In this talk, I will discuss some models and algorithms from
two different fields: (1) machine learning, including logistic
regression and deep learning, and (2) numerical PDEs, including
multigrid methods. I will explore mathematical relationships between
these models and algorithms and demonstrate how such relationships can
be used to understand, study and improve the model structures,
mathematical properties and relevant training algorithms for deep
neural networks. In particular, I will demonstrate how a new
convolutional neural network known as MgNet, can be derived by making
very minor modifications of a classic geometric multigrid method for
the Poisson equation and then explore the theoretical and practical
potentials of MgNet.

The talk presents a quick review on Landau damping for Vlasov-Poisson system near Penrose stable data, followed by a joint work with D. Han-Kwan and F. Rousset, where the damping, with screening potential, is proved for data with (essentially) $C^{1}$ regularity on the whole space.

A surface has geometric free-boundary in a barrier hypersurface if its boundary meets the barrier orthogonally, like a bubble on a bathtub. We extend Brakke's weak notion of mean curvature flow to have a free-boundary condition, and using toy examples we show why this extension is necessary. Contrary to the classical flow, for which the barrier is ``invisible,'' the weak flow allows for the surfaces to ``pop'' upon tangential contact with the barrier. When the initial surface is mean-convex, we generalize White's partial regularity and structure theory to the free-boundary setting. This is in part joint work with Robert Haslhofer, Mohammad Ivaki, and Jonathan Zhu.

Khovanov homology, as defined by Mikhail Khovanov based on the representation theory of quantum groups, is a powerful invariant for knots. It categorifies the famous Jones polynomial, meaning that Jones polynomial can be recovered as its Euler characteristic. Despite its simple combinatorial definition, Khovanov homology has deep relation with quantum topology, gauge theory and representation theory. In this talk, I will recall the definition of Khovanov homology and introduce some of its important properties and applications (including Rassmussen's combinatorial proof of the Milnor conjecture). No previous knowledge on knot theory will be assumed.

Low rank approximation has been extensively studied in the past. It is most suitable to reproduce rectangular like structures in the data. In this talk I introduce a generalization using ``shifted'' rank-1 matrices to approximate $A\in\mathbb{C}^{M\times N}$. These matrices are of the form $S_{\lambda}(uv^*)$ where $u\in\mathbb{C}^M$, $v\in\mathbb{C}^N$ and $\lambda\in\mathbb{Z}^N$. The operator $S_{\lambda}$ circularly shifts the $k$-th column of $uv^*$ by $\lambda_k$.

These kind of shifts naturally appear in applications, where an object $u$ is observed in $N$ measurements at different positions indicated by the shift $\lambda$. The vector $v$ gives the observation intensity. This model holds e.g., for seismic waves that are recorded at $N$ sensors at different times $\lambda$.

The main difficulty of the above stated problem lies in finding a suitable shift vector $\lambda$. Once the shift is known, a simple singular value decomposition can be applied to reconstruct $u$ and $v$. We propose a greedy method to reconstruct $\lambda$ and validate our approach in numerical examples.

Reference:

F. Boss mann, J. Ma, Enhanced image approximation using shifted rank-1 reconstruction, Inverse Problems and Imaging, accepted 2019, https://arxiv.org/abs/1810.01681

The KPZ equation, a model for the random growth of rough interfaces, has been the subject of great physical and mathematical interest since its introduction in 1986. By simple changes of variables, it is closely related to the stochastic heat equation, which models the partition function of a random walk in a random environment, and the stochastic Burgers equation, a simple model for turbulence. I will explain several recent results about the existence, classification, and properties of spacetime-stationary solutions to these equations on ${\bf R}^d$ for various values of $d$. These solutions thus represent the behavior of the models in large domains on long time scales. Most of the results are joint work with various combinations of C. Graham, Y. Gu, L. Ryzhik, and O. Zeitouni.

