori's theorem to the case of a
variety over an algebraically closed field of positive characteristic, with
Betti constructible sheaves replaced by $\ell$adic sheaves.
This is the case $p=0$ of the general problem which asks whether the bounded
derived category of $p$perverse sheaves is equivalent to $D(X)$, resolved
affirmatively for the middle perversity by Beilinson. 
04/23/21
Yajnaseni Dutta  University of Bonn
Holomorphic 1forms and geometry
AbstractIn this talk I will discuss various topological and geometric consequences of the existence of zeros of global holomorphic 1forms on smooth projective varieties. Such consequences have been indicated by a plethora of results. I will present some old and new results in this direction. One highlight of the topic is an interesting connection between two sets of such 1forms, one that arises out of the generic vanishing theory and the other that falls out of Hodge theory of algebraic maps.
\\
\\
This is joint work with Feng Hao and Yongqiang Liu. 
04/23/21
Brian Tran  UC San Diego
Variational Discretizations of Lagrangian and Hamiltonian Field Theories
AbstractIn this talk, after discussing the necessary background material, I will discuss the main results of my recent work \emph{Variational Structures in Cochain Projection Based Discretizations of Lagrangian PDEs} and \emph{Multisymplectic Hamiltonian Variational Integrators}; this is joint work with Prof. Melvin Leok.
\\
\\
Building on these ideas, I will conclude by discussing future research directions. 
04/27/21
Prasuna Bandi  Tata Institute of Fundamental Research
Density at integer points of an inhomogeneous quadratic form and linear form
AbstractIn 1987, Margulis solved an old conjecture of Oppenheim which states that for a nondegenerate, indefinite and irrational quadratic form $Q$ in $n \geq 3$ variables, $Q(\mathbb{Z}^n)$ is dense in $\mathbb{R}$. Following this, Dani and Margulis proved the simultaneous density at integer points for a pair consisting of quadratic and linear form in $3$ variables when certain conditions are satisfied. We prove an analogue of this for the case of an inhomogeneous quadratic form and a linear form.
\\
\\
This is based on joint work with Anish Ghosh. 
04/27/21
Shuang Liu  UC San Diego
A parallel cutcell algorithm for the freeboundary GradShafranov problem
AbstractA parallel cutcell algorithm is described to solve the free boundary problem of the GradShafranov equation.
The algorithm reformulates the freeboundary problem in an irregular bounded domain and its important aspects include a
searching algorithm for the magnetic axis and separatrix, a surface integral along the irregular boundary to determine the
boundary values, an approach to optimize the coil current based on a targeting plasma shape, Picard iterations with Aitken's
acceleration for the resulting nonlinear problem and a Cartesian grid embedded boundary method to handle the complex
geometry. The algorithm is implemented in parallel using a standard domaindecomposition approach and a good parallel
scaling is observed. Numerical results verify the accuracy and efficiency of the freeboundary GradShafranov solver. 
04/27/21
Ryan Mike  UC San Diego
Rational Tangles and the Square Dance
AbstractAn $n$tangle is a proper embedding of the disjoint union of n arcs into a $3$ball, in such a way that the endpoints are mapped to $2n$ marked points on the ballâ€™s boundary. In 1967, Conway developed the theory of a special class of $2$tangles, called ``rational tangles," leading to important results on the classification of knots. Rational tangles themselves have an elegant classification which relates to continued fraction expansions of rational numbers. We explore this connection in the context of a fun activity, which was developed by Conway in order to demonstrate some aspects of the theory.

04/27/21
Takahiro Hasebe  Hokkaido University
The eigenvalues of principal submatrices in rotationally invariant hermitian random matrices and the MarkovKrein Correspondence
AbstractThis talk establishes a concentration phenomenon on the empirical eigenvalue distribution (EED) of the principal submatrix in a random hermitian matrix whose distribution is invariant under unitary conjugacy. More precisely, if the EED of the whole matrix converges to some deterministic probability measure ð”ª, then its difference from the EED of its principal submatrix, after a rescaling, concentrates at the Rayleigh measure (in general, a Schwartz distribution) associated with ð”ª by the MarkovKrein correspondence. For the proof, we use the moment method with Weingarten calculus and free probability. At some stage of calculations, the proof requires a relation between the moments of the Rayleigh measure and free cumulants of ð”ª. This formula is more or less known, but we provide a different proof by observing a combinatorial structure of noncrossing partitions.
\\
\\
This is a joint work with Katsunori Fujie. 
04/28/21
Peter Petersen  UCLA
Rigidity of Homogeneous Gradient Soliton Metrics and Related Equations
AbstractThis is joint work with Will Wylie. The goal is to classify, if possible, the homogeneous geometric solitons. Here a geometric soliton is the soliton for a geometric flow. The Ricci flow is the most prominent example of such a flow, but there are man others where the Ricci tensor is replaced with some other tensor that depends in a natural way on the Riemannian structure. We will also consider some more general problems showing that our techniques can be used for other geometric problems.

04/29/21
Anne Gelb  Dartmouth College
Empirical Bayesian Inference using Joint Sparsity
AbstractWe develop a new empirical Bayesian inference algorithm for solving a linear inverse problem given multiple measurement vectors (MMV) of undersampled and noisy observable data. Specifically, by exploiting the joint sparsity across the multiple measurements in the sparse domain of the underlying signal or image, we construct a new support informed sparsity promoting prior. Several applications can be modeled using this framework. Our numerical experiments demonstrate that using this new prior not only improves accuracy of the recovery, but also reduces the uncertainty in the posterior when compared to standard sparsity producing priors.
\\
\\
This is joint work with Theresa Scarnati formerly of the Air Force Research Lab Wright Patterson and now working at Qualis Corporation in Huntsville, AL, and Jack Zhang, recent bachelor degree recipient at Dartmouth College and now enrolled at University of Minnesotaâ€™s PhD program in mathematics. 
04/29/21
March Boedihardjo  UCLA
Spectral norms of Gaussian matrices with correlated entries
AbstractWe give a nonasymptotic bound on the spectral norm of a $d\times d$ matrix $X$ with centered jointly Gaussian entries in terms of the covariance matrix of the entries. In some cases, this estimate is sharp and removes the $\sqrt{\log d}$ factor in the noncommutative Khintchine inequality.
\\
\\
Joint work with Afonso Bandeira 
04/29/21
Christian Klevdal  University of Utah
Integrality of Glocal systems
AbstractSimpson conjectured that for a reductive group $G$, rigid
$G$local systems on a smooth projective complex variety are integral. I
will discuss a proof of integrality for cohomologically rigid $G$local
systems. This generalizes and is inspired by work of Esnault and
Groechenig for $GL_n$. Surprisingly, the main tools used in the proof
(for general $G$ and $GL_n$) are the work of L. Lafforgue on the
Langlands program for curves over function fields, and work of Drinfeld
on companions of $\ell$adic sheaves. The major differences between
general $G$ and $GL_n$ are first to make sense of companions for
$G$local systems, and second to show that the monodromy group of a
rigid Glocal system is semisimple.
\\
\\
All work is joint with Stefan Patrikis. 
04/29/21
Yueyang Zhong  University of Chicago
BehaviorAware Queueing: The FiniteBuffer Setting with Many Strategic Servers
May

05/04/21
Lingfu Zhang  Princeton University
Factor of IID for the free Ising model on the dregular tree
AbstractIt is known that there are factors of IID for the free Ising model on the dregular tree when it has a unique Gibbs measure and not when reconstruction holds (when it is not extremal). We construct a factor of IID for the free Ising model on the dregular tree in (part of) its intermediate regime, where there is nonuniqueness but still extremality. The construction is via the limit of a system of stochastic differential equations.
\\
\\
This is a joint work with Danny Nam and Allan Sly. 
05/04/21
N. Christopher Phillips  University of Oregon
Lower bounds on the radius of comparison of the crossed product by a minimal homeomorphism.
AbstractLet $X$ be a compact metric space, and let $h$ be a homeomorphism
of $X$. The mean dimension of $h$ is an invariant invented by people
in topological dynamics, with no consideration of C*algebras. The
shift on the product of copies of $[0, 1]^d$ indexed by $\mathbb{Z}$ has mean
dimension $d$.
\\
\\
The radius of comparison of a C*algebra $A$ is an invariant
introduced with no consideration of dynamics, and originally applied
to C*algebras which are not given as crossed products. It is a
numerical measure of bad behavior in the Cuntz semigroup of $A$, and
its original use was to distinguish counterexamples to the original
formulation of the Elliott conjecture.
\\
\\
It is conjectured that if $h$ is a minimal homeomorphism of a compact
metric space, then the radius of comparison of $C* (Z, X, h)$ is equal
to half the mean dimension of $h$. There is a generalization to
countable amenable groups. Considerable progress has been made on
proving that the radius of comparison of $C* (Z, X, h)$ is at most
half the mean dimension; in particular, this is known in full
generality for minimal homeomorphisms. We give the first systematic
results for the opposite inequality. We do not get the exact expected
lower bound, but, for many known examples of actions of amenable
groups with large mean dimension, we come close.
\\
\\
The methods depend on ``mean cohomological independence dimension,"
Cech cohomology, and the Chern character.
\\
\\
This is joint work with Ilan Hirshberg. 
05/04/21
Sui Tang  UC Santa Barbara
Datadriven discovery of interaction laws in multiagent systems
AbstractMultiagent systems are ubiquitous in science, from the modeling of particles in Physics to preypredator in Biology, to opinion dynamics in economics and social sciences, where the interaction law between agents yields a rich variety of collective dynamics. We consider the following inference problem for a system of interacting particles or agents: given only observed trajectories of the agents in the system, can we learn what the laws of interactions are? We would like to do this without assuming any particular form for the interaction laws, i.e. they might be ``any" function of pairwise distances.
\\
\\
In this talk, we consider this problem in the case of a finite number of agents, with an increasing number of observations. We cast this as an inverse problem, and study it in the case where the interaction is governed by an (unknown) function of pairwise distances. We discuss when this problem is wellposed, and we construct estimators for the interaction kernels with provably good statistical and computational properties. We measure their performance on various examples, that include extensions to agent systems with different types of agents, secondorder systems, and stochastic systems. We also conduct numerical experiments to test the large time behavior of these systems, especially in the cases where they exhibit emergent behavior.
\\
\\
This talk is based on the joint work with Fei Lu, Mauro Maggioni, Jason Miller, and Ming Zhong. 
05/05/21
Yury Ustinovskiy  CIMS and NYU
Hermitian curvature flow and uniformization of nonnegatively curved Hermitian manifolds
AbstractThe classical Gauss's Theorema Egregium and the Uniformization theorem for the Riemann surfaces are illustrations of a prominent theme in geometry  control of the global topology/geometry of a manifold through the bounds on its curvature. In the last decades, with the development of new analytic tools (Yamabe equation, meancurvature flow etc), this idea has found numerous applications in classification problems. Application of geometric flows (specifically the Ricci flow) turned out to be particularly fruitful in the context of Kaehler (and projective algebraic) geometry. At the same time there are very few efficient analytic methods available in nonKahler complex geometry. In this talk we will introduce the Hermitian Curvature Flow on an arbitrary compact complex manifold. We will prove a delicate version of the maximum principle for tensors along this flow and present applications to the classification problems for the complex/algebraic manifolds admitting a compatible metric with ``semipositive curvature."

05/05/21
Jingrui Cheng  Stony Brook University
On the existence question of constant scalar curvature Kahler metrics
AbstractI will explain the a priori estimates for the cscK equation on a compact manifold, and how to use these estimates to obtain existence when the associated energy functional is ``coercive." If time permits, I will also explain how we can hope to get existence from a more ``algebraic" condition, which might be easier to check in practice.

05/05/21
Mareike Dressler  UC San Diego
Multiple typical ranks in matrix completion
AbstractIn a matrix completion problem, one has access to a subset of entries of a matrix and wishes to determine the missing entries subject to some constraint (e.g. a rank bound). Such problems appear in computer vision, collaborative filtering, and many other applications. In this talk, I will discuss how certain notions from nonlinear algebra can be used to understand the structure underlying various types of matrix completion problems.

05/05/21
David Baraglia  University of Adelaide
Nontrivial smooth families of K3 surfaces
AbstractLet X be a compact, smooth manifold and Diff(X) the diffeomorphism group. The topology of Diff(X) and of the classifying space BDiff(X) are of great interest. For instance, the kth homotopy group of BDiff(X) corresponds to smooth families over the ksphere with fibres diffeomorphic to X. By a recent result of Bustamante, Krannich and Kupers, if X has even dimension not equal to 4 and finite fundamental group, then the homotopy groups of BDiff(X) are all finitely generated. In contrast we will show that when X is a K3 surface, the second homotopy group of BDiff(X) contains a free abelian group of countably infinite rank as a direct summand. Our families are constructed using the moduli space of Einstein metrics on K3. Their nontriviality is detected using families SeibergWitten invariants.

05/06/21
Maria Fox  University of Oregon
Supersingular Loci of Some Unitary Shimura Varieties
AbstractUnitary Shimura varieties are moduli spaces of abelian
varieties with an action of a quadratic imaginary field, and extra
structure. In this talk, we'll discuss specific examples of unitary
Shimura varieties whose supersingular loci can be concretely described
in terms of DeligneLusztig varieties. By RapoportZink uniformization,
much of the structure of these supersingular loci can be understood by
studying an associated moduli space of pdivisible groups (a
RapoportZink space). We'll discuss the geometric structure of these
associated RapoportZink spaces as well as some techniques for studying
them. 
05/06/21
Angelos Aveklouris  University of Chicago
Matching demand and supply in service platforms.
AbstractService platforms must determine rules for matching heterogeneous demand (customers) and supply (workers) that arrive randomly over time and may be lost if forced to wait too long for a match. We show how to balance the tradeoff between making a less good match quickly and waiting for a better match, at the risk of losing impatient customers and/or workers. When the objective is to maximize the cumulative value of matches over a finitetime horizon, we propose discretereview matching policies, both for the case in which the platform has access to arrival rate parameter information and the case in which the platform does not. We show that both the blind and nonblind policies are asymptotically optimal in a highvolume setting. However, the blind policy requires frequent resolving of a linear program. For that reason, we also investigate a blind policy that makes decisions in a greedy manner, and we are able to establish an asymptotic lower bound for the greedy, blind policy that depends on the matching values and is always higher than half of the value of an optimal policy. Next, we develop a fluid model that approximates the evolution of the stochastic model and captures explicitly the nonlinear dependence between the amount of demand and supply waiting and the distribution of their patience times. We establish a fluid limit theorem and show that the fluid limit converges to its equilibrium. Based on the fluid analysis, we propose a policy for a more general objective that additionally penalizes queue buildup.

05/07/21
Jakub Witaszek  University of Michigan
Global +regularity and the Minimal Model Program for arithmetic threefolds
AbstractIn this talk, I will explain a mixed characteristic
analogue of Frobenius regularity and how it can be used to establish the
Minimal Model Program for threefolds in mixed characteristic.
\\
\\
This is based on a joint work with Bhargav Bhatt, Linquan Ma, Zsolt Patakfalvi,
Karl Schwede, Kevin Tucker, and Joe Waldron. 
05/11/21
Cagri Sert  University of Zurich
Expanding measures: Random walks and rigidity on homogeneous spaces
AbstractWe will start by reviewing recent developments in random walks on homogeneous spaces. In a second part, we will discuss the notion of a $H$expanding probability measure on a connected semisimple Lie group $H$. As we shall see, for a $H$expanding $\mu$ with $H < G$, on the one hand, one can obtain a description of $\mu$stationary probability measures on the homogeneous space $G / \Lambda$ ($G$ Lie group, $\Lambda$ lattice) using the measure classification results of EskinLindenstrauss, and on the other hand, the recurrence techniques of BenoistQuint and EskinMirzakhaniMohammadi can be adapted to this setting. With some further work, these allow us to deduce equidistribution and orbit closure description results simultaneously for a class of subgroups which contains Zariskidense subgroups and further epimorphic subgroups of $H$. If time allows, we will see how, utilizing an idea of SimmonsWeiss, these also allow us to deduce Birkhoff genericity of a class of fractal measures with respect to certain diagonal flows, which, in turn, has applications in diophantine approximation problems. \\
\\
Joint work with Roland Prohaska and Ronggang Shi. 
05/11/21
Jonathan Siegel  Postdoctoral Scholar  Pennsylvania State University
Approximation Rates and Metric Entropy of Shallow Neural Networks
AbstractWe consider the problem of approximating high dimensional functions using shallow neural networks, and more generally by sparse linear combinations of elements of a dictionary. We begin by introducing natural spaces of functions which can be efficiently approximated in this way. Then, we derive the metric entropy of the unit balls in these spaces, which allows us to calculate optimal approximation rates for approximation by shallow neural networks. This gives a precise measure of how large this class of functions is and how well shallow neural networks overcome the curse of dimensionality. Finally, we describe an algorithm which can be used to solve highdimensional PDEs using this space of functions.