Symmetric functions are highly ubiquitous in algebraic combinatorics, with connections to the representation theory of $GL_n$ and $S_n$, the geometry of Grassmannians, and more. There is a classical way to 'compose' symmetric functions called plethysm which has a nice representation-theoretic interpretation. We will describe a related operation called Chern plethysm which has inputs given by a symmetric function F and a vector bundle E and outputs a symmetric polynomial. Chern plethysm provides numerous Schur-positivity results, indicating a representation-theoretic connection, but finding this connection remains an open problem. Joint with Sara Billey and Vasu Tewari.

The moduli space M of complex curves of fixed topology is an orbifold classifying space for surface bundles. As such the cohomology rings of M and its various orbifold covers give characteristic classes for surface bundles. I will discuss the Steinberg module which is central to the duality present in these cohomology rings. I will then explain current joint work with T. Brendle and A. Putman on surfaces with marked points which expands on results of N. Fullarton and A. Putman for surfaces without marked points. We show that certain finite-sheeted orbifold covers M[l] of M have large nontrivial Q-cohomology in their cohomological dimension.

Given a graph $G$, one can compute the eigenvalues of its adjacency matrix $A_G$. Remarkably, these eigenvalues can tell us quite a bit about $G$. 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 $G$. This talk is the first in a series of introductory talks to the subject of spectral graph theory. In particular, we'll be discussing the adjacency matrix and the information encoded by its eigenvalues.

First stated in 1973, the Cycle Double Cover Conjecture states that every bridgeless graph has a collection of cycles which cover every edge exactly twice. The problem remains wide open to this day despite much work. This talk is to present my partial results on the problem, explain why the tutte polynomial is awesome, and to draw some of the beautiful proofs by picture that arose in my work.

Nonlocal models are experiencing a firm upswing recently as more realistic alternatives to the conventional local models for studying various phenomena from physics and biology to materials and social sciences. In this talk, I will describe our recent effort in taming the computational challenges for nonlocal models. I will first highlight a family of numerical schemes - the asymptotically compatible schemes - for nonlocal models that are robust with the modeling parameter approaching an asymptotic limit. Second, I will discuss nonlocal-to-local coupling techniques so as to improve the computational efficiency of using nonlocal models. This also motivates the development of new mathematical results - for instance, a new trace theorem that extends the classical results.

Although new nonlocal models have been gaining popularity in various applications, they often appear as phenomenological models, such as the peridynamics model in fracture mechanics. Here I will illustrate how to characterize the origin of nonlocality through homogenization of wave propagation in periodic media.

Rational varieties are some of the simplest examples of varieties, e.g. most of their points can be parametrized by affine space. It is natural to ask (1) How can we determine when a variety is rational? and (2) If a variety is not rational, can we measure how far it is from being rational? A famous particular case of this problem is when the variety is a smooth hypersurface in projective space. This problem has attracted a great deal of attention both classically and recently. The interesting case is when the degree of the hypersurface is at most the dimension of the projective space (the ``Fano''range), these hypersurfaces share many of the properties of projective space. In this talk, we present recent work with Nathan Chen which says that smooth Fano hypersurfaces of large dimension can have arbitrarily large degrees of irrationality, i.e. they can be arbitrarily far from being rational.

A ridge in a data cloud is a low-dimensional geometric feature that generalizes the concept of local modes, in the sense that ridge points are local maxima constrained in some subspace. In this talk we give nonparametric confidence regions for $r$-dimensional ridges of probability density functions on ${R}^d$ , where $1 \leq r \< d$. We view ridges as the intersections of level sets of some special functions. The vertical variation of the plug-in kernel estimators for these functions constrained on the ridges is used as the measure of maximal deviation for ridge estimation. Two types of confidence regions for density ridges will be presented: one is based on the asymptotic distribution of the maximal vertical deviation, which is established by utilizing the extreme value theory of nonstationary $\chi^2$-fields indexed by manifolds; and the other is a bootstrap approach (including multiplier bootstrap and empirical bootstrap), the theoretical validity of which leverages the recent study in the literature on the Gaussian approximation of suprema of empirical processes.

Khovanov homology is a powerful invariant of knots which has many connections to other areas of mathematics, such as representation theory and gauge theory. However, it is strangely difficult to extract topological information from this invariant. After discussing this invariant, we will show that it is able to detect the figure-eight knot.

As in complex analysis, Liouville theorems in PDEs assert that any solution of a specific equation is trivial. The meaning of
``trivial'' depends on the context. In this lecture, we will discuss Liouville theorems for superlinear parabolic PDEs. An overview, a typical application, and recent results will be presented.

We describe an ongoing program to resolve certain problems at the interface of Diophantine approximation and homogenous dynamics. Highlights include computing the Hausdorff and packing dimensions of the set of singular systems of linear forms and show they are equal, thereby resolving a conjecture of Kadyrov-Kleinbock-Lindenstrauss-Margulis (2014) as well as answering a question of Bugeaud-Cheung-Chevallier (2016). As a corollary of the Dani correspondence principle, this implies that the set of divergent trajectories of a one-parameter diagonal action on the space of unimodular lattices with exactly two Lyapunov exponents with opposite signs has equal Hausdorff and packing dimensions. Other applications include dimension formulas with respect to the uniform exponent of irrationality for simultaneous and dual approximation in two dimensions. This is joint work with David Simmons, Lior Fishman, and Mariusz Urbanski. The reduction of various problems to questions about certain combinatorial objects that we call templates along with a variant of Schmidt's game allows us to answer some of these problems, while leaving plenty that remain open. The talk will be accessible to students and faculty interested in some convex combination of homogeneous dynamics, Diophantine approximation and geometric measure theory.

Can one emulate the simple random walk without actually doing anything random? This talk will be about a deterministic version of random walk called rotor walk, and we will measure its performance in emulating the simple random walk with respect to different parameters, e.g., the shape of the trajectory, number of returns to the origin, etc. In particular, we will see that the number of returns to the origin for the rotor walk can be made equal to the same number for the simple random walk. This resolves a conjecture of Florescu, Ganguly, Levine, and Peres (2014).

In Perelman's well-known paper, he claimed the unqueness of the 3D \verb=\=kappa-noncollpased steady GRS (without giving any proof or ideas) and conjectured that 3D noncompact \verb=\=kappa-solution with positive sectional curvature must be the Ricci flow generated the Bryant soliton. Perelman's claim and conjecture has been proved by Simon Brendle in 2012 and 2018, respectively. The classification of 3D \verb=\=kappa-noncollpased steady GRS plays an important role in his proof of the conjecture. Brendle's work is based on the observation that 3D \verb=\=kappa-noncollpased steady GRS must be asymptotically cylindercial. In higher dimensions, the asymptotical geometry of steady GRS is much more complicated. In this talk, we will talk about some recent progress on the asymptotical geometry of 4D steady GRS. This is a joint work with Prof. Bennett Chow.

A triangulation of a topological space is a homeomorphism from this space to a simplicial complex. A famous problem in topology is whether all manifolds are triangulable. Surprisingly, the answer is no when dimension is at least 4. In this talk, I will explain the beautiful work of Galewski-Stern and Matumoto, which provides an obstruction theory for triangulating manifolds. I will also explain Manolescu's disproof of triangulation conjecture in all dimensions greater than 4.

The growth of bacterial colony exhibits striking complex patterns and robust scaling laws. Understanding the principles that underlie such growth has far-reaching consequences in biological and health sciences. In this work, we develop a mechanical theory of cell-cell and cell-environmental interactions and construct a hybrid three-dimensional computational model for the growth of {\it E.~coli} colony on a hard agar surface. Our model consists of microscopic descriptions of the growth, division, and movement of individual cells, and macroscopic diffusion equations for the nutrients. The cell movement is driven by the cellular mechanical interactions. Our unique treatment of the force arising from the liquid-air surface tension is applicable to both the monolayer (discrete) and multilayer (continuum) growth regimes. Our large-scale simulations and analysis predict the linear growth of the colony in both the radial and vertical directions, conforming the experimental observations. This work is the first step toward detailed computational modeling of bacterial growth with mechanical and biochemical interactions. This is joint work with Mya Warren, Hui Sun, Yue Yan, Jonas Cremer, and Terence Hwa.