05/11/21
Boyu Zhang  Princeton
Irreducible representations of link groups in SU(2)
AbstractSuppose L is a link in S\^{}3, we show that $\pi_1(S^3L)$ admits an irreducible meridiantraceless representation in SU(2) if and only if L is not the unknot, the Hopf link, or a connected sum of Hopf links. As a corollary, $\pi_1(S^3L)$ admits a (not necessarily meridiantraceless) irreducible representation in SU(2) if and only if L is neither the unknot nor the Hopf link. This result generalizes a theorem of Kronheimer and Mrowka to the case of links. The proof is based on singular instanton Floer theory and an observation about finite simple graphs.
\\
\\
This is joint work with Yi Xie. 
05/12/21
Scott Mahan  UC San Diego
Approximation Theory for Equivariant Neural Networks

05/12/21
Daniela De Silva  Columbia University
On the Boundary Harnack Principle
AbstractWe present a direct unified analytic proof of the classical Boundary Harnack Principle for solutions to linear uniformly elliptic equations in either divergence or nondivergence form. The proof extends also to (appropriate) H\"older domains. The strategy also applies to the parabolic context. Applications of the BHP to free boundary problems are discussed."

05/13/21
Sourav Chatterjee  Stanford
New results for surface growth
AbstractThe growth of random surfaces has attracted a lot of attention in probability theory in the last ten years, especially in the context of the KardarParisiZhang (KPZ) equation. Most of the available results are for exactly solvable onedimensional models. In this talk I will present some recent results for models that are not exactly solvable. In particular, I will talk about the universality of deterministic KPZ growth in arbitrary dimensions, and if time permits, a necessary and sufficient condition for superconcentration in a class of growing random surfaces.

05/13/21
Simone Brugiapaglia  Concordia University
The curse of dimensionality and the blessings of sparsity and Monte Carlo sampling
AbstractFrom polynomial to deep neural network approximation in high dimensions approximating multidimensional functions from pointwise samples is a ubiquitous task in data science and scientific computing. This task is made intrinsically difficult by the presence of four contemporary challenges: (i) the target function is usually defined over a high or infinitedimensional domain; (ii) generating samples can be very expensive; (iii) samples are corrupted by unknown sources of errors; (iv) the target function might take values in a function space. In this talk, we will show how these challenges can be substantially overcome thanks to the ``blessings" of sparsity and Monte Carlo sampling.
\\
\\
First, we will consider the case of sparse polynomial approximation via compressed sensing. Focusing on the case where the target function is smooth (e.g., holomorphic), but possibly highly anisotropic, we will show how to obtain sample complexity bounds only mildly affected by the curse of dimensionality, nearoptimal accuracy guarantees, stability to unknown errors corrupting the data, and rigorous convergence rates of algebraic and exponential type. Then, we will illustrate how the mathematical toolkit of sparse polynomial approximation via compressed sensing can be employed to obtain a practical existence theorem for Deep Neural Network (DNN) approximation of highdimensional Hilbertvalued functions. This result shows not only the existence of a DNN with desirable approximation properties, but also how to compute it via a suitable training procedure in order to achieve bestinclass performance guarantees. We will conclude by discussing open research questions.
\\
\\
The presentation is based on joint work with Ben Adcock, Casie Bao, Nick Dexter, Sebastian Moraga, and Clayton G. Webster. 
05/13/21
Sean Howe  University of Utah
Bialgebraicity in local Shimura varieties
AbstractA classical transcendence result of Schneider on the modular
$j$invariant states that, for $\tau \in \mathbb{H}$, both $\tau$ and
$j(\tau)$ are in $\overline{\mathbb{Q}}$ if and only if $\tau$ is
contained in an imaginary quadratic extension of $\mathbb{Q}$. The space
$\mathbb{H}$ has a natural interpretation as a parameter space for
$\mathbb{Q}$Hodge structures (or, in this case, elliptic curves), and
from this perspective the imaginary quadratic points are distinguished
as corresponding to objects with maximal symmetry. This result has been
generalized by Cohen and ShigaWolfart to more general moduli of Hodge
structures (corresponding to abeliantype Shimura varieties), and by
UllmoYafaev to higher dimensional loci of extra symmetry (special
subvarieties), where bialgebraicity is intimately connected with the
PilaZannier approach to the AndreOort conjecture.
\\
\\
In this talk, I will discuss work in progress with Christian Klevdal on
an analogous bialgebraicity characterization of special subvarieties in
Scholze's local Shimura varieties and more general diamond moduli of
$p$adic Hodge structures. 
05/13/21
Justin Mulvany  USC
Fair Scheduling of Heterogeneous Customer Populations.

05/14/21
Jiawang Nie  UC San Diego
Optimization, Positivstellensatz and Convex Algebraic Geometry
AbstractThis talk is about optimizing polynomial functions under constraints. A general method is to apply the MomentSOS hierarchy of semidefinite programming relaxations. The convergence is based on various Positivstellensatz. Closely related polynomial optimization is convex algebraic geometry.
\\
\\
It concerns geometric properties of convex semialgebraic sets through semidefinite programming. We are going to review basic results for these topics. 
05/17/21
Nandagopal Ramachandran  UC San Diego
RubinStark units and Equivariant Iwasawa Theory

05/18/21
Philipp Kunde  University of Hamburg
Anticlassification results for the Kakutani equivalence relation
AbstractDating back to the foundational paper by John von Neumann, a fundamental theme in ergodic theory is the \emph{isomorphism problem} to classify invertible measurepreserving transformations (MPT's) up to isomorphism. In a series of papers, Matthew Foreman, Daniel Rudolph and Benjamin Weiss have shown in a rigorous way that such a classification is impossible. Besides isomorphism, Kakutani equivalence is the best known and most natural equivalence relation on ergodic MPT's for which the classification problem can be considered. In joint work with Marlies Gerber we prove that the Kakutani equivalence relation of ergodic MPT's is not a Borel set. This shows in a precise way that the problem of classifying such transformations up to Kakutani equivalence is also intractable.

05/18/21
O\v{g}uz \c{S}avk  Bo\v{g}azi\c{c}i University
Classical and new plumbings bounding contractible manifolds and homology balls
AbstractA central problem in lowdimensional topology asks which homology 3spheres bound contractible 4manifolds and homology 4balls. In this talk, we address this problem for plumbed 3manifolds and we
present the classical and new results together. Our approach is based on
Mazurâ€™s famous argument which provides a unification of all results in a
fairly simple way. 
05/18/21
Tatiana Shulman  Chalmers University of Gothenburg
Central sequence algebras via nilpotent elements
AbstractA central sequence in a $C^*$algebra is a sequence (x\_n) of elements such that [x\_n, a] converges to zero, for any element a of the $C^*$algebra. In von Neumann algebra setting one typically means the convergence with respect to tracial norms, while in $C^*$theory it is with respect to the $C^*$norm. In this talk we will consider the $C^*$theory version of central sequences. We will discuss properties of central sequence algebras and in particular address a question of J. Phillips and of Ando and Kirchberg of which separable $C^*$algebras have abelian central sequence algebras.
\\
\\
Joint work with Dominic Enders. 
05/18/21
Tingting Tang  San Diego State University and SDSU Imperial Valley
Application of numerical algebraic geometry in parametric semidefinite programming
AbstractIn this talk, we study the property of the solution of semidefinite programs with multidimensional perturbation variables using the Davidenko di erential equations. Under the assumptions of strict complementary and nondegeneracy, we aim to find the a priori unknown maximal convex permissible perturbation set where the semidefinite program has a unique optimum and the optimum is analytic. A sweeping euler numerical method is developed to approximate this a priori unknown perturbation set and solve the semidefinite program within this set. We prove local and global error bounds for this secondorder sweeping Euler scheme and demonstrate results on several examples.

05/19/21
Jun Lau  UC San Diego
Coleman integration and modular curves

05/19/21
Hui Yu  Columbia
Contact points with integer frequencies in the thin obstacle problem
AbstractThe thin obstacle problem is a classical free boundary problem arising from the study of an elastic membrane resting on a lowerdimensional obstacle. Concerning the behavior of the solution near a contact point between the membrane and the obstacle, many important questions remain open. In this talk, we discuss a unified method that leads to a rate of convergence to `tangent cones' at contact points with integer frequencies.
\\
\\
This talk is based on a recent joint work with Ovidiu Savin. 
05/19/21
Yingjia Fu  UC San Diego
Asymptotic Behavior of a Fluid Model for Bandwidth Sharing with General File Size Distributions

05/19/21
Maddie Weinstein  UC Berkeley
Metric Algebraic Geometry
AbstractMetric algebraic geometry is a term proposed for the study of properties of real algebraic varieties that depend on a distance metric. The distance metric can be the Euclidean metric in the ambient space or a metric intrinsic to the variety. In this talk, we introduce metric algebraic geometry through a discussion of Voronoi cells, bottlenecks, offset hypersurfaces, and the reach of an algebraic variety. We also show applications to the computational study of the geometry of data with nonlinear models.

05/20/21
Nahid Walji  University of British Columbia
On the conjectural decomposition of symmetric powers of automorphic representations for GL(3) and GL(4)
AbstractLet $\Pi$ be a cuspidal automorphic representation for GL(3) over a
number field. We fix an integer $k \geq 2$ and we assume that the
symmetric $m$th power lifts of $\Pi$ are automorphic for $m \leq k$,
cuspidal for $m < k$, and that certain associated Rankinâ€“Selberg
products are automorphic. In this setting, we bound the number of
cuspidal isobaric summands in the $k$th symmetric power lift. In
particular, we show it is bounded above by 3 for $k \geq 7$, and bounded
above by 2 when $k \geq 19$ with $k$ congruent to 1 mod 3. We will also
discuss the analogous problem for GL(4). 
05/20/21
Felipe Campos  UC San Diego
Comparison Methods for Markov Processes.
AbstractThe talk will be an overview of Comparison Methods for Stochastic Models and Risks by Muller and Stoyan, and Stochastic Orderings for Markov Processes on Partially Ordered Sets, by William Massey.

05/21/21
Olivier Martin  Stonybrook University
Effective zerocycles and the BlochBeilinson filtration
AbstractLet $X$ be a smooth projective variety whose algebra of
holomorphic forms is generated in degree at most $2$, for instance an
abelian variety or a hyperK\"ahler manifold. A recent conjecture of Voisin 
05/24/21
Yizhe Zhu  UC San Diego
Spectral analysis of sparse random graphs and hypergraphs

05/25/21
Josh Frisch  California Institute of Technology
TBA

05/25/21
Matthew Kennedy  University of Waterloo
TBA

05/25/21
Ben Nassau, Lecturer  Rowan University
The Case for Inquirybased Learning
AbstractHow do we make mathematicians? How do we become them ourselves? And why is it rare that we teach that way? In this talk, we will discuss the pedagogy of inquirybased learning (IBL) and how it can make for a more holistic, equitable, and successful classroom. Bring questions and an open mind  there may be a quiz!

05/26/21
Jiewon Park  Caltech
Geometric applications of the Laplace equation on Ricciflat manifolds
AbstractWe will study complete Ricciflat manifolds with Euclidean volume growth. In the case when a tangent cone at infinity of the manifold has smooth cross section, the Green function for the Laplace equation can be used to define a functional which measures how fast the manifold converges to the tangent cone. Using the Ã…Âojasiewicz inequality of ColdingMinicozzi for this functional, we describe how two arbitrarily far apart scales in the manifold can be identified in a natural way. I will also discuss a matrix Harnack inequality for the Green function when there is an additional condition on sectional curvature, which is an analogue of various matrix Harnack inequalities obtained by Hamilton and LiCao in timedependent settings.

05/26/21
Junyu Cao  The University of Texas at Austin
Adaptive Data Acquisition for Personalized Recommender Systems with Optimality Guarantees on ShortForm Video Platforms
AbstractShortfirm video (SFV) has been exploding on digital platforms recently. The vast amount of videos and fastevolving trends on digital platforms pose technical challenges in making personalized recommendations. In this work, we introduce a new pure exploration problem on SFV platforms. We propose an adaptive data acquisition method, called Adaptive Acquisition Tree (AAT), to jointly account for heterogeneity in user preferences and highdimensional product characteristics. We adaptively divide users based on preference similarity and then learn a personalized transductive bandit policy that can be used on partially or even unobserved arms to accommodate the fastevolving and emerging trends on SFV platforms. We analytically characterize the prediction error, which is determined by both the sample size and the impurity of parameters within a group. We further derive the sample complexity for identifying an optimal set for a single user and for all users. We evaluate the algorithm via numerical experiments on data collected from the NetEase platform. Our result demonstrates that the proposed policy, compared with several stateoftheart benchmarks, performs significantly better in four transductive scenarios for both spotlight recommendation (i.e., bestarm identification) and topK recommendations. With the potential to improve the expected view time by 2025\%, our method pertains to both academic and practical values, given the increasing popularity of shortform videos and, more broadly, online usercontent generation platforms.

05/27/21
Akram Aldroubi  Vanderbilt University
Dynamical Sampling: A Sampling Tour
AbstractDynamical sampling is a term describing an emerging set of problems related to recovering signals and evolution operators from spacetime samples. For example, one problem is to recover a function f from spacetime samples $\{(A_{t_i}f)(x_i)\}$ of its time evolution $f_t = (A_t f)(x)$, where $\{A_t\}_{t \in T}$ is a known evolution operator and $(x_i,t_i) \in R^d \times R^+$.
\\
\\
Another example is a system identification problem when both $f$ and the evolution family $\{A_t\}_{t\in T}$ are unknown. Applications of dynamical sampling include inverse problems in sensor networks, and source term recovery from physically driven evolutionary systems. Dynamical sampling problems are tightly connected to frame theory as well as more classical areas of mathematics such as approximation theory, and functional analysis. In this talk, I will describe a few problems in dynamical sampling, some results and open problems. 
05/27/21
Amol Aggarwal  Columbia University
Spectral Statistics of L\'{e}vy Matrices
AbstractL\'{e}vy matrices are symmetric matrices whose entries are random variables with infinite variance; they are governed by a parameter $\alpha \in (0, 2)$ dictating the power law decay of their entries. For $\alpha <1$, they are believed to serve as one of the few examples of a matrix model exhibiting a mobility edge, also called an Anderson transition, that separates chaotic (GOE) eigenvalue spacing statistics from ordered (Poisson) ones. In this talk we describe results concerning the statistics for the eigenvalue spacings and eigenvector entries of L\'{e}vy matrices. In particular, for $\alpha \in (1, 2)$ their eigenvalue statistics asymptotically follow those of the GOE throughout the spectrum, and for $\alpha < 1$ the same statement holds around small eigenvalues.
\\
\\
These describe joint works with Patrick Lopatto, Jake Marcinek, and HorngTzer Yau. 
05/27/21
Evan O'Dorney  Princeton University
Arithmetic statistics of $H^1(K, T)$
AbstractCoclasses in a Galois cohomology group $H^1(K, T)$ parametrize
extensions of a number field with certain Galois group. It is natural to
want to count these coclasses with general local conditions and with
respect to a discriminantlike invariant. In joint work with Brandon
Alberts, I present a novel tool for studying this: harmonic analysis on
adelic cohomology, modeled after the celebrated use of harmonic analysis
on the adeles in Tate's thesis. This leads to a more illuminating
explanation of a fact previously noticed by Alberts, namely that the
Dirichlet series counting such coclasses is a finite sum of Euler
products; and we nail down the asymptotic count of coclasses in
satisfying generality. 
05/27/21
Nandagopal Ramachandran  UC San Diego
Hilbert's 12th problem and BrumerStark units
AbstractThe goal will be to talk about Dasgupta and Kakde's recent results on understanding the maximal abelian extensions for totally real number fields.

05/28/21
Cameron Cinel  UC San Diego
Sofic Groups
AbstractI will give a brief overview of sofic groups, some of their equivalent characterizations, and their relation to the surjunctivity conjecture. Time permitting, I will discuss other similar groups, such as hyperlinear or linearly sofic groups.

05/28/21
Jacob Keller  UC San Diego
Moduli Spaces and KStability
AbstractThis talk will discuss moduli spaces of algebraic varieties and will introduce Kstability as a tool to construct moduli spaces of Fano varieties.

05/28/21
Jun Bo Lau  UC San Diego
Periods
AbstractWe discuss periods in the classical sense, with motivation from transcendental number theory, and relate them to the study of differential equations and values of Lfunctions.

05/28/21
Rachel Webb  UC Berkeley
Abelianization and quantum lefschetz for orbifold Ifunctions
AbstractLet G be a connected reductive group with maximal torus T, and let V
and E be two representations of G. Then E defines a vector bundle on
the orbifold V//G; let X//G be the zero locus of a regular section.
The quasimap Ifunction of X//G encodes the geometry of maps from $P^1$
to X//G and is related to GromovWitten invariants of X//G. By
directly analyzing these maps from $P^1$, we explain how to relate the
Ifunction of X//G to that of V//T. Our formulas prove a mirror
symmetry conjecture of OnetoPetracci that relates the quantum period
of X//G to a certain Laurent polynomial defined by a Fano polytope.
Finally, we describe a large class of examples to which our formulas
apply, examples that are the orbifold analog of quiver flag varieties.
Question for the audience: what else can one investigate with these
examples? 
05/31/21
Yucheng Tu  UC San Diego
Eigenvalue Estimates, Minimal Hypersurfaces and Isoperimetric Inequalities.
Jun

06/01/21
Osama Khalil, Postdoctoral Scholar  University of Utah
On the MozesShah phenomenon for horocycle flows on moduli spaces
AbstractThe MozesShah phenomenon on homogeneous spaces of Lie groups asserts that the space of ergodic measures under the action by subgroups generated by unipotents is closed. A key input to their work is Ratner's fundamental rigidity theorems. Beyond its intrinsic interest, this result has many applications to counting problems in number theory. The problem of counting saddle connections on flat surfaces has motivated the search for analogous phenomena for horocycle flows on moduli spaces of flat structures. In this setting, Eskin, Mirzakhani, and Mohammadi showed that this property is enjoyed by the space of ergodic measures under the action of (the full upper triangular subgroup of) $\mathrm{SL}(2,\mathbb{R})$. We will discuss joint work with Jon Chaika and John Smillie showing that this phenomenon fails to hold for the horocycle flow on the stratum of genus two flat surfaces with one cone point. As a corollary, we show that a dense set of horocycle flow orbits are not generic for any measure; in contrast with Ratner's genericity theorem.

06/01/21
Di Fang  UC Berkeley
Timedependent unbounded Hamiltonian simulation with vector norm scaling
AbstractHamiltonian simulation is a basic task in quantum computation. The accuracy of such simulation is usually measured by the error of the unitary evolution operator in the operator norm, which in turn depends on certain norm of the Hamiltonian. For unbounded operators, after suitable discretization, the norm of the Hamiltonian can be very large, which significantly increases the simulation cost. However, the operator norm measures the worstcase error of the quantum simulation, while practical simulation concerns the error with respect to a given initial vector at hand. We demonstrate that under suitable assumptions of the Hamiltonian and the initial vector, if the error is measured in terms of the vector norm, the computational cost may not increase at all as the norm of the Hamiltonian increases using Trotter type methods. In this sense, our result outperforms all previous error bounds in the quantum simulation literature. Our result extends that of [Jahnke, Lubich, BIT Numer. Math. 2000] to the timedependent setting. We also clarify the existence and the importance of commutator scalings of Trotter and generalized Trotter methods for timedependent Hamiltonian simulations.

06/01/21
Sayan Das  University of California Riverside
On Connes' rigidity conjecture
AbstractA bold conjecture due to Connes (1980) predicts that the group von Neumann algebra of an i.c.c. property (T) group completely remembers the group. The strong form of Connes' conjecture, due to Popa (2005) predicts that the factors arising from property (T) groups have trivial fundamental group. In this talk I shall discuss recent progress towards these conjectures, and present the first examples of property (T) factors with trivial fundamental group. This talk is based on a joint work with Ionut Chifan, Cyril Houdayer, and Krishnendu Khan.

06/02/21
Lutz Warnke  Georgia Tech
Random Graphs: Phase Transition and Beyond
AbstractRandom graphs play a central role across several branches of mathematics and applied sciences.
The phase transition of random graphs, where the component structure suddenly changes from `many small' components to `one giant' component, is an intriguing phenomenon with rich connections to percolation theory and mathematical physics.
In computer science and extremal combinatorics, random graphs also provide strong probabilistic guarantees for hardtoanswer deterministic questions, such as the construction of interesting combinatorial objects.
In this talk we illustrate these two complementary aspects of random graphs, highlighting that their analysis often brings together tools and techniques from different areas (including combinatorics, probability and differential equations).
\\
\\
We will first focus on Achlioptas processes, which are variants of classical ErdosRenyi random graphs that are difficult to analyze. Settling a number of conjectures and open problems, we show that the phase transition of socalled `boundedsize' Achlioptas processes have the same key features as the Erd\"o{}sR\'{e}nyi reference model (which in the language of mathematical physics means that they are in the same `universality class'). 
06/02/21
Cedric L. M. Josz  Columbia University
Sparse polynomial interpolation: sparse recovery, super resolution, or Prony?
AbstractWe show that the sparse polynomial interpolation problem reduces to a discrete superresolution problem on the ndimensional torus. Therefore, the semidefinite programming approach initiated by CandÃ¨s and FernandezGranda (Commun. Pure Appl. Math. 67(6) 906956, 2014) in the univariate case can be applied. We extend their result to the multivariate case, i.e., we show that exact recovery is guaranteed provided that a geometric spacing condition on the supports holds and evaluations are sufficiently many (but not many). It also turns out that the sparse recovery LPformulation of â„“1norm minimization is also guaranteed to provide exact recovery provided that the evaluations are made in a certain manner and even though the restricted isometry property for exact recovery is not satisfied. (A naive sparse recovery LP approach does not offer such a guarantee.) Finally, we also describe the algebraic Prony method for sparse interpolation, which also recovers the exact decomposition but from less point evaluations and with no geometric spacing condition. We provide two sets of numerical experiments, one in which the superresolution technique and Prony's method seem to cope equally well with noise, and another in which the superresolution technique seems to cope with noise better than Prony's method, at the cost of an extra computational burden (i.e., a semidefinite optimization).

06/03/21
Gerlind PlonkaHoch  University of G\"{o}ttingen"
Recovery of sparse signals from their Fourier coefficients
AbstractIn this talk, we study a new recovery procedure for nonharmonic signals, or more generally for extended exponential sums y(t), which are determined by a finite number of parameters. For the reconstruction we employ a finite set of classical Fourier coefficients of y(t). Our new recovery method is based on the observation that the Fourier coefficients of y(t) possess a special rational structure. We apply the recently proposed AAA algorithm by Nakatsukasa et al. (2018) to recover this rational structure in a stable way. If a sufficiently large set of Fourier coefficients of y(t) is available, then our recovery method automatically detects the correct number of terms M of the exponential sums y(t) and reconstructs all unknown parameters of the signal model. Our method provides a new stable alternative to the known numerical approaches for the recovery of exponential sums that are based on Prony's method.
\\
\\
These results have been obtained jointly with Markus Petz and Nadiia Derevianko. 
06/03/21
Duncan Dauvergne  Princeton University
The directed landscape
AbstractThe directed landscape is a random `directed metric' on the spacetime plane that arises as the scaling limit of integrable models of last passage percolation. It is expected to be the universal scaling limit for all models in the KPZ universality class for random growth. In this talk, I will describe its construction in terms of the Airy line ensemble via an isometric property of the RobinsonSchenstedKnuth correspondence, and discuss some surprising Brownian structures that arise from this construction.
\\
\\
Based on joint work with M. Nica, J. Ortmann, B. Virag, and L. Zhang. 
06/03/21
Kelly Isham  University of California Irvine
Asymptotic growth of orders in a fixed number field via subrings in $\mathbb{Z}^n$
AbstractLet $K$ be a number field of degree $n$ and $\mathcal{O}_K$ be
its ring of integers. An order in $\mathcal{O}_K$ is a finite index
subring that contains the identity. A major open question in arithmetic
statistics asks for the asymptotic growth of orders in $K$. In this
talk, we will give the best known lower bound for this asymptotic
growth. The main strategy is to relate orders in $\mathcal{O}_K$ to
subrings in $\mathbb{Z}^n$ via zeta functions. Along the way, we will
give lower bounds for the asymptotic growth of subrings in
$\mathbb{Z}^n$ and for the number of index $p^e$ subrings in
$\mathbb{Z}^n$. We will also discuss analytic properties of these zeta
functions. 
06/03/21
Pooja Agarwal  UC San Diego
InfiniteDimensional Scaling Limits of ManyServer Stochastic Networks
AbstractStochastic networks arise in a variety of real world applications including telecommunications, service systems such as call centers, computer networks, health care services and biological systems. Leaving aside some very simple examples, it is usually infeasible to perform an exact analysis of these networks. A useful alternative is instead to identify a suitable approximation that provides insight into performance and can be shown to be accurate in a relevant asymptotic regime. This talk looks at two different types of scaling limits of two classes of manyserver stochastic networks. In both cases, the scaling limits are infinitedimensional and require the development of new techniques for their analysis.

06/03/21
Tianyi Yu  UC San Diego
Harmonic bases for generalized coinvariant algebras
AbstractS. Griffin introduced a quotient $R_{n,\lambda}$ of the polynomial ring $\mathbb{Q}$ with $\mathbb{Q}$. It simultaneously generalizes the Delta Conjecture coinvariant rings of HaglundRhoadesShimozono and the cohomology rings of Springer fibers studied by Tanisaki and GarsiaProcesi. We describe the space $V_{n,\lambda}$ of harmonics attached to $R_{n,\lambda}$ and produce a harmonic basis of $R_{n,\lambda}$ indexed by certain ordered set partitions $\mathcal{OP}_{n,\lambda}$. Our description of $V_{n,\lambda}$ involves injective tableaux and Vandermonde determinants. Combinatorics of our harmonic basis is governed by a new extension of the Lemher code.
\\
\\
This is a joint work with Brendon Rhoades and Zehong Zhao. 
06/03/21
Ruth Williams  UC San Diego
Stochastic Networks: Bottlenecks, Entrainment and Reflection
AbstractStochastic models of complex networks with limited resources arise in a wide variety of applications in science and engineering, e.g., in manufacturing, transportation, telecommunications, computer systems, customer service facilities, and systems biology. Bottlenecks in such networks cause congestion, leading to queueing and delay. Sharing of resources can lead to entrainment effects. Understanding the dynamic behavior of such modern stochastic networks present challenging mathematical problems.
\\
\\
This talk will describe some developments in this area. A key feature will be dimension reduction, resulting from entrainment due to resource sharing. An example of bandwidth sharing in a data network will be featured. 
06/03/21
Srivatsa Srinivas  UC San Diego
What is the opposite of a vector space?
AbstractI will try to convince you that there is an opposite of a vector space! This talk will involve additive combinatorics and fourier analysis on finite, not infinite groups...

06/04/21
Jacob Keller  UC San Diego
K Stability and Moduli Spaces of Vector Bundles
AbstractRecently there has been great progress in constructing moduli spaces of Kstable Fano Varieties, and there are many questions to ask about the geometry of these moduli spaces. The question I am considering is whether they are uniruled. My goal is to show that there are components of these moduli spaces that are general type, answering that question in the negative. I wish to do this by studying moduli spaces of vector bundles on curves, showing they are Kstable, and studying the components of the moduli space of Kstable Fano varieties that parametrize them.

06/04/21
Wei Yin  UC San Diego
Dirichlet's Theorem on Arithmetic Progression and Chebotarev's Density Theorem
AbstractThis is a piece of beautiful math involving basic theory of algebraic number fields and Lfunctions. Aimed for those who did not take the 204 Series.

06/04/21
Mingjie Chen  UC San Diego
IsogenyBased Cryptography
AbstractIsogenybased cryptography is becoming an increasingly wellestablished subject within the postquantum cryptography landscape. In this talk, we introduce supersingular isogeny graphs and study the hard problems in them. We also introduce some modernday isogenybased cryptosystems.

06/04/21
Greg Patchell  UC San Diego
A Reluctant Groupie
AbstractAs an analyst, I used to think I didn't like groups. And when studying generalized Bernoulli actions, I came across an example of a horrible combinatorial argument to prove a fact about groups. Luckily, there is another way: functional analysis, which I'd like to share with you. Now I realize I just don't like finite groups, which usually aren't amenable to analytic methods.

06/10/21
Alisa Knizel  University of Chicago
Stationary measure for the open KPZ equation
AbstractThe KardarParisiZhang (KPZ) equation is the stochastic partial differential equation that models interface growth. In the talk I will present the construction of a stationary measure for the KPZ equation on a bounded interval with general inhomogeneous Neumann boundary conditions. Along the way, we will encounter classical orthogonal polynomials, the asymmetric simple exclusion process, and precise asymptotics of qGamma functions.
\\
\\
This construction is a joint work with Ivan Corwin. 
06/10/21
Piotr Indyk  MIT
LearningBased Sampling and Streaming
AbstractClassical algorithms typically provide "one size fits all" performance, and do not leverage properties or patterns in their inputs. A recent line of work aims to address this issue by developing algorithms that use machine learning predictions to improve their performance. In this talk I will present two examples of this type, in the context of streaming and sampling algorithms. In particular, I will show how to use machine learning predictions to improve the performance of (a) lowmemory streaming algorithms for frequency estimation (ICLRâ€™19), and (b) sampling algorithms for estimating the support size of a distribution (ICLRâ€™21). Both algorithms use an MLbased predictor that, given a data item, estimates the number of times the item occurs in the input data set.
\\
\\
The talk will cover material from papers coauthored with T Eden, CY Hsu, D Katabi, S Narayanan, R Rubinfeld, S Silwal, T Wagner and A Vakilian. 
06/17/21
Wenjing Liao  Georgia Tech
Regression and doubly robust offpolicy learning on lowdimensional manifolds by neural networks
AbstractMany data in realworld applications are in a highdimensional space but exhibit lowdimensional structures. In mathematics, these data can be modeled as random samples on a lowdimensional manifold. Our goal is to estimate a target function or learn an optimal policy using neural networks. This talk is based on an efficient approximation theory of deep ReLU networks for functions supported on a lowdimensional manifold. We further establish the sample complexity for regression and offpolicy learning with finite samples of data. When data are sampled on a lowdimensional manifold, the sample complexity crucially depends on the intrinsic dimension of the manifold instead of the ambient dimension of the data. These results demonstrate that deep neural networks are adaptive to lowdimensional geometric structures of data sets.
\\
\\
This is a joint work with Minshuo Chen, Haoming Jiang, Liu Hao, Tuo Zhao at Georgia Institute of Technology.
Jul

07/15/21
Qi (Rose) Yu  UC San Diego
Equivariant Neural Networks for Learning Spatiotemporal Dynamics
AbstractApplications such as climate science and transportation require learning complex dynamics from largescale spatiotemporal data. Existing machine learning frameworks are still insufficient to learn spatiotemporal dynamics as they often fail to exploit the underlying physics principles. Representation theory can be used to describe and exploit the symmetry of the dynamical system. We will show how to design neural networks that are equivariant to various symmetries for learning spatiotemporal dynamics. Our methods demonstrate significant improvement in prediction accuracy, generalization, and sample efficiency in forecasting turbulent flows and predicting realworld trajectories. This is joint work with Robin Walters, Rui Wang, and Jinxi Li.