Motivated by quantum information theory, we study log-Sobolev inequalities for matrix valued functions on finite graphs and compact Lie groups. Using tools from noncommutative geometry we find a surprising link between graph Laplacians and sublaplacians on the orthogonal group. Simple combinatorial tools, namely the existence of a spanning tree, then allows us to find concrete lower bounds for spectral gaps. Joint work with H. Li and N. LaRacuente.

In this talk, we will be discussing the discrete fourier transform in one
and two dimensions, including some of the transform's properties, as well
as various strategies for efficiently evaluating the discrete fourier transform
via several fast fourier transform algorithms. The algorithms discussed
will include the Cooley-Turkey algorithm, the radix-2 decimation in time
strategy, the split-radix algorithm, the mixed-radix algorithm, the prime
factor algorithm, and Rader's algorithm. Finally, we will discuss various
applications of the fast fourier transform, as well as considerations for
using it in practical applications.

Recently there has been a lot of work on the construction of the K-moduli space, i.e. a good moduli space parametrizing K-polystable Fano varieties. It is conjectured that this moduli space is projective and the polarization is given by a natural line bundle, the Chow-Mumford (CM) line bundle. In this talk, I will present a recent joint work with Chenyang Xu where we show the CM line bundle is ample on any proper subspace parametrizing reduced uniformly K-stable Fano varieties, which conjecturally should be the entire K-moduli. As an application, we prove that the moduli space parametrizing smoothable K-polystable Fano varieties is projective.

Suppose $L$ is a link with n components and the rank of $Kh(L;Z/2)$ is $2^n$, we show that $L$ can be obtained by disjoint unions and connected sums of Hopf links and unknots. This result gives a positive answer to a question asked by Batson-Seed, and generalizes the unlink detection theorem of Khovanov homology by Hedden-Ni and Batson-Seed. The proof relies on a new excision formula for the singular instanton Floer homology introduced by Kronheimer and Mrowka. This is joint work with Yi Xie.

In this talk I will survey recent work with Kleiner in which we verify two topological conjectures using Ricci flow. First, we classify the homotopy type of every 3-dimensional spherical space form. This proves the Generalized Smale Conjecture and gives an alternative proof of the Smale Conjecture, which was originally due to Hatcher. Second, we show that the space of metrics with positive scalar curvature on every 3-manifold is either contractible or empty. This completes work initiated by Marques. Our proof is based on a new uniqueness theorem for singular Ricci flows, which I have previously obtained with Kleiner. Singular Ricci flows were inspired by Perelman's proof of the Poincar\'e and Geometrization Conjectures, which relied on a flow in which singularities were removed by a certain surgery construction. Since this surgery construction depended on various auxiliary parameters, the resulting flow was not uniquely determined by its initial data. Perelman therefore conjectured that there must be a canonical, weak Ricci flow that automatically ``flows through its singularities'' at an infinitesimal scale. Our work on the uniqueness of singular Ricci flows gives an affirmative answer to Perelman's conjecture and allows the study of continuous families of singular Ricci flows leading to the topological applications mentioned above.

The study of stochastic processes and their expected suprema arises in many natural contexts. Some examples include understanding the modulus of continuity for Brownian motion, bounding the maximum singular value of a random matrix, or quantifying discrepancies between a distribution and its corresponding empirical distribution. Early attempts at understanding Gaussian processes dates back as far as Kolmogorov and more recently to Dudley, Fernique, Pisier, and Marcus, among many others. Michel Talagrand has provided a powerful and elegant framework known as the Generic Chaining which unifies and extends the work of these mathematicians to give optimal bounds on the expected suprema of Gaussian processes. The aspirational goal of this talk is to outline chaining \'a la Talagrand while focusing on understanding and not getting lost in the details of the set-up. Material is drawn mostly from Chapter 2 of Talagrand's text ``Upper and Lower Bounds for Stochastic Processes.''