07/22/21
Yaniv Plan  University of British Columbia
A family of measurement matrices for generalized compressed sensing
AbstractWe consider the problem of recovering a structured signal x that lies close to a subset of interest T in $R^n$, from its random noisy linear measurements y = B A x + w, i.e., a generalization of the classic compressed sensing problem. Above, B is a fixed matrix and A has independent subgaussian rows. By varying B, and the subgaussian distribution of A, this gives a family of measurement matrices which may have heavy tails, dependent rows and columns, and singular values with large dynamic range. Typically, generalized compressed sensing assumes a random measurement matrix with nearly uniform singular values (with high probability), and asks: How many measurements are needed to recover x? In our setting, this question becomes: What properties of B allow good recovery? We show that the â€œeffective rank'' of B may be used as a surrogate for the number of measurements, and if this exceeds the squared Gaussian complexity of TT then accurate recovery is guaranteed. We also derive the optimal dependence on the subgaussian norm of the rows of A, to our knowledge this was not known previously even in the case when B is the identity. We allow model mismatch and inexact optimization with the aim of creating an easily accessible theory that specializes well to the case when T is the range of an expansive neural net.
Aug

08/03/21
Denise Rava  UC San Diego
Survival Analysis and Causal Inference: from Marginal Structural Cox to Additive Hazards Model and beyond.

08/03/21
Jacqueline Warren  UC San Diego
Horospherical flows in infinite volume rank one homogeneous spaces: effective equidistribution and applications

08/12/21
Andrea Bertozzi  UCLA
The challenges of modeling pandemic spread with early time data, finite size populations, and opinion dynamics
AbstractThe coronavirus disease 2019 (COVID19) pandemic placed epidemic modeling at the forefront of worldwide public policy making. Nonetheless, modeling and forecasting the spread of COVID19 remain a challenge. This talk begins with a review of the historical use of epidemic models and addresses the challenges of choosing a model in the early stages of a worldwide pandemic. The spread of COVID19 has illustrated the heterogeneity of disease spread at different population levels. With finite size populations, chance variations may lead to significant departures from the mean. In reallife applications, finite size effects arise from the variance of individual realizations of an epidemic course about its fluid limit. I will illustrate how to model this variance with a martingale formulation consisting of a deterministic and a stochastic component, along with estimates for the size of the variance compared to real world data and simulations. Another cause of heterogeneities is the differing attitudes at the population level for control measures such as maskwearing and physical distancing. Often, individuals form opinions about their behaviors from social network opinions. I will show some results from a twolayer multiplex network for the coupled spread of a disease and conflicting opinions. We model each process as a contagion. We develop approximations of meanfield type by generalizing monolayer pair approximations to multilayer networks; these approximations agree well with Monte Carlo simulations for a broad range of parameters and several network structures. We find that lengthening the duration that individuals hold an opinion may help suppress disease transmission, and we demonstrate that increasing the crosslayer correlations or intralayer correlations of node degrees may lead to fewer individuals becoming infected with the disease.

08/13/21
Zhichao Wang  Department of Mathematics, UCSD
Structured and nonlinear random matrix theory
AbstractIn this talk, we will introduce first introduce Toeplitz structured random matrices and review some asymptotic results of these matrices. We will briefly show the limiting Toeplitz law and recent result of central limit theorem for linear statistics of a specific strucetured Toeplitz matrix based on moment methods. Some challenges, future directions and applications of random Toeplitz matrices will also be mentioned in this talk. Secondly, we will introduce the nonlinear random matrices in random neural networks. In the linear proportional regime, the limiting eigenvalue distributions of conjugate matrices and empirical neural tangent kernels at initial have been studied via Stieltjes transform, which will help us better understand the deep neural networks. We will finally present a recent result of nonlinear random matrix theory beyond linear regime, where a deformed a Wigner law will appear.
Sep

09/09/21
Anna Ma  UC Irvine
The Kaczmarz Algorithm: Greed, Randomness, and Tensors
AbstractThe Kaczmarz algorithm is an iterative method for solving linear systems of equations of the form Ax=y. Owing to its low memory footprint, the Kaczmarz algorithm has gained popularity for its practicality in applications to largescale data, acting only on single rows of A at a time. In this talk, we discuss selecting rows of A randomly (Randomized Kaczmarz), selecting rows in a greedy fashion (Motzkin's Method), and selecting rows in a partially greedy fashion (Sampling KaczmarzMotzkin algorithm). Despite their variable computational costs, these algorithms have been proven to have the same theoretical upper bound on the convergence rate. Here we present an improvement upon previous known convergence bounds of the Sampling KaczmarzMotzkin algorithm, capturing the benefit of partially greedy selection schemes. Time permitting, we also will discuss an extension of the Kaczmarz algorithm to the setting where data takes on the form of a tensor and make connections between the new Tensor Kaczmarz algorithm and previously established algorithms. \\ \\ This presentation contains joint work with Jamie Haddock and Denali Molitor.

09/23/21
Joel Tropp  Caltech
Scalable semidefinite programming
AbstractSemidefinite programming (SDP) is a powerful framework from convex optimization that has striking potential for data science applications. This talk describes a provably correct randomized algorithm for solving large, weakly constrained SDP problems by economizing on the storage and arithmetic costs. Numerical evidence shows that the method is effective for a range of applications, including relaxations of MaxCut, abstract phase retrieval, and quadratic assignment problems. Running on a laptop equivalent, the algorithm can handle SDP instances where the matrix variable has over $10^{14}$ entries. This talk will highlight the ideas behind the algorithm in a streamlined setting. The insights include a careful problem formulation, design of a bespoke optimization method, and use of randomized matrix computations. Joint work with Alp Yurtsever, Olivier Fercoq, Madeleine Udell, and Volkan Cevher. Based on arXiv 1912.02949 (Scalable SDP, SIMODS 2021) and other papers (SketchyCGM in AISTATS 2017, Nystr\"{o}m sketch in NeurIPS 2017)."

09/23/21
Felipe GarciaRamos  Universidad Aut\'{o}noma de San Luis Potos\'{\i}
Local entropy theory and descriptive complexity
AbstractWe will give an introduction to local entropy theory and we will trace the descriptive complexity of different families of topological dynamical systems with completely positive entropy (CPE) and uniform positive entropy (UPE). Joint work with Udayan B. Darji.

09/23/21

09/28/21
Nick Georgakopoulos  University of Chicago
The RO(G) graded cohomology of Gequivariant classifying spaces
AbstractThe cohomology of classifying spaces is an important classical topic in algebraic topology. However, much less is known in the equivariant setting, where one wants to know the RO(G)graded cohomology of classifying Gspaces. The problem is that RO(G)graded cohomology is notoriously difficult to compute even when G is cyclic.In this talk, I will explain my computations in the case of cyclic 2groups G while keeping technical details to a minimum. The main goal is to understand rational equivariant characteristic classes, but I will also discuss some mod 2 computations and their relevance to the equivariant dual Steenrod algebra.

09/28/21

09/28/21
Isaac Goldbring  University of California, Irvine
The Connes Embedding Problem, MIP*=RE, and the Completeness Theorem
AbstractThe Connes Embedding Problem (CEP) is arguably one of the most famous open problems in operator algebras. Roughly, it asks if every tracial von Neumann algebra can be approximated by matrix algebras. Earlier this year, a group of computer scientists proved a landmark result in complexity theory called MIP*=RE, and, as a corollary, gave a negative solution to the CEP. However, the derivation of the negative solution of the CEP from MIP*=RE involves several very complicated detours through C*algebra theory and quantum information theory. In this talk, I will present joint work with Bradd Hart where we show how some relatively simple modeltheoretic arguments can yield a direct proof of the failure of the CEP from MIP*=RE while simultaneously yielding a stronger, GÃ¶delianstyle refutation of CEP as well as the existence of â€œmanyâ€ counterexamples to CEP. No prior background in any of these areas will be assumed.

09/28/21
Sam Spiro  UCSD
The Count of Monte Carlo
AbstractMany of the questions asked during the birth of probability (e.g. what's the probability of getting a certain hand in poker?) are equivalent to basic counting problems, and since then there have been numerous applications of combinatorics to probability (e.g. moment method proofs for the semicircle law). In this talk, probability strikes back with a vengeance by solving some (nontrivial) counting problems.

09/28/21
Ruth Luo  UCSD
A Dirac Theorem for hamiltonian hypergraphs
AbstractDirac proved that every $n$vertex graph with minimum degree at least $n/2$ contains a hamiltonian cycle. We prove an analogue for hypergraphs: we give exact bounds for the minimum degree of a uniform hypergraph that implies the existence of hamiltonian Berge cycles. \\ \\ This is joint work with Alexandr Kostochka and Grace McCourt.

09/30/21
Yariv Aizenbud  Yale University
NonParametric Estimation of Manifolds from Noisy Data
AbstractA common task in many datadriven applications is to find a low dimensional manifold that describes the data accurately. Estimating a manifold from noisy samples has proven to be a challenging task. Indeed, even after decades of research, there is no (computationally tractable) algorithm that accurately estimates a manifold from noisy samples with a constant level of noise. In this talk, we will present a method that estimates a manifold and its tangent in the ambient space. Moreover, we establish rigorous convergence rates, which are essentially as good as existing convergence rates for function estimation. This is a joint work with Barak Sober.

09/30/21
Forte Shinko  California Institute of Technology
Realizations of equivalence relations and subshifts
AbstractEvery continuous action of a countable group on a Polish space induces a Borel equivalence relation. We are interested in the problem of realizing (i.e. finding a Borel isomorphic copy of) these equivalence relations as continuous actions on compact spaces. We provide a number of positive results for variants of this problem, and we investigate the connection to subshifts.

09/30/21
Mauricio del Razo Sarmina  Univ. of Amsterdam, Netherlands
Multiscale Molecular Kinetics by Coupling Markov State Models and Reaction Diffusion Dynamics
AbstractA novel approach to simulate simple proteinÂligand systems at large timeÂ and lengthÂscales is to couple Markov state models (MSMs) of molecular kinetics with particleÂbased reactionÂdiffusion (PBRD) simulations; this approach is named MSM/RD. Current formulations of MSM/RD lack an underlying mathematical framework to derive coupling schemes; they are limited to proteinÂligand systems, where the ligand orientation and conformation switching are not taken into account; and they lack multiÂparticle extensions. In this work, we develop a general MSM/RD framework by coarseÂgraining molecular dynamics into hybrid switching diffusion processes, a class of stochastic processes that integrate continuous dynamics and discrete events into the same process. With this MSM/RD framework, it is possiÂble to derive MSM/RD coupling schemes as discretizations of the underlying equations. It also allows conformation switching and the inclusion of all the rotational degrees of freedom. Given enough data to parametrize the model, it is capable of modeling proteinÂprotein interactions over large timeÂ and lengthÂscales, and it can be extended to hanÂdle multiple molecules. We derive the MSM/RD framework, and we implement and verify it for two proteinÂprotein benchmark systems and one multiparticle implementation to model the formation of pentameric ring molecules.
Oct

10/05/21
Ang Li  University of Kentucky
A comparison between $C_2$equivariant and classical squaring operations
AbstractFor any $C_2$equivariant spectrum, we can functorially assign two nonequivariant spectra  the underlying spectrum and the geometric fixed point spectrum. They both induce maps from $RO(C_2)$graded cohomology to classical cohomology. In this talk, I will compare the $RO(C_2)$graded squaring operations with the classical squaring operations along the induced maps. This is joint work with Prasit Bhattacharya and Bertrand Guillou.

10/05/21
Yunan Yang  Simons Institute for the Theory of Computing
The interchange between Lagrangian and Eulerian perspectives for solving Kinetic inverse problems
AbstractI will talk about two recent projects on solving inverse problems for kinetic models, where a change of perspective between Lagrangian and Eulerian is highly beneficial. In the first project, we are interested in recovering the initial temperature of the nonlinear Boltzmann equation given macroscopic quantities observed at a later time. With the problem formulated as constrained optimization, our proposed adjoint DSMC method, together with the wellknown (forward) DSMC method, makes it possible to evaluate Boltzmannconstrained gradient within seconds, independent of the size of the parameter. In the second project, we are interested in calibrating the parameter in the chaotic dynamic system. Transforming the longtime trajectories to an invariant measure significantly improves the inverse problem's illposedness. It also turns the original ODE model into a PDE model (continuity equation or FokkerPlanck equation), allowing efficient gradient calculation for the resulting PDEconstrained optimization problem.

10/05/21
Srivatsav Kunnawalkam Elayavalli  Vanderbilt University
Strong 1boundedness and Property (T)
AbstractAbstract: Strong 1boundedness is a notion introduced by Kenley Jung which captures (among Connesembeddable von Neumann algebras) the property of having "a small amount of matrix approximations". Some examples are diffuse hyperfinite von Neumann algebras, vNa's with Property Gamma, vNa's that are non prime, vNa's that have a diffuse hyperfinite regular subalgebra and so on. This is a von Neumann algebra invariant, and was used to prove in a unified approach remarkable rigidity theorems. One such is the following: A non trivial free product of von Neumann algebras can never be generated by two strongly 1bounded von Neumann subalgebras with diffuse intersection. This result cannot be recovered by any other methods for even hyperfinite subalgebras. In this talk, I will present joint work with David Jekel and Ben Hayes, settling an open question from 2005, whether Property (T) II$_1$ factors are strongly 1bounded, in the affirmative. Time permitting, I will discuss some insights and future directions.

10/05/21
Jason O'Neill  University of California, San Diego
Towards supersaturation for oddtown and eventown
AbstractIn an $n$resident town, Oddtown, all of their clubs must satisfy the following properties: all clubs must have an odd number of members and amongst any two distinct clubs, there must be an even number of residents in common. The classical oddtown theorem states that any such town can have at most $n$ clubs. In this talk, we explore how the residents can have $n+1$ clubs of odd size and minimize the chance of the town catching them. That is, we'd like to minimize the number of pairs of clubs with an odd number of members in common. We will also explore a similar problem with Eventown.

10/05/21
Thomas Grubb  UCSD
White box image classification via topological data analysis
AbstractThe letters D and O are topologically indistinguishable (both are circles). However, after superimposing each symbol with their reflections across several axes, one \emph{can} distinguish between them. The curvature in the O results in a different evolution in first homology when compared to the angled D. In this talk we will expand on this idea by explaining a white box classification algorithm which classifies an image as one of the 26 letters in the (capitalized) English alphabet. The driving force is the theory of persistent homology, as implemented in the Ripser package. This technique is less powerful than traditional techniques of machine learning (such as a neural net), but it is much more explainable.

10/05/21
Sam Spiro  UCSD
Maximal Independent Sets in Cliquefree Graphs
AbstractAn independent set $I$ of a graph $G$ is said to be a maximal independent set (MIS) if it is maximal with respect to set inclusion. Nielsen proved that the maximum number of MIS's of size $k$ in an $n$vertex graph is asymptotic to $(n/k)^k$, with the extremal construction being a disjoint union of $k$ cliques with sizes as close to $n/k$ as possible. In this talk we study how many MIS's of size $k$ an $n$vertex graph $G$ can have if $G$ does not contain a clique $K_t$. We prove for all fixed $k$ and $t$ that there exist such graphs with $n^{\lfloor\frac{(t2)k}{t1}\rflooro(1)}$ MIS's of size $k$ by utilizing recent work of Gowers and B. Janzer on a generalization of the RuzsaSzemer\'edi problem. We prove that this bound is essentially best possible for trianglefree graphs when $k\le 4$. \medskip This is joint work with Xiaoyu He and Jiaxi Nie.

10/06/21
Connor Mooney  UC Irvine
Solutions to the MongeAmpere equation with polyhedral and Yshaped singularities
AbstractThe MongeAmpere equation det$(D^2u) = 1$ arises in prescribed curvature problems and in optimal transport. An interesting feature of the equation is that it admits singular solutions. We will discuss new examples of convex functions on $R^n$ that solve the MongeAmpere equation away from finitely many points, but contain polyhedral and Yshaped singular structures. Along the way we will discuss geometric motivations for constructing such examples, as well as their connection to a certain obstacle problem.