Graph distances have proven quite useful in machine learning/statistics, particularly in the estimation of Euclidean or geodesic distances. The talk will include a partial review of the literature, and then present more recent developments on the estimation of curvature-constrained distances on a surface, and well as on the estimation of Euclidean distances based on an unweighted and noisy neighborhood graph.

Using the Torelli theorem for K3 surfaces of Pyatetskii-Shapiro and
Shafarevich one can describe the automorphism group of a K3 surface over
${\mathbb C}$ up to finite error as the quotient of the orthogonal group
of its Picard lattice by the subgroup generated by reflections in
classes of square -2. We will give a similar description valid over an
arbitrary field in which the reflection group is replaced by a certain
subgroup. We will then illustrate this description by giving several
examples of interesting behaviour of the automorphism group, and by
showing that the automorphism groups of two families of K3 surfaces that
arise from Diophantine problems are finite. This is joint work with
Martin Bright and Ronald van Luijk (University of Leiden).

Thromboembolism, one of the leading causes of morbidity and mortality worldwide, is characterized by formation of obstructive intravascular clots (thrombi) and their mechanical breakage (embolization). A novel two-dimensional multi-phase computational model will be described that simulates active interactions between the main components of the clot, including platelets and fibrin. It can be used for studying the impact of various physiologically relevant blood shear flow conditions on deformation and embolization of a partially obstructive clot with variable permeability. Simulations provide new insights into mechanisms underlying clot stability and embolization that cannot be studied experimentally at this time. In particular, multi-phase model simulations, calibrated using experimental intravital imaging of an established arteriolar clot, show that flow-induced changes in size, shape and internal structure of the clot are largely determined by two shear-dependent mechanisms: reversible attachment of platelets to the exterior of the clot and removal of large clot pieces [1]. Model simulations also predict that blood clots with higher permeability are more prone to embolization with enhanced disintegration under increasing shear rate. In contrast, less permeable clots are more resistant to rupture due to shear rate dependent clot stiffening originating from enhanced platelet adhesion and aggregation. Role of platelets-fibrin network mechanical interactions in determining shape of a clot will be also discussed and quantified using analysis of experimental data [2,3]. These results can be used in future to predict risk of thromboembolism based on the data about composition, permeability and deformability of a clot under specific local hemodynamic conditions.

1. Xu S, Xu Z, Kim OV, Litvinov RI, Weisel JW, Alber M. Model predictions of deformation, embolization and permeability of partially obstructive blood clots under variable shear flow. J. R. Soc. Interface 14 (2017) 20170441.

2. Oleg V. Kim, Rustem I. Litvinov, Mark S. Alber & John W. Weisel, Quantitative structural mechanobiology of platelet driven blood clot contraction, Nature Communications 8 (2017) 1274.

3. Samuel Britton, Oleg Kim, Francesco Pancaldi, Zhiliang Xu, Rustem I. Litvinov, John W.Weisel, Mark Alber [2019], Contribution of nascent cohesive fiber-fiber interactions to the non-linear elasticity of fibrin networks under tensile load, Acta Biomaterialia 94 (2019) 514--523.

Combining a multi-armed bandit task and Bayesian computational modeling, we find that humans systematically under-estimate reward availability in the environment. This apparent pessimism turns out to be an optimism bias in disguise, and one that compensates for other idiosyncrasies in human learning and decision-making under uncertainty, such as a default tendency to assume non-stationarity in environmental statistics as well as the adoption of a simplistic decision policy. In particular, reward rate underestimation discourages the decision-maker from switching away from a ``good'' option, thus achieving near-optimal behavior (which never switches away after a win). Furthermore, we demonstrate that the Bayesian model that best predicts human behavior is equivalent to a particular class of reinforcement learning models, thus giving statistical, normative grounding to phenomenological models of human behavior.

By a ``modular form'' for a reductive group $G$ we mean an
automorphic form that has some sort of very nice Fourier expansion. The
classic example are the holomorphic Siegel modular forms, which are
special automorphic functions for the group $\mathrm{Sp}_{2g}$.
Following work of Gan, Gross, Savin, and Wallach, it turns out that
there is a notion of modular forms on certain real forms of the
exceptional groups. I will define these objects and explain what is
known about them.

We introduce the normalized Laplacian matrix and study how its eigenvalues relate to random walks on graphs and isoperimetric inequalities.

In the early '60s, Abraham Robinson scrapped his standard transmission
calc-mobile and built a non-standard (aka automatic) model. His new model
allowed him and others to shift many proofs into a more intuitive gear.
We'll discuss the first time he took his model through the
car- \L o\'s; and d-rive the transfer principle. We'll also get
infinitely close to talking about the hyperreal numbers.

I will discuss compactifications of the moduli space of smooth plane curves of degree d at least 4. We will regard a plane curve as a log Fano pair $(\mathbb{P}^2,aC)$, where a is a rational number, and study the compactifications coming from K stability for general log Fano pairs. We establish a wall crossing framework to study these spaces as a varies and show that, when a is small, the moduli space coming from K stability is isomorphic to the GIT moduli space. We describe all wall crossings for degree 4, 5, and 6 plane curves and discuss the picture for general Q-Gorenstein smoothable log Fano pairs. This is joint work with Kenneth Ascher and Yuchen Liu.

We will talk about Kronheimer and Mrowka's knot concordance
invariant, $s^\sharp$. We compute the invariant for various knots. Our
computations reveal some unexpected phenomena, including that
$s^\sharp$ differs from Rasmussen's invariant $s$, and that it is not
additive under connected sums. We also generalize the definition of
$s^\sharp$ to links by giving a new characterization of the invariant
in terms of immersed cobordisms.

We will discuss central sequence algebras of von Neumann algebras, and provide a class of II$_1$ factors whose central sequence algebra is not the ``tail'' algebra associated to any decreasing sequence of von Neumann subalgebras. This settles a question of McDuff from 1969. We will also discuss an application of these techniques to the notion of tracial stability for groups. This is based on joint work with Adrian Ioana.

The a-function on a Coxeter group W is a function a : W $\rightarrow$ N defined by Lusztig which is intimately related to the partition of W into Kazhdan--Lusztig cells and to the representation theory of the Hecke algebra of W. It is known that the identity element of W is the only element with a-value 0, while a non-identity element has a-value 1 if and only if it has a unique reduced word. However, as its definition relies on the Kazhdan--Lusztig basis of the Hecke algebra, thea-function is often difficult to compute for general elements.
In this talk we will focus on elements of a-value 2, or a-2 elements. We show that a-2 elements arefully commutative in the sense of Stembridge, which allows us to associate to them certain posets called heaps and, in many cases, certain generalized Temperley--Lieb diagrams. Using heaps and Temperley--Lieb diagrams, we conjecture a combinatorial characterization of a-2 elements, classify all Coxeter groups with finitely many a-2 elements, and enumerate a-2 elements for all groups from the classification. Joint with Richard Green.

We will talk about several methods of proving non-asymptotic bounds on
the spectral norm of random matrices including epsilon-net argument and
Slepian's lemma. The main idea is to relate the spectral norm of a
random matrix to the supremum of a random process.

In his thesis, M. Bhargava proved parameterizations and identified local
conditions which he used to give asymptotic counts for $S_4$ quartic and
quintic number fields, ordered by discriminant. This talk will discuss
results in an ongoing project to add detail to Bhargava's work by
considering in addition to the field discriminant, the lattice shape of
the ring of integers in the canonical embedding, and by giving strong
rates with lower order terms in the asymptotics. These results build on
earlier work of Taniguchi-Thorne, Bhargava-Shankar-Tsimerman and
Bhargava-Harron.

I will discuss certain methods used to find so called ``cyclotomic
solutions'' to a system of polynomial equations. These are solutions in
which each coordinate is a root of unity. I will then give a simple real
world example in which these methods come in handy, coming from joint work
with Christian Woll.