10/06/21
Rusiru Gambheera  UCSD
An Unconditional Equivariant Main Conjecture in Iwasawa Theory

10/06/21
Julia Lindberg  University of WisconsinMadison
The Method of Moments for Gaussian Mixture Models
AbstractThe method of moments is a statistical method for density estimation that equates sample moments to moment equations for a given family of densities. When the underlying distribution is assumed to be a convex combination of Gaussian densities, the resulting moment equations are polynomial in the density parameters. We examine the asymptotic behavior of the variety stemming from these equations as the number of components and the dimension of each component increases. This is joint work with Jose Israel Rodriguez and Carlos Amendola.

10/07/21
Brett Kolesnik  UC San Diego
Hpercolation

10/07/21
Riley Thornton  UCLA
Cayley Diagrams and Factors of IID Processes
AbstractA Cayley diagram is a labeling of a graph $G$ that encodes an action of a group which induces $G$. For instance, a $d$edge coloring of a $d$regular tree is a Cayley diagram for the group $(\mathbb{Z}/2\mathbb{Z})^{*d}$. In this talk, we will investigate when a Cayley graph $G=(\Gamma, E)$ admits an $\operatorname{Aut}(G)$f.i.i.d. Cayley diagram and show that $\Gamma$f.i.i.d. solutions to local labeling problems for such graphs lift to $\operatorname{Aut}(G)$f.i.i.d. solutions.

10/07/21
Kiran Kedlaya  UCSD
Orders of abelian varieties over $\mathbb{F}_2$
AbstractWe describe several recent results on orders of abelian varieties over $\mathbb{F}_2$: every positive integer occurs as the order of an ordinary abelian variety over $\mathbb{F}_2$ (joint with E. Howe); every positive integer occurs infinitely often as the order of a simple abelian variety over $\mathbb{F}_2$; the geometric decomposition of the simple abelian varieties over $\mathbb{F}_2$ can be described explicitly (joint with T. D'NellyWarady); and the relative class number one problem for function fields is reduced to a finite computation (work in progress). All of these results rely on the relationship between isogeny classes of abelian varieties over finite fields and Weil polynomials given by the work of Weil and HondaTate. With these results in hand, most of the work is to construct algebraic integers satisfying suitable archimedean constraints.

10/08/21
Johannes Schmitt  University of Z\"urich"
Strata of kdifferentials and double ramification cycles
AbstractThe moduli space of stable curves parameterizes tuples $(C,p_1,...,p_n)$ of a compact, complex curve $C$ together with distinct marked points $p_1,\dots, p_n$. Inside this moduli space, there are natural subsets, called the strata of $k$differentials, defined by the condition that there exists a meromorphic $k$differential on $C$ with zeros and poles of some fixed multiplicities at the points $p_i$. I will discuss basic properties of these strata and explain a conjecture relating their fundamental class to the socalled double ramification cycles on the moduli space. I explain the idea of the proof of this conjecture and some ongoing work with Costantini and Sauvaget on how to use this relation to compute intersection numbers of the strata with $\psi$classes on the moduli of curves.

10/12/21
Yunze Lu  University of Michigan
Calculations in nonabelian equivariant cohomology
AbstractCalculating the coefficients of equivariant generalized cohomology theories has been a fundamental question for equivariant homotopy theory. In this talk, I will talk about some calculations when the group is nonabelian. Examples include $RO(G)$graded EilenbergMacLane cohomology of a point with constant coefficient when $G$ is a dihedral group of order $2p$ or the quaternion group $Q_8$, and coefficient ring of $\Sigma_3$equivariant complex cobordism. I will discuss techniques in such computations: isotropy separation, cellular structures and dualities. This is joint work with Po Hu and Igor Kriz.

10/12/21
Adrian Ioana  University of California, San Diego
Almost commuting matrices and stability for product groups.
AbstractI will present a recent result showing that the direct product group $\Gamma = \mathbb F_2 \times \mathbb F_2$ is not HilbertSchmidt stable. Specifically, $\Gamma$ admits a sequence of asymptotic homomorphisms (with respect to the normalized HilbertSchmidt norm) which are not perturbations of genuine homomorphisms. As we will explain, while this result concerns unitary matrices, its proof relies on techniques and ideas from the theory of von Neumann algebras. We will also explain how this result can be used to settle in the negative a natural version of an old question of Rosenthal concerning almost commuting matrices. More precisely, we derive the existence of contraction matrices $A,B$ such that $A$ almost commutes with $B$ and $B^*$ (in the normalized HilbertSchmidt norm), but there are no matrices $Aâ€™,Bâ€™$ close to $A,B$ such that $Aâ€™$ commutes with $Bâ€™$ and $Bâ€™^*$.

10/12/21
Johannes Brust  UCSD
Scalable Computational Methods with Recent Applications
AbstractFor computations with many variables in optimization or solving large systems in numerical linear algebra, developing efficient methods is highly desirable. This talk introduces an approach for largescale optimization with sparse linear equality constraints that exploits computationally efficient orthogonal projections. For approximately solving large linear systems, (randomized) sketching methods are becoming increasingly popular. By recursively augmenting a deterministic sketching matrix, we develop a method with a finite termination property that compares favorably to randomized methods. Moreover, we describe the construction of logical linear systems that can be used in e.g., COVID19 pooling tests, and a nonlinear leastsquares method that addresses large data sizes in machine learning.

10/12/21
Yuming Paul Zhang  UCSD
Homogenization for combustion in random media
AbstractWe study long time dynamics of combustive processes in random media, modeled by reactiondiffusion equations with random ignition reactions. One expects that under reasonable hypotheses on the randomness, large scale dynamics of solutions to these equations is almost surely governed by a homogeneous HamiltonJacobi equation. While this was previously shown in one dimension as well as for radially symmetric reactions in several dimensions, we prove this phenomenon in the general nonisotropic multidimensional setting. We also show that the rate of convergence of solutions to the HamiltonJacobi dynamics is at least algebraic in the relevant spacetime scales when the initial data is close to an indicator function of a convex set. This talk is based on joint work with Andrej Zlato\v{s}.

10/12/21
Sam Spiro  University of California, San Diego
Slow Fibonacci Walks
AbstractWe define a Fibonacci walk to be any sequence of positive integers satisfying the recurrence $w_{k+2}=w_{k+2}=w_{k+1}+w_k$, and we say that a sequence is an $n$Fibonacci walk if $w_k=n$ for some $k$. Note that every $n$ has a number of (boring) $n$Fibonacci walks, e.g. the sequence starting $n,n,2n,\ldots$. To make things interesting, we consider $n$Fibonacci walks which have $w_k=n$ with $k$ as large as possible, and we call this an $n$slow Fibonacci walk. For example, the two 6slow Fibonacci walks start 2, 2, 4, 6 and 4, 1, 5, 6. In this talk we discuss a number of properties about $n$slow Fibonacci walks, such as the number of slow walks a given $n$ can have, as well as how many $n$ have a given number of walks. We also discuss slow walks that follow more general recurrence relations. This is joint work with Fan Chung and Ron Graham.

10/12/21
Max Johnson  UCSD
Making Something Out of Nothing: Secondary Operations in Algebra and Homotopy
AbstractHave you ever thought to yourself ``Homological Algebra is great, but I wish there were more technicalities and operations to keep track of"? Do you ever worry that the chain complex feels left out after you take cohomology? Does it keep you up at night that elements whose products are 0 get less of a say in the cohomology ring's structure? Has your topologist friend ever ignored nullhomotopic maps, making them feel excluded and unheard? In this talk, Max will be explaining two closely related constructions, in homological algebra and in homotopy, that arise only when other operations give a trivial output. These so called ``secondary operations" play a large role in computational algebraic topology, and can be an indispensable tool for proving results both highly abstract (see Tyler Lawson's ``BP is not $E_\infty$") and very concrete (see Massey's ``Higher Order Linking Numbers").

10/12/21
Tianyi Yu  UCSD
GrothendiecktoLascoux Expansions
AbstractWe establish the conjecture of Reiner and Yong for an explicit combinatorial formula for the expansion of a Grothendieck polynomial into the basis of Lascoux polynomials. This expansion is a subtle refinement of its symmetric function version due to Buch, Kresch, Shimozono, Tamvakis, and Yong, which gives the expansion of stable Grothendieck polynomials indexed by permutations into Grassmannian stable Grothendieck polynomials. Our expansion is the Ktheoretic analogue of that of a Schubert polynomial into Demazure characters, whose symmetric analogue is the expansion of a Stanley symmetric function into Schur functions. This is a joint work with Mark Shimozono.

10/13/21
Yumeng Zhu  UCSD
Convex relaxation methods for phase retrieval problems

10/14/21
Wolfgang K\"{o}nig"  Weierstrass Institute Berlin (WIAS) and TU Berlin
A largedeviations principle for all the components in a sparse inhomogeneous Erd\H{o}sR\'{e}nyi graph

10/14/21
Anton Bernshteyn  Georgia Tech
Equivariant maps to free and almost free subshifts
AbstractLet $\Gamma$ be a countably infinite group. Seward and TuckerDrob proved that every free Borel action of $\Gamma$ on a Polish space $X$ admits a Borel equivariant map $\pi$ to the free part of the Bernoulli shift $k^\Gamma$, for any $k \geq 2$. Our goal in this talk is to generalize this result by putting extra restrictions on the image of $\pi$. For instance, can we ensure that $\pi(x)$ is a proper coloring of the Cayley graph of $\Gamma$ for all $x \in X$? More generally, can we guarantee that the image of $\pi$ is contained in a given subshift of finite type? The main result of this talk is a positive answer to this question in a very broad (and, in some sense, optimal) setting. The main tool used in the proof of our result is a probabilistic technique for constructing continuous functions with desirable properties, namely a continuous version of the Lov\'{a}sz Local Lemma.

10/14/21
Jeff Lagarias  University of Michigan
Complex Equiangular Lines and the Stark Conjectures
AbstractThis talk is expository. It describes the history of an exciting connection made by physicists between an unsolved problem in combinatorial design theory the existence of maximal sets of $d^2$ complex equiangular lines in ${\mathbb C}^d$ rephrased as a problem in quantum information theory, and topics in algebraic number theory involving class fields of real quadratic fields. Work of my former student Gene Kopp recently uncovered a surprising, deep (unproved!) connection with the Stark conjectures. For infinitely many dimensions $d$ he predicts the existence of maximal equiangular sets, constructible by a specific recipe starting from suitable Stark units, in the rank one case. Numerically computing special values at $s=0$ of suitable Lfunctions then permits recovering the units numerically to high precision, then reconstructing them exactly, then testing they satisfy suitable extra algebraic identities to yield a construction of the set of equiangular lines. It has been carried out for $d=5, 11, 17$ and $23$.

10/14/21
Lev Tsimring  The BioCircuit Institute, UCSD
Fate Decision Landscape In SingleCell Aging

10/14/21
Eliza O'Reilly  Caltech
Random Tessellation Features and Forests
AbstractThe Mondrian process in machine learning is a Markov partition process that recursively divides space with random axisaligned cuts. This process is used to build random forests for regression and classification as well as Laplace kernel approximations. The construction allows for efficient online algorithms, but the restriction to axisaligned cuts does not capture dependencies between features. By viewing the Mondrian as a special case of the stable under iterated (STIT) process in stochastic geometry, we resolve open questions about the generalization of cut directions. We utilize the theory of stationary random tessellations to show that STIT processes approximate a large class of stationary kernels and STIT random forests achieve minimax rates for Lipschitz functions (forests and trees) and $C^2$ functions (forests only). This work opens many new questions at the intersection of stochastic geometry and statistical learning theory. Based on joint work with Ngoc Mai Tran.

10/15/21
Nikolas Kuhn  Institut Mittag Leffler
Blowup formulas for virtual sheaftheoretic invariants on projective surfaces
AbstractFor a smooth projective surface X, natural objects of study are its moduli spaces of (semi) stable coherent sheaves. In rank one, their structural invariants are wellunderstood, starting with G\"{o}ttsche's famous formula for the Betti numbers of the Hilbert schemes of points of X in terms of the Betti numbers of X itself. Even for rank two

10/19/21
Araminta Amabel  MIT
Deformation Theory and Supersymmetric Quantum Mechanics
AbstractThere is a deep relationship between deformation theory for symplectic manifolds and quantizing field theories. In this talk, I'll discuss this story for symplectic supermanifolds and supersymmetric mechanics. We will approach these questions using modern descent techniques that work more generally for factorization algebras associated to higherdimensional field theories. Relations to manifold invariants such as the Lgenus will also be discussed. No physics knowledge is required.

10/19/21
Jorge Garza Vargas  University of California, Berkeley
Asymptotics of polynomials via free probability
AbstractSince the seminal work of Voiculescu in the early 90â€™s, the connection between the asymptotic behavior of random matrices and free probability has been extensively studied. More recently, in relation to the solution of the KadisonSinger problem, Marcus, Spielman, and Srivastava discovered a deep connection between certain classical polynomial convolutions and free probability. Soon after, this connection was further understood by Marcus, who introduced the notion of finite free probability.
In this talk I will present recent results on finite free probability with applications to the asymptotic analysis of realrooted polynomials. Our approach is based on a careful combinatorial analysis of the finite free cumulants, and allows us to study the asymptotic dynamics of the root distribution of polynomials after repeated differentiation, as well as the fluctuations of the root distributions of polynomials around their limiting measure. This is joint work with Octavio Arizmendi and Daniel Perales: arXiv:2108.08489.

10/19/21
Mohandas Pillai  UCSD
Global, nonscattering solutions to the quintic, focusing semilinear wave equation on $\mathbb{R}^{1+3}$
AbstractWe consider the quintic, focusing semilinear wave equation on $\mathbb{R}^{1+3}$, in the radially symmetric setting, and construct infinite time blowup, relaxation, and intermediate types of solutions. More precisely, we first define an admissible class of timedependent length scales, which includes a symbol class of functions. Then, we construct solutions which can be decomposed, for all sufficiently large time, into an AubinTalentini (soliton) solution, rescaled by an admissible length scale, plus radiation (which solves the free 3 dimensional wave equation), plus corrections which decay as time approaches infinity. The solutions include infinite time blowup and relaxation with rates including, but not limited to, positive and negative powers of time, with exponents sufficiently small in absolute value. We also obtain solutions whose soliton component has oscillatory length scales, including ones which converge to zero along one sequence of times approaching infinity, but which diverge to infinity along another such sequence of times. The method of proof is similar to a recent wave maps work of the author, which is itself inspired by matched asymptotic expansions.

10/19/21
David Kamensky  UCSD
Beyond FE with FEniCS: Automating isogeometric and immersed methods for numerical PDEs
AbstractThe opensource FEniCS Project (https://fenicsproject.org/) has proven to be a popular and successful finite element (FE) automation tool, applicable to many problem domains involving partial differential equations (PDEs). (CCoM seminar regulars may recall a 2017 talk by L. Ridgway Scott on FEniCS and its implications for pedagogy.) The present talk discusses recent work extending FEniCS to numerical methods other than traditional FE methods. The library tIGAr (https://github.com/davidkamensky/tIGAr) extends FEniCS to isogeometric analysis (IGA), where splinebased geometries from design and graphics replace the meshes of traditional FE analysis. This library retains a similar workflow to traditional FE analysis with FEniCS, while using objectoriented abstractions to separate PDE solution from geometry creation. This design permits analysis of many different PDEs, using a wide variety of existing spline types, and provides an interface to add support for future sp line constructions. This talk surveys several example applications of tIGAr, including divergenceconforming IGA of incompressible flow, KirchhoffLove shell analysis, and nonlocal contact mechanics. Going further beyond standard FE analysis, we consider immersedboundary methods, which present more complicated challenges for automation software. Some initial results on combining FEniCS and tIGAr for immersed fluidstructure interaction will be presented, along with recent work coupling tIGArbased isogeometric shell analysis at intersection curves of separatelyparameterized structural components. Lastly, we discuss the ongoing development of generalpurpose tools for immersed FE analysis.