Belyi's theorem says that on one hand, a curve over a field of
characteristic 0 that admits a finite map to $\mathbf{P}^1$ ramified
over at most three points must descend to a subfield algebraic over
$\mathbb{Q}$, and on the other hand any curve over such a subfield does
indeed admit such a morphism (without any further base extension). One
might ask whether a similar statement holds over a field of
characteristic $p$, replacing $\mathbb{Q}$ with $\mathbb{F}_p$. For
general morphisms this is false, but it becomes true if we restrict to
tamely ramified morphisms to $\mathbf{P}^1$. Such a statement was
originally given by Saidi, in which the ``other hand'' assertion was made
conditional on the existence of some tamely ramified morphism from the
given curve to $\mathbf{P}^1$.

In the pre-talk, we will discuss how to establish existence of a tamely
ramified morphism in characteristic \mbox{$p\>2$}. This is ``classical'' over an
infinite algebraic extension of $\mathbb{F}_p$; to do it over a fixed
finite field requires a density statement in the style of Poonen's
finite field Bertini theorem.

In the talk proper, we will discuss work of Sugiyama-Yasuda that
establishes the existence of a tamely ramified morphism when the base
field is algebraically closed of characteristic 2. The case where the
base field is finite of characteristic 2 requires a further geometric
reinterpretation of the key construction of Sugiyama-Yasuda; this is
joint work with Daniel Litt and Jakub Witaszek.

Abel's theorem (1824) that the generic polynomial of degree n is
solvable in radicals if and only if $n$ $\<$ 5 is well-known. However, the
classical works of Bring (1786) and Klein (1884) give solutions of the
generic quintic polynomial by allowing certain other ``nice'' algebraic
functions of one variable. For the sextic, it is conjectured that any
solution requires algebraic functions of two variables. In this talk, we
will examine and relate the intrinsic geometries of the known solutions
of the sextic in two variables, extending the work of Green (1978).

An important result due to Sanderson and Ion says that characters of level one Demazure modules are specialized Macdonald polynomials. In this talk, I will introduce a new class of symmetric polynomials indexed by a pair of dominant weights in $\mathfrak{sl}_{n+1}$ which is expressed as a linear combination of specialized symmetric Macdonald polynomials with coefficients defined recursively. These polynomials arose in my own work while investigating the characters of higher level Demazure modules. Using representation theory, we will see that this new family of polynomials interpolates between characters of level one and level two Demazure modules for affine $\mathfrak{sl}_{n+1}$ and give rise to new results in the representation theory of current algebras as a corollary. This is based on joint work with Vyjayanthi Chari, Peri Shereen, and Jeffrey Wand.

In this talk, we will first briefly review the generic chaining method and Dudley's inequality. Then, we will show the application of chaining methods in random dimensionality reduction. We will mainly consider dimensionality reduction with Johnson-Lindenstrauss type embeddings, which are random matrix constructions to reduce the dimension while approximately preserving Euclidean inter-point distances in the data set. In particular, we will consider ``fast'' and ``sparse'' Johnson-Lindenstrauss embeddings. Finally, we summarize some applications of chaining methods in data streaming algorithms and dictionary learning. The material of this talk is based on some lecture notes by Jelani Nelson and some research work by Sjoerd Dirksen, Jean Bourgain and Jelani Nelson.

A key question in population biology is understanding the conditions under which the species of an ecosystem persist or go extinct. Theoretical and empirical studies have shown that persistence can be facilitated or negated by both biotic interactions and environmental fluctuations. We study the dynamics of n interacting species that live in a stochastic environment. Our models are described by $n$ dimensional piecewise deterministic Markov processes. These are processes $(X(t), r(t))$ where the vector $X$ denotes the density of the $n$ species and $r(t)$ is a finite state space process which keeps track of the environment. In any fixed environment the process follows the flow given by a system of ordinary differential equations. The randomness comes from the changes or switches in the environment, which happen at random times. We give sharp conditions under which the populations persist as well as conditions under which some populations go extinct exponentially fast. As an example we look at the competitive exclusion principle from ecology and show how the random switching can `rescue' species from extinction.