10/20/21
Zi Yang  UCSB
The MultiObjective Optimization

10/21/21
Pratyush Sarkar  Yale University
Generalization of Selberg's 3/16 theorem for convex cocompact thin subgroups of SO(n, 1)
AbstractSelbergâ€™s 3/16 theorem for congruence covers of the modular surface is a beautiful theorem which has a natural dynamical interpretation as uniform exponential mixing. BourgainGamburdSarnak's breakthrough works initiated many recent developments to generalize Selberg's theorem for infinite volume hyperbolic manifolds. One such result is by OhWinter establishing uniform exponential mixing for convex cocompact hyperbolic surfaces. These are not only interesting in and of itself but can also be used for a wide range of applications including uniform resonance free regions for the resolvent of the Laplacian, affine sieve, and prime geodesic theorems. I will present a further generalization to higher dimensions and some of these immediate consequences.

10/21/21
Thomas Grubb  UCSD
A cutbycurves criterion for overconvergence of $F$isocrystals
AbstractLet $X$ be a smooth, geometrically irreducible scheme over a finite field of characteristic $p > 0$. With respect to rigid cohomology, $p$adic coefficient objects on $X$ come in two types: convergent $F$isocrystals and the subcategory of overconvergent $F$isocrystals. Overconvergent isocrystals are related to $\ell$adic etale objects ($\ell\neq p$) via companions theory, and as such it is desirable to understand when an isocrystal is overconvergent. We show (under a geometric tameness hypothesis) that a convergent $F$isocrystal $E$ is overconvergent if and only if its restriction to all smooth curves on $X$ is. The technique reduces to an algebraic setting where we use skeleton sheaves and crystalline companions to compare $E$ to an isocrystal which is patently overconvergent. Joint with Kiran Kedlaya and James Upton.

10/21/21
Wasiur Khuda Bukhsh  University of Nottingham, UK
Incorporating age and delay into models for biophysical systems
AbstractIn many biological systems, chemical reactions or changes in a physical state are assumed to occur instantaneously. For describing the dynamics of those systems, Markov models that require exponentially distributed interevent times have been used widely. However, some biophysical processes such as gene transcription and translation are known to have a significant gap between the initiation and the completion of the processes, which renders the usual assumption of exponential distribution untenable. In this talk, we consider relaxing this assumption by incorporating agedependent random time delays (distributed according to a given probability distribution) into the system dynamics. We do so by constructing a measurevalued Markov process on a more abstract state space, which allows us to keep track of the 'ages' of molecules participating in a chemical reaction. We study the largevolume limit of such agestructured systems. We show that, when appropriately scaled, the stochastic system can be approximated by a system of partial differential equations (PDEs) in the largevolume limit, as opposed to ordinary differential equations (ODEs) in the classical theory. We show how the limiting PDE system can be used for the purpose of further model reductions and for devising efficient simulation algorithms. To describe the ideas, we will use a simple transcription process as a running example.

10/22/21
Ming Zhang  UCSD
Equivariant Verlinde algebra and quantum Ktheory of the moduli space of vortices
AbstractIn studying complex ChernSimons theory on a Seifert manifold, GukovPei proposed an equivariant Verlinde formula, a oneparameter deformation of the celebrated Verlinde formula. It computes, among many things, the graded dimension of the space of holomorphic sections of (powers of) a natural determinant line bundle over the Hitchin moduli space. GukovPei conjectured that the equivariant Verlinde numbers are equal to the equivariant quantum Kinvariants of a noncompact (Kahler) quotient space studied by HananyTong. In this talk, I will explain the setup of this conjecture and its proof via wallcrossing of moduli spaces of (parabolic) BradlowHiggs triples. It is based on work in progress with Wei Gu and Du Pei.

10/26/21
Prasit Bhattacharya  University of Notre Dame
Equivariant Steenrod Operations
AbstractClassical Steenrod algebra is one of the most fundamental algebraic gadgets in stable homotopy theory. It led to the theory of characteristic classes, which is key to some of the most celebrated applications of homotopy theory to geometry. The Gequivariant Steenrod algebra is not known beyond the group of order 2. In this talk, I will recall a geometric construction of the classical Steenrod algebra and generalize it to construct Gequivariant Steenrod operations. Time permitting, I will discuss potential applications to equivariant geometry.

10/26/21
Jonas Hirsch  University of Leipzig
On bounded solutions of linear elliptic operators with measurable coefficients  De Giorgiâ€™s theorem revisited
AbstractWe consider the classical framework of the famous DeGiorgiNashMoser theorem: $div(A(x)\nabla u)=f$, where $A(x)$ is a symmetric, elliptic matrix field, $f$ is given and $u:U\subset \mathbb{R}^n\to\mathbb{R}$ is the unknown. N. Trudinger was the first one to relax the assumptions on the coefficients matrix $A(x)$. He was able to derive boundedness results if the matrix is barely integrable in the right spaces. In particular he was able to show that if $\lambda(x)\xi^2\leq \xi\cdot A(x)\xi\leq \Lambda(x)\xi^2,\quad \forall x$ and the $\lambda^{1}\in L^p, \Lambda\in L^q$ satisfying $\frac{1}{p}+\frac{1}{q}<\frac{2}{n}$. The integrability condition had been considerably improved by P. Bella and M. Schaffner in the framework of the Moseriteration to $\frac{1}{p}+\frac{1}{q}<\frac{2}{n1}$. A counterexample had been constructed by Franchi, Serapioni, and Serra Cassano under $\frac{1}{p}+\frac{1}{q}>\frac{2}{n}$. The aim of this talk is to revisit De Giorgiâ€™s original approach having in mind the question concerning the optimal integrability assumption on the coefficient field. We will present how this question is surprisingly linked to a question in linear programming with an infinite horizon. This talk will be about my ongoing project with M. Schaffner, hence about work in progress.

10/26/21
Tai Melcher  University of Virginia
Some nice measures in infinitedimensions
AbstractGaussian measures have long been recognized as the appropriate measures to use in infinitedimensional analysis. Their regularity properties have allowed the development of a calculus on these measure spaces that has become an invaluable tool in the analysis of stochastic processes and their applications. Gaussian measures arise naturally in the context of random diffusions, specifically as the end point distribution of Brownian motion, and one may see their regularity as arising from nice properties of the generator of the diffusion. More particularly, in finite dimensions, hypoellipticity of the generator is a standard assumption required for regularity of the associated measure. However, in infinite dimensions it has remained elusive to demonstrate that hypoellipticity is a sufficient condition for regularity. Using techniques first developed by Bruce Driver and Masha Gordina, there has been some recent success in proving regularity for some natural infinitedimensional hypoelliptic models. These techniques rely on establishing uniform bounds on coefficients appearing in certain functional analytic inequalities for semigroups on finitedimensional approximations. We will discuss some of these successful applications, including more recent work studying models satisfying only a weak notion of hypoellipticity. This includes joint works with Fabrice Baudoin, Dan Dobbs, Bruce Driver, Nate Eldredge, and Masha Gordina.

10/26/21
Anders Forsgren  KTH Royal Institute of Technology
QuasiNewton methods for minimizing a quadratic function
AbstractWe discuss quasiNewton methods for minimizing a strictly convex quadratic function that may be subject to errors in the evaluation of the gradients. The methods all give identical behavior in exact arithmetic, generating minimizers of Krylov subspaces of increasing dimensions, thereby having finite termination. In exact arithmetic, the method of conjugate gradients gives identical iterates and has less computational cost. In a framework of small errors, e.g., finite precision arithmetic, the performance of the method of conjugate gradients is expected to deteriorate, whereas a BFGS quasiNewton method is empirically known to behave very well. We discuss the behavior of limitedmemory quasiNewton methods, balancing the good performance of a BFGS method to the low computational cost of the method of conjugate gradients. We also discuss largeerror scenarios, in which the expected behavior is not so clear. In particular, we are interested in the behavior of quasiNewton matrices that differ from the identity by a lowrank matrix, such as a memoryless BFGS method. Our numerical results indicate that for large errors, a memoryless quasiNewton method often outperforms a BFGS method. We also consider a more advanced model for generating search directions, based on solving a chanceconstrained optimization problem. Our results indicate that such a model often gives a slight advantage in final accuracy, although the computational cost is significantly higher. The talk is based on joint work with Tove Odland, David Ek, Gianpiero Canessa and Shen Peng.

10/26/21
Nathaniel ``Tanny" Liberman  UCSD
Every Counterexample In Topology Appearing In The Book ``Counterexamples In Topology'' by Lynn Steen and J. Arthur Seebach, Jr. (And Whether Or Not Each One Is Compact)
AbstractWe present all 143 counterexamples in topology appearing in the book ``Counterexamples In Topology", by Lynn Steen and J. Arthur Seebach, Jr. For each such counterexample, we state whether or not it is compact.

10/26/21
Cameron Cinel  UCSD
Sofic Lie Algebras

10/27/21
Yang Zheng  UCSD
Chordal Graphs, Semidefinite Optimization, and Sumofsquares Matrices
AbstractSemidefinite optimization is a type of convex optimization problems over the cone of positive semidefinite (PSD) matrices, and sumofsquares (SOS) optimization is another type of convex optimization problems concerned with the cone of SOS polynomials. Both semidefinite and SOS optimization have found a wide range of applications, including control theory, fluid dynamics, machine learning, and power systems. In theory, they can be solved in polynomial time using interiorpoint methods, but these methods are only practical for small to mediumsized problem instances. In this talk, I will introduce decomposition methods for semidefinite optimization and SOS optimization with chordal sparsity, which scale more favorably to largescale problem instances. It is known that chordal decomposition allows one to equivalently decompose a PSD cone into a set of smaller and coupled cones. In the first part, I will apply this fact to reformulate a sparse semidefinite program (SDP) into an equivalent SDP with smaller PSD constraints that is suitable for the application of firstorder operatorsplitting methods. The resulting algorithms have been implemented in the opensource solver CDCS. In the second part, I will extend the classical chordal decomposition to the case of sparse polynomial matrices that are positive (semi)definite globally or locally on a semialgebraic set. The extended decomposition results can be viewed as sparsityexploiting versions of the HilbertArtin, Reznick, Putinar, and PutinarVasilescu PositivstellensÃ¤tze. This talk is based on our work: https://arxiv.org/abs/1707.05058 and https://arxiv.org/abs/2007.11410

10/28/21
Gaultier Lambert  University of Zurich
The Edge Scaling Limit of the Characteristic Polynomial of the Gaussian $\beta$Ensembles

10/28/21
Rongrong Wang  Michigan State University
Sigma Delta quantization on images, manifolds, and graphs
AbstractIn digital signal processing, quantization is the step of converting a signal's realvalued samples into a finite string of bits. As the first step in digital processing, it plays a crucial role in determining the information conversion rate and the reconstruction accuracy. Compared to nonadaptive quantizers, the adaptive ones are known to be more efficient in quantizing bandlimited signals, especially when the bitbudget is small (e.g.,1 bit) and noises are present. However, adaptive quantizers are currently only designed for 1D functions/signals. In this talk, I will discuss challenges in extending it to high dimensions and present our proposed solutions. Specifically, we design new adaptive quantization schemes to quantize images/videos as well as functions defined on 2D surface manifolds and general graphs, which are common objects in signal processing and machine learning. Mathematically, we start from the 1D SigmaDelta quantization, extend them to highdimensions and build suitable decoders. The discussed theory would be useful in natural image acquisition, medical imaging, 3D printing, and graph embedding.

10/28/21
Wooyeon Kim  ETH Zurich
Effective equidistribution of expanding translates in $ASL_d(\mathbb{R})/ASL_d(\mathbb{Z})$
AbstractIn this talk, we discuss effective versions of Ratnerâ€™s theorems in the space of affine lattices. For $d \geq 2$, let $Y=ASL_d(\mathbb{R})/ASL_d(\mathbb{Z})$, $H$ be a minimal horospherical group of $SL_d(\mathbb{R})$ embedded in $ASL_d(\mathbb{R})$, and $a_t$ be the corresponding diagonal flow. Then $(a_t)$pushforwards of a piece of $H$orbit become equidistributed with a polynomial error rate under certain Diophantine condition of the initial point of the orbit. This generalizes the previous results of StrÃ¶mbergsson for $d = 2$ and of Prinyasart for $d = 3$.

10/28/21
Rahul Dalal  Johns Hopkins
Counting level1, quaternionic automorphic representations on $G_2$
AbstractQuaternionic automorphic representations are one attempt to generalize to other groups the special place holomorphic modular forms have among automorphic representations of $GL_2$. Like holomorphic modular forms, they are defined by having their real component be one of a particularly nice class (in this case, called quaternionic discrete series). We count quaternionic automorphic representations on the exceptional group $G_2$ by developing a $G_2$ version of the classical EichlerSelberg trace formula for holomorphic modular forms. There are two main technical difficulties. First, quaternionic discrete series come in Lpackets with nonquaternionic members and standard invariant trace formula techniques cannot easily distinguish between discrete series with real component in the same Lpacket. Using the more modern stable trace formula resolves this issue. Second, quaternionic discrete series do not satisfy a technical condition of being ``regular", so the trace formula can a priori pick up unwanted contributions from automorphic representations with nontempered components at infinity. Applying some computations of Mundy, this miraculously does not happen for our specific case of quaternionic representations on $G_2$. Finally, we are only studying level1 forms, so we can apply some tricks of Chenevier and TaÃ¯bi to reduce the problem to counting representations on the compact form of $G_2$ and certain pairs of modular forms. This avoids involved computations on the geometric side of the trace formula.

10/28/21
Miroslav Krstic  UCSD
Stabilization of a Hyperbolic PDE of a Bioreactor With Distributed Age
AbstractFor an advectionreaction PDE model of population, with a nonlocal boundary condition modeling ``birth", and with a multiplicative input whose nature is the ``harvesting rate", we design a feedback law that stabilizes a desired equilibrium profile (of population density vs. age). Without feedback the system has one eigenvalue at the origin and the remainder of its infinite spectrum has negative real parts, i.e., the systems is, as engineers call it, ``neutrally stable". Hence, a feedback is needed to move one eigenvalue to the left without making any of the other ones spill to the right of the imaginary axis. This control design objective is achieved by transforming the system into a controltheoretic canonical form consisting of a firstorder ODE in which the input is present and whose eigenvalue needs to be made negative by feedback, and an infinitedimensional inputfree system called the ``zero dynamics", which we prove to be exponentially stable. The key feature of the overall PDE system and its feedback control law is the positivity of both the population density state and the harvesting rate input, which is a key element of the analysis, captured by a``control Lyapunov functional" which blows up when the population density or control approach zero.

10/29/21
Justin Lacini  University of Kansas
Logarithmic bounds on Fujita's conjecture
AbstractA longstanding conjecture of T. Fujita asserts that if X is a smooth complex projective variety of dimension n and if L is an ample line bundle, then $K_X+mL$ is basepoint free for $m>=n+1$. The conjecture is known up to dimension five by work of Reider, Ein, Lazarsfeld, Kawamata, Ye and Zhu. In higher dimensions, breakthrough work of Angehrn, Siu, Helmke and others showed that the conjecture holds if m is larger than a quadratic function in n. We show that for $n>=2$ the conjecture holds for m larger than $n(loglog(n)+3)$. This is joint work with L. Ghidelli.
Nov

11/01/21
Nir Avni  Northwestern University
Model theory of higher rank arithmetic groups
AbstractI'll describe a new rigidity phenomenon of lattices in higher rank semisimple groups. Specifically, I'll explain why the theories of such groups can't have (finitely generated) deformations, why these groups have a very rich collection of definable subgroups, and finish by discussing a conjecture saying that being a higher rank arithmetic lattice is a firstorder property. Based on joint works with Alex Lubotzky and Chen Meiri.

11/02/21
Felix Parraud  KTH Royal Institute of Technology (Stockholm)
Free probability and random matrices: the asymptotic behaviour of polynomials in independent random matrices
AbstractIt has been known for a long time that as their size grow to infinity, many models of random matrices behave as free operators. This link was first explicited by Voiculescu in 1991 in a paper in which he proved that the trace of polynomials in independent GUE matrices converges towards the trace of the same polynomial evaluated in free semicircular variables. In 2005, Haagerup and Thorbjornsen proved the convergence of the norm instead of the trace. The main difficulty of their proof was to prove a sharp enough upper bound of the difference between the trace of random matrices and their free limit. They managed to do so with the help of the socalled linearization trick which allows to relate the spectrum of a polynomial of any degree with scalar coefficients with a polynomial of degree 1 with matrix coefficients. A drawback of this method is that it does not give easily good quantitative estimates. In arXiv:1912.04588, we introduced a new strategy to approach those questions which does not rely on the linearization trick and instead is based on free stochastic calculus. In this talk, I will first focus on the paper arXiv:2011.04146, in which we proved an asymptotic expansion for traces of smooth functions evaluated in independent GUE random matrices, whose coefficients are defined through free probability. And then I will talk about arXiv:2005.1383, in which we adapted the previous method to the case of Haar unitary matrices.

11/02/21
JeanMichel Roquejoffre  University of Toulouse
Sharp location of the level sets in some reactiondiffusion equations
AbstractIn a large class of reactiondiffusion equations, the solution starting from a compactly supported initial datum develops a transition between two rest states, that moves at an asymptotically linear rate in time, and whose thickness remains asymptotically bounded in time. The issue is its precise location in time, that is, up to terms that are o(1) as time goes to infinity. This question is well understood in one space dimension; I will discuss what happens in the less well settled multidimensional framework. Joint works with L. Rossi and V. Roussier.

11/02/21
Nicholas Sieger  UCSD
PseudoQuasiRandom Boolean Functions
AbstractWhat makes the digits 645751311064590590501615753639260425710 and 1010100101010011111111010100111010010111 so special? These digits look as if they were chosen at random, yet they are entirely deterministic (take the fractional part of the square root of 7). In this talk, I will explore the theory of quasirandomness, which characterizes ``randomlike" sequences, graphs, sets, and many other objects. In particular, I will present a theory of quasirandomness for Boolean functions and show how random Boolean functions lead to a very challenging open problem: the Inverse Theory of the Gowers Norms.

11/03/21
Kisun Lee  UCSD
Polyhedral homotopy method for Nash equilibrium problem
AbstractIn this talk, we discuss the problem of finding generalized Nash equilibria (GNE) in the viewpoint of sparse polynomials. To obtain optimality conditions for GNE, we consider the KarushKuhnTucker (KKT) system using the Lagrange multiplier. We discuss that if all objectives and constraints polynomials are generic, the number of solutions of the KKT system equals its mixed volume, and so the polyhedral homotopy method can be optimal for finding GNEs. Lastly, comparisons with existing methods will be given.

11/03/21
Yiren Wang  UCSD
Towards a Theoretical Foundation of the Modelfree Bootstrap

11/04/21
Weilin Li  New York University
Superresolution, subspace methods, and Fourier matrices
AbstractThis talk is concerned with the inverse problem of recovering a discrete measure on the torus given a finite number of its noisy Fourier coefficients. We focus on the diffraction limited regime where at least two atoms are closer together than the Rayleigh length. We show that the fundamental limits of this problem and the stability of subspace (algebraic) methods, such as ESPRIT and MUSIC, are closely connected to the minimum singular value of nonharmonic Fourier matrices. We provide novel bounds for the latter in the case where the atoms are located in clumps. We also provide an analogous theory for a statistical model, where the measure is timedependent and Fourier measurements are collected over at various times. Joint work with Wenjing Liao, Albert Fannjiang, Zengying Zhu, and Weiguo Gao.

11/04/21
Evgeni Dimitrov  Columbia University
Gibbsian line ensembles and betacorners processes

11/04/21
Aaron Calderon  Yale University
Random hyperbolic surfaces via random flat surfaces
AbstractWhat does it mean to pick a ``random'' hyperbolic surface, and how does one even go about ``picking'' one? Mirzakhani gave an inductive answer to this question by gluing together smaller random surfaces along long curves; this is equivalent to studying the equidistribution of certain sets inside the moduli space of hyperbolic surfaces. Starting from first concepts, in this talk Iâ€™ll explain a new method for building random hyperbolic surfaces by building random \emph{flat} ones. As time permits, we will also discuss the application of this technique to Mirzakhaniâ€™s ``twist torus conjecture.'' This is joint work (in progress) with James Farre.

11/04/21
Nida Obatake  Institute of Defense Analyses and UCSD
Algebraic and Geometric Methods in the Study of Chemical Reaction Networks
AbstractChemical Reaction Network theory is an area of mathematics that analyzes the behaviors of chemical processes. One major problem concerns the stability of steady states of these networks. Does a given chemical reaction network have the capacity for Hopf bifurcations (an important unstable steadystate that yields periodic oscillations)? Our first contribution is a novel procedure for constructing a Hopf bifurcation of a chemical reaction network. This algorithm  our Newtonpolytope method  gives an easytocheck condition for the existence of a Hopf bifurcation and explicitly constructs one if it exists. Another important invariant of a chemical reaction network is its maximum number of steady states. This number, however, is in general difficult to compute, as it translates to counting positive real solutions of parametrized polynomial systems. To this end, we introduce an upper bound on this number  namely, a network's mixed volume  that is easy to compute. In this talk, we apply our two new tools to an important biologicalsignaling network, called the ERK network. Rubinstein et al. (2016) showed that the ERK network exhibits multiple steady states, bistability, and periodic oscillations (for a very particular choice of initial conditions). Conradi and Shiu (2015) proved that when certain reactions are omitted, the ERK network reduces to the processive dualsite phosphorylation network, which has a unique, stable steadystate (for any initial conditions). This stark contrast in dynamics prompted Rubinstein et al.'s question, "How are bistability and oscillations lost as reactions are removed from the ERK network?" By analyzing subnetworks of the ERK network, we systematically answer this question and demonstrate that bistability and oscillations persist even after we greatly simplify the model (by making reactions irreversible and removing intermediate species).

11/04/21
Linus Hamann  Princeton
Compatibility of the FarguesScholze and GanTakeda local Langlands
AbstractGiven a prime $p$, a finite extension $L/\mathbb{Q}_{p}$, a
connected $p$adic reductive group $G/L$, and a smooth irreducible
representation $\pi$ of $G(L)$, FarguesScholze recently attached a
semisimple Weil parameter to such $\pi$, giving a general candidate for
the local Langlands correspondence. It is natural to ask whether this
construction is compatible with known instances of the correspondence
after semisimplification. For $G = GL_{n}$ and its inner forms,
FarguesScholze and HansenKalethaWeinstein show that the
correspondence is compatible with the correspondence of
HarrisTaylor/Henniart. We verify a similar compatibility for $G =
GSp_{4}$ and its unique nonsplit inner form $G = GU_{2}(D)$, where $D$
is the quaternion division algebra over $L$, assuming that
$L/\mathbb{Q}_{p}$ is unramified and $p > 2$. In this case, the local
Langlands correspondence has been constructed by GanTakeda and
GanTantono. Analogous to the case of $GL_{n}$ and its inner forms, this
compatibility is proven by describing the Weil group action on the
cohomology of a local Shimura variety associated to $GSp_{4}$, using
basic uniformization of abelian type Shimura varieties due to Shen,
combined with various global results of KretShin and Sorensen on Galois
representations in the cohomology of global Shimura varieties associated
to inner forms of $GSp_{4}$ over a totally real field. After showing the
parameters are the same, we apply some ideas from the geometry of the
FarguesScholze construction explored recently by Hansen, to give a more
precise description of the cohomology of this local Shimura variety,
verifying a strong form of the Kottwitz conjecture in the process. 
11/05/21
Yi Hu  University of Arizona
Local Resolution of Singularities
AbstractMnev's universality theorem asserts that every singularity type over the ring of integers appears in some thin Schubert cell of the Grassmannian Gr(3,E) for some vector space E. We construct sequential blowups of Gr(3,E) such that certain induced birational transforms of all thin Schubert cells become smooth over prime fields. This implies that every singular variety X defined over a prime field admits local resolutions. For a singular variety X over a general perfect field k, we spread it out and deduce that X/k admits local resolution as well.

11/08/21
Fran\c{c}ois Thilmany  University of Louvain
On the connections between discreteness of arithmetic groups and the Lehmer conjecture
AbstractThe famous Lehmer problem asks whether there is a gap between 1 and the Mahler measure of algebraic integers which are not roots of unity. Asked in 1933, this deep question concerning number theory has since then been connected to several other subjects. After introducing the concepts involved, we will briefly describe a few of these connections with the theory of linear groups. Then, we will discuss the equivalence of a weak form of the Lehmer conjecture and the ``uniform discreteness" of cocompact lattices in semisimple Lie groups (conjectured by Margulis). (Joint work with Lam Pham.)

11/09/21
Arseniy Kryazhev  UCSD
Introduction to spectra, part 2

11/09/21
Hugo Lavenant  Bocconi University
Dynamical Optimal Transport: discretization and convergence
AbstractI will present the dynamical formulation of optimal transport (a.k.a BenamouBrenier formulation): it consists in writing the optimal transport problem as the minimization of a convex functional under a PDE constraint, and can handle a priori a vast class of cost functions and geometries. It is one of the oldest numerical method to solve the problem, and it is also the basis for a lot of extensions and generalizations of the optimal transport problem.
The optimization problem is then discretized to end up with a finite dimensional convex optimization problem. I will illustrate this method by presenting a discretization when the ground space is a surface. Although much effort has been devoted to solve efficently the discretized problem, the study of convergence under mesh refinement of the solution of the approximate problems has only been tackled recently. I will present an abstract framework guaranteeing convergence under mesh refinement, with no condition on the relative scale of the spacial and temporal mesh sizes, and even if the densities are very singular.

11/09/21
Ian Charlesworth  KU Leuven
Recent work on free Stein dimension
AbstractFree information theory is largely concerned with the following question: given a tuple of noncommutative random variables, what regularity properties of the algebra they generate can be inferred from assumptions about their joint distribution? This can include von Neumann algebraic properties, such as factoriality or absence of Cartan subalgebras, and free probabilistic properties, such as a lack of noncommutative rational relations.
After giving some background, I will talk on free Stein dimension, a quantity which measures the ease of defining derivations on a tuple of noncommutative variables which turns out to be a $*$algebra invariant. I will mention some recent results on its theory, including behaviour in the presence of algebraic relations as well as under direct sum and amplification of algebras. I will also mention some recent attempts to
adapt its utility from polynomial algebras to W*algebras, and time permitting, some cases where explicit estimates can be found on the Stein dimension of generating tuples of von Neumann algebras. This project is joint work with Brent Nelson. 
11/09/21
Bjoern Bringmann  IAS
Invariant Gibbs measures for the threedimensional wave equation with a Hartree nonlinearity
AbstractIn this talk, we discuss the construction and invariance of the Gibbs measure for a threedimensional wave equation with a Hartreenonlinearity.
In the first part of the talk, we construct the Gibbs measure and examine its properties. We discuss the
mutual singularity of the Gibbs measure and the socalled Gaussian free field. In contrast, the Gibbs
measure for one or twodimensional wave equations is absolutely continuous with respect to the Gaussian free field.In the second part of the talk, we discuss the probabilistic wellposedness of the corresponding nonlinear wave equation, which is needed in the proof of invariance. This was the first theorem proving the invariance of a singular Gibbs measure for any dispersive equation.

11/09/21
Nathan Wenger  UCSD
Geometric Constructions Relating to Vector Bundles
AbstractWe provide a brief introduction to the theory of vector bundles and present a few useful and interesting constructions with a geometric flavor.

11/10/21
Shangzhi Zeng  The University of Hong Kong
Difference of convex algorithms for bilevel programs with applications in hyperparameter selection

11/15/21
Junho Peter Whang  Seoul National University
Diophantine study of Stokes matrices
AbstractStokes matrices (i.e. unipotent upper triangular matrices) and their nonlinear braid group actions arise naturally in a number of geometric and algebraic contexts. Integral Stokes matrices are often of particular interest, motivating their reduction theory. After reviewing classical work of Markoff treating the case of 3by3 matrices, we describe joint work with YuWei Fan for the 4by4 case by establishing an exceptional connection to SL2character varieties of surfaces. This will also serve as an opportunity to present our recent work on effective finite generation of integral points on the latter moduli spaces. Time permitting, we finish by presenting new results (and problems) for Stokes matrices of larger dimension.

11/16/21
Shangjie Zhang  UCSD
Models of spectra

11/16/21
Priyanga Ganesan  Texas A \& M University
Spectral bounds for chromatic number of quantum graphs
AbstractQuantum graphs are an operator space generalization of classical graphs that have appeared in different branches of mathematics including operator systems theory, noncommutative topology and quantum information theory. In this talk, I will review the different perspectives to quantum graphs and introduce a chromatic number for quantum graphs using a nonlocal game with quantum inputs and classical outputs. I will then show that many spectral lower bounds for chromatic numbers in the classical case (such as Hoffmanâ€™s bound) also hold in the setting of quantum graphs. This is achieved using an algebraic formulation of quantum graph coloring and tools from linear algebra.

11/16/21
Elizabeth Wong  UCSD
Nonlinear Optimization for Optimal Power Flow Problems
AbstractIn this talk, we present an algorithm created for Challenge 2 of the Grid Optimization Competition from ARPAE (Advanced Research Projects AgencyEnergy). The competition considered a security constrained optimal power flow (SCOPF) problem whose solution determines optimal dispatch and control settings for power generation and grid control equipment and maximizes the market surplus associated with the operation of the grid, subject to pre and postcontingency constraints. We will discuss the practical issues associated with the challenge and describe the approach and heuristics established to enhance performance of the algorithm submitted to the competition. Results from the competition will also be presented.
This is joint work with Frank E. Curtis (Lehigh University), Daniel Molzahn (Georgia Tech), Andreas W\"{a}chter and Ermin Wei (Northwestern University)."

11/16/21
Brian Tran  UCSD
Multisymplectic Variational Integrators for Hamiltonian PDEs
AbstractWe will begin by discussing the multisymplectic structure associated to Hamiltonian PDEs as a generalization of the symplectic structure associated to Hamilton's equations in classical mechanics. Subsequently, we will turn to the question of how to computationally model such Hamiltonian PDEs while preserving the multisymplectic structure at the discrete level. This will lead us to the notion of a multisymplectic integrator. The analogue of these integrators in Hamiltonian mechanics are known as symplectic integrators, which are extremely wellstudied and have proven to provide extremely robust and physically faithful simulations of mechanical systems. Multisymplectic integration, on the other hand, is still in its relative infancy.
After establishing the necessary background, we will introduce our construction of variational integrators for Hamiltonian PDEs which automatically yield multisymplectic integrators. This construction gives a systematic framework for constructing such multisymplectic integrators, based on the notion of a Type II generating functional. As an application of our framework, we will derive the class of multisymplectic partitioned RungeKutta methods and provide a numerical example with the family of sineGordon soliton solutions. This is joint work with Prof. Melvin Leok.

11/16/21
Josh Swanson  USC
DUSTPAN distributions as limit laws for Mahonian statistics on forests
AbstractBuilding on work of Stanley and Bj\"{o}rnerWachs

11/17/21
Sylvia Herbert  UCSD
HamiltonJacobi Reachability Analysis
AbstractIn this talk I will introduce HamiltonJacobi (HJ) Reachability Analysis for dynamic systems. HJ reachability uses level set methods to compute the set of initial conditions from which a dynamical system is guaranteed to reach its goal and/or avoid unsafe regions despite worstcase conditions. I will discuss challenges, recent advances, and applications.