Change point detection is a popular tool for identifying locations in a data sequence where an abrupt change occurs in the data distribution and has been widely studied for Euclidean data. Modern data very often is non- Euclidean, for example distribution valued data or network data. Change point detection is a challenging problem when the underlying data space is a metric space where one does not have basic algebraic operations like addition of the data points and scalar multiplication.
In this talk, I propose a method to infer the presence and location of change points in the distribution of a sequence of independent data taking values in a general metric space. Change points are viewed as locations at which the distribution of the data sequence changes abruptly in terms of either its Fr\'echet mean or Fr\'echet variance or both. The proposed method is based on comparisons of Fr\'echet variances before and after putative change point locations. First, I will establish that under the null hypothesis of no change point the limit distribution of the proposed scan function is the square of a standardized Brownian Bridge. It is well known that such con- vergence is rather slow in moderate to high dimensions. For more accurate results in finite sample applications, I will provide a theoretically justified bootstrap-based scheme for testing the presence of change points. Next, I will show that when a change point exists, (1) the proposed test is con- sistent under contiguous alternatives and (2) the estimated location of the change-point is consistent. All of the above results hold for a broad class of metric spaces under mild entropy conditions. Examples include the space of univariate probability distributions and the space of graph Laplacians for networks. I will illustrate the efficacy of the proposed approach in empirical studies and in real data applications with sequences of maternal fertility distributions. Finally, I will talk about some future extensions and other related research directions, for instance, when one has samples of dynamic metric space data. This talk is based on joint work with Prof. Hans-Georg M"uller.

Let $X$ be a smooth affine variety over a finite field of characteristic
$p$. The Dwork-Monsky trace formula is a fundamental tool in
understanding the $L$-functions of $p$-adic representations of
$\pi_1(X)$. We extend this result to the study of representations valued
in a higher-dimensional local ring $R$. The special case
$R=\mathbb{Z}_p[[T]]$ arises naturally in the study
of \'etale $\mathbb{Z}_p$-towers over $X$. Time permitting, we discuss
some spectral-halo type results and conjectures describing the $p$-adic
variation of slopes in certain $\mathbb{Z}_p$-towers.

The complement system plays a major role in the immune system to recognize and clear invading pathogens. Although activation of the complement system is tightly controlled, dysregulation leads to a cascade of events that is implicated in autoimmune disorders and infectious diseases. Here, to gain a systems-level understanding of the complement system, we developed two comprehensive quantitative models that describe the biochemical reactions of the complement system under a renal disorder known as C3 glomerulopathy (C3G) and bacterial infection by Neisseria meningitidis. Our C3G model is composed of (ODEs) 290 ordinary differential equations with 142 kinetic parameters that describe the state of complement system under homeostasis and disease (C3G). Furthermore, we introduced therapy states by modeling known inhibitors of the complement system, a compstatin variant (C3 inhibitor) and eculizumab (C5 inhibitor). We then evaluate our system by generating concentration-time profiles of biomarkers such as C3, C3a-desArg, C5, and fC5b-9. Our model shows compstatin treatment to have strong restorative effects on early-stage biomarkers such as C3 and C3a-desArg, whereas eculizumab has strong restorative effects on late-stage biomarkers C5 and fC5b-9. These results also implicate the need for patient-tailored therapies that target early stage complement activation under C3G and that treatment may depend on the specific manifestations of a patient's genetic profile in complement regulatory function. After our modeling efforts in C3G, we continued complement modeling for Neisseria meningitidis. This pathogen can cause meningococcal disease and studies have shown individuals with deficiencies in the complement system, notably the membrane attack complex (MAC), have a 7,000- to 10,000-fold higher risk of developing meningococcal disease. We subsequently developed a quantitative biochemical model to assess dynamics of MAC production. bacterial infection by composed of 670 ODEs with 328 kinetic parameters Our model shows highest MAC deposition on Neisseria meningitidis is mainly dependent on a concentration barrier where immune activators are at least three orders of magnitude higher than regulators. This makes rising levels of immune regulators as early intervention markers for the sporadic meningococcal disease