11/18/21
Johannes Alt  University of Geneva and Courant Institute
Localization and delocalization in Erd\H{o}sR\'{e}nyi graphs

11/18/21
Sunrose Shrestha  Wesleyan University
Periodic straightline flows on the Mucube
AbstractThe dynamics of straightline flows on compact translation surfaces (surfaces formed by gluing Euclidean polygons edgeto edge via translations) has been widely studied due to connections to polygonal billiards and TeichmÃ¼ller theory. However, much less is known regarding straightline flows on noncompact infinite translation surfaces. In this talk we will review work on straight line flows on infinite translation surfaces and consider such a flow on the Mucube â€“ an infinite $\mathbb{Z}^3$ periodic halftranslation squaretiled surface â€“ first discovered by Coxeter and Petrie and more recently studied by AthreyaLee. We will give a complete characterization of the periodic directions for the straightline flow on the Mucube â€“ in terms of a subgroup of $\mathrm{SL}_2 \mathbb{Z}$. We will use the latter characterization to obtain the group of derivatives of affine diffeomorphisms of the Mucube. This is joint work (in progress) with Andre P. Oliveira, Felipe A. RamÃrez and Chandrika Sadanand.

11/18/21
Gabriel DorfsmanHopkins  UC Berkeley
Untilting Line Bundles on Perfectoid Spaces
AbstractLet $X$ be a perfectoid space with tilt $X^\flat$. We build a natural map $\theta:\mathrm{Pic} X^\flat\to\lim\mathrm{Pic} X$ where the (inverse) limit is taken over the $p$power map, and show that $\theta$ is an isomorphism if $R = \Gamma(X,\mathcal{O}_X)$ is a perfectoid ring.
As a consequence we obtain a characterization of when the Picard groups of $X$ and $X^\flat$ agree in terms of the $p$divisibility of $\mathrm{Pic} X$. The main technical ingredient is the vanishing of higher derived limits of the unit group $R^*$, whence the main result follows from the Grothendieck spectral sequence.

11/18/21
Varun Khurana  UCSD
Learning Sheared Distributions with Linearized Optimal Transport
AbstractIn this paper, we study supervised learning tasks on the space of probability measures. We approach this problem by embedding the space of probability measures into $L^2$ spaces using the optimal transport framework. In the embedding spaces, regular machine learning techniques are used to achieve linear separability. This idea has proved successful in applications and when the classes to be separated are generated by shifts and scalings of a fixed measure. This paper extends the class of elementary transformations suitable for the framework to families of shearings, describing conditions under which two classes of sheared distributions can be linearly separated. We furthermore give necessary bounds on the transformations to achieve a prespecified separation level, and show how multiple embeddings can be used to allow for larger families of transformations. We demonstrate our results on image classification tasks.
Based on joint work with Caroline Moosmueller, Harish Kannan, and Alex Cloninger.

11/18/21
Nicolas Monod  EPFL, Switzerland
Safe spaces in geometry and analysis
AbstractIn the Wild West of geometry and groups, some familiar objects feel like home: Euclidean spaces, hyperbolic geometry, and more generally all the geometries described by semisimple Lie groups or similar matrix groups over local fields.
Harmonic analysis and operator algebras have their own wild seas, but again with a small safe haven: the ``Type I", home of the commutative world, compact groups, generalizations of Fourier analysis. In a precise sense, objects that can be ``described".
I will present a connection between these two worlds and show how it leads to previously unexpected classification results.

11/19/21
Valery Alexeev  University of Georgia
Compact moduli spaces of K3 surfaces
AbstractI will explain recent results on modular, geometrically meaningful compactifications of moduli spaces of K3 surfaces, most of which are joint with Philip Engel. A key notion is that of a recognizable divisor: a canonical choice of a divisor in a multiple of the polarization that can be canonically extended to any Kulikov degeneration. For a moduli of latticepolarized K3s with a recognizable divisor we construct a canonical stable slc pair (KSBA) compactification and prove that it is semi toroidal. We prove that the rational curve divisor is recognizable, and give many other examples.

11/22/21
Keivan MallahiKarai  Jacobs University
Optimal linear sofic approximation of countable groups
AbstractVarious notions of metric approximation for countable groups have been introduced and studied in the last decade, with sofic and hyperlinear approximations being two notable examples among them. The class of linear sofic groups was introduced by Glebsky and Rivera and was subsequently studied by Arzhantseva and Paunescu. This mode of approximation uses the general linear group over, say, the field of complex numbers as model groups, equipped with the distance defined using the normalized rank. Among their other interesting results, Arzhantseva and Paunescu prove that every linear sofic group is 1/4linear sofic, where the constant 1/4 quantifies how well nonidentity elements can be separated from the identity matrix. In this talk, which is based on joint work with Maryam Mohammadi Yekta, we will address the question of optimality of the constant 1/4 and report on some progress in this direction.

11/23/21
Maxwell Johnson  UCSD
The Adams spectral sequence

11/23/21
Amudhan KrishnaswamyUsha  Delft University of Technology
Multilinear Fourier multipliers on noncommutative groups
AbstractFor a function $m$ on the real line, its Fourier multiplier $T_m$ is the operator which acts on a function $f$ by first multiplying the Fourier transform of $f$ by $m$, and then taking the inverse Fourier transform of the product. These are wellstudied objects in classical harmonic analysis. Of particular interest is when the Fourier multiplier defines a bounded operator on $L_p$. Fourier multipliers can be generalized to arbitrary locally compact groups. If the group is nonabelian, the $L_p$ spaces involved are now the noncommutative $L_p$ spaces associated with the group von Neumann algebra. Fourier multipliers also have a natural extension to the multilinear setting. However, their behaviour can differ markedly from the linear case, and there is much that is unknown even about multilinear Fourier multipliers on the reals.
One question of interest is this: If $m$ is a function on a group $G$ which defines a bounded $L_p$ multiplier, is the restriction of m to a subgroup $H$ also the symbol of a bounded $L_p$ multiplier on $H$? De Leeuw proved that the answer is yes, when $G$ is $\mathbb{R}^n$. This was extended to the commutative case by Saeki and to the noncommutative case (provided the group $G$ is sufficiently nice) by Caspers, Parcet, Perrin and Ricard. In this talk, I will show how to extend these De Leeuw type theorems to multilinear Fourier multipliers on noncommutative groups. This is part of joint work with Martijn Caspers, Bas Janssens and Lukas Miaskiwskyi.

11/23/21
Francois Hamel  AixMarseille University
Symmetry properties for the Euler equations and related semilinear elliptic equations
AbstractIn this talk, I will discuss radial and onedimensional symmetry properties for the stationary incompressible Euler equations in dimension 2 and some related semilinear elliptic equations. I will show that a steady flow of an ideal incompressible fluid with no stagnation point and tangential boundary conditions in an annulus is a circular flow. The same conclusion holds in complements of disks as well as in punctured disks and in the punctured plane, with some suitable conditions at infinity or at the origin. If possible, I will also discuss the case of parallel flows in twodimensional strips, in the halfplane and in the whole plane. The proofs are based on the study of the geometric properties of the streamlines of the flow and on radial and onedimensional symmetry results for the solutions of some elliptic equations satisfied by the stream function. The talk is based on joint works with N. Nadirashvili.

11/23/21
Anton Bernshteyn  Carnegie Mellon University
Weak degeneracy of graphs
AbstractMotivated by the study of greedy algorithms for graph coloring, we introduce a new graph parameter, which we call weak degeneracy. This notion formalizes a particularly simple way of ``saving" colors while coloring a graph greedily. It turns out that many upper bounds on chromatic numbers follow from corresponding bounds on weak degeneracy. In this talk I will survey some of these bounds as well as state a number of open problems. This is joint work with Eugene Lee.

11/29/21
Dan Popovici  University of Toulouse
Hermitiansymplectic and K\"ahler Metrics on Compact Complex Manifolds"
AbstractThis is joint work with S. Dinew (Krakow). We investigate the class of compact complex Hermitiansymplectic manifolds $X$. For each Hermitiansymplectic metric $\omega$ on $X$, we introduce a functional acting on the metrics in a certain cohomology class of $\omega$ and prove that its critical points (if any) must be K\"ahler when X is 3dimensional.

11/30/21
Marwa Banna  NYU Abu Dhabi
BerryEsseen Bounds for Operatorvalued Free Limit Theorems
AbstractThe development of free probability theory has drawn much inspiration from its deep and far reaching analogy with classical probability theory. The same holds for its operatorvalued extension, where the fundamental notion of free independence is generalized to free independence with amalgamation as a kind of conditional version of the former. Its development naturally led to operatorvalued free analogues of key and fundamental limiting theorems such as the operatorvalued free Central Limit Theorem due to Voiculescu and results about the asymptotic behaviour of distributions of matrices with operatorvalued entries.
In this talk, we show BerryEsseen bounds for such limit theorems. The estimates are on the level of operatorvalued Cauchy transforms and the L{\'e}vy distance. We address also the multivariate setting for which we consider linear matrix pencils and noncommutative polynomials as test functions. The estimates are in terms of operatorvalued moments and yield the first quantitative bounds on the L{\'e}vy distance for the operatorvalued free CLT. This also yields quantitative estimates on joint noncommutative distributions of operatorvalued matrices having a general covariance profile.
This is a joint work with Tobias Mai.

11/30/21
Jordan Benson  UCSD
The May Spectral Sequence

11/30/21
William Feldman  University of Utah
Limit shapes of Bernoullitype free boundaries in periodic media
AbstractI will discuss some simplified models for the shape of liquid droplets on rough solid surfaces, especially Bernoullitype free boundary problems. In these models small scale roughness leads to large scale nonuniqueness, hysteresis, and anisotropies. In technical terms we need to understand laminating/foliating families of planelike solutions, this is related to ideas of AubryMather theory, but, unlike most results in that area, we need to consider local (but not global) energy minimizers.

11/30/21
Shubhankar Sahai  UCSD
An informal introduction to descent
AbstractIn geometry, one often starts with a base space (e.g. a manifold, or a variety, etc) and is interested in constructing global objects over the base. One of the ways to construct these global objects is to glue them over the base from simpler local data.
For example, one builds global vector bundles by first describing them locally as products and then gluing them via isomorphisms satisfying certain cocycle conditions. Said more abstractly, one tries to recover the `category' of vector bundles on the base by looking at the `category' of vector bundles on `small open' sets on the base. The fact that one can do this is succinctly summarised by saying that the `category' of vector bundles satisfies $\emph{descent}$ on open sets over topological spaces. More provocatively, one says that the `category' of vector bundles is a $\emph{stack}$ over the base. This abstraction is not just decorativeâ€”the analogous statement fails for isomorphism classes of vector bundles!
In this talk I will discuss this and other ideas which go under the collective name of descent. The first half will be an informal introduction to descent with minimal prerequisites. The second half will discuss counterparts of these ideas in the context of algebraic geometry and commutative algebra, although again with minimal prerequisites.
Dec

12/01/21
Zach Higgins  UCSD
Iwasawa Theory of Taelman Class Modules
AbstractRecently, Taelman defined a ``class module" associated to any Drinfeld module defined over a function field. In the spirit of Iwasawa theory, we will study the structure of these class modules in certain padic towers of fields. Using the Equivariant Tamagawa Number Formula for Drinfeld modules, we will propose an Iwasawa main conjecture for these class modules.

12/01/21
Amir Mohammadi  UCSD
Finitary analysis in homogeneous spaces and applications
AbstractRigidity phenomena in homogeneous dynamics have been extensively studied over the past few decades with several striking results and applications.
In this talk, we will give an overview of recent activities related to quantitative aspect of the analysis in this context; we will also highlight some applications. 
12/01/21
Xindong Tang  Hong Kong Polytech University
Rational Generalized Nash Equilibrium Problems

12/02/21
Nike Sun  MIT
Generalized Ising perceptron models

12/02/21
Josh Southerland  University of Washington
Towards a shrinking target property for primitive squaretiled surfaces
AbstractIn this talk, I will discuss ongoing work to develop a method for proving a shrinking target property on primitive squaretiled surfaces that comes from the action of a subgroup $G$ of its Veech group. Our main tool is the construction of a Fourierlike transform which we can use to relate the $L^2$norm of the Koopman operator induced by $G$ to the $L^2$norm of a Markov operator related to a random walk on $G$.

12/02/21
James Upton  UC San Diego
Newton Polygons of Abelian $L$Functions on Curves
AbstractLet $X$ be a smooth, affine, geometrically connected curve
over a finite field of characteristic $p > 2$. Let $\rho:\pi_1(X) \to
\mathbb{C}^\times$ be a character of finite order $p^n$. If $\rho\neq
1$, then the Artin $L$function $L(\rho,s)$ is a polynomial, and a
theorem of KramerMiller states that its $p$adic Newton polygon
$\mathrm{NP}(\rho)$ is bounded below by a certain Hodge polygon
$\mathrm{HP}(\rho)$ which is defined in terms of local monodromy
invariants. In this talk we discuss the interaction between the polygons
$\mathrm{NP}(\rho)$ and $\mathrm{HP}(\rho)$. Our main result states that
if $X$ is ordinary, then $\mathrm{NP}(\rho)$ and $\mathrm{HP}(\rho)$
share a vertex if and only if there is a corresponding vertex shared by
certain ``local" Newton and Hodge polygons associated to each ramified
point of $\rho$. As an application, we give a local criterion that is
necessary and sufficient for $\mathrm{NP}(\rho)$ and $\mathrm{HP}(\rho)$
to coincide. This is joint work with Joe KramerMiller. 
12/02/21

12/06/21
Alex Kontorovich  Rutgers University
Asymptotic Length Saturation for Zariski Dense Surfaces
AbstractThe lengths of closed geodesics on a hyperbolic manifold are determined by the traces of its fundamental group. When the latter is a Zariski dense subgroup of an arithmetic group, the trace set is contained in the ring of integers of a number field, and may have some local obstructions. We say that the surface's length set ``saturates" (resp. ``asymptotically saturates") if every (resp. almost every) sufficiently large admissible trace appears. In joint work with Xin Zhang, we prove the first instance of asymptotic length saturation for punctured covers of the modular surface, in the full range of critical exponent exceeding onehalf (below which saturation is impossible).

12/07/21

12/07/21
Evangelos ``Vaki" Nikitopoulos  UCSD
Choose Your Own Functional Calculus Adventure
AbstractLet $A$ be a (unital) algebra over the complex numbers and $a \in A$. At a very high level, the term \textit{functional calculus} refers to constructions of the form, ``Take some collection $\mathcal{F}$ of scalar functions, and, for all $f \in \mathcal{F}$, define $f(a) \in A$ in a sensible way." One can always take $\mathcal{F} = \mathbb{C}[t]$ with the obvious definition of $p(a) \in A$ for $p \in \mathbb{C}[t]$, but this is pretty much the end of the construction when $A$ has no additional structure. When $A$ has some analytic structure  as is frequently the case in functional analysis and operator algebras  one can construct functional calculi for much larger classes of functions. In this slightly experimental talk, it is possible that I will discuss functional calculus in Banach algebras, $C^*$algebras, and/or von Neumann algebras. The talk will be in a ``choose your own adventure" style, so the audience will decide the exact trajectory of the talk democratically. (I offer my thanks to Max Johnson for the idea to give this kind of talk.) Prerequisites will be minimal: passing familiarity with norms, inner products, bounded/continuous linear maps, completeness, etc. should suffice.

12/07/21
Nicola Tarasca  Virginia Commonwealth University
Incident varieties of algebraic curves and canonical divisors
AbstractThe theory of canonical divisors on curves has witnessed an explosion of interest in recent years, motivated by the recent developments in the study of limits of canonical divisors on nodal curves. Imposing conditions on canonical divisors allows one to construct natural geometric subvarieties of moduli spaces of pointed curves, called strata of canonical divisors. The strata are in fact the projection on moduli spaces of curves of incidence varieties in the projectivized Hodge bundle. I will present a graph formula for the class of the restriction of such incident varieties over the locus of pointed curves with rational tails. The formula is expressed as a linear combination of tautological classes indexed by decorated stable graphs, with coefficients enumerating appropriate weightings of decorated stable graphs. I will conclude with some applications. Joint work with Iulia Gheorghita.

12/07/21
Jaeseong Oh  Korea Institute for Advanced Study
A combinatorial model for the transition matrix between the Specht and web bases
AbstractWe introduce a new class of permutations, called web permutations. Using these permutations, we provide a combinatorial interpretation for entries of the transition matrix between the Specht and web bases, which answers Rhoades's question. Furthermore, we study enumerative properties of these permutations. This is based on the work with ByungHak Hwang and Jihyeug Jang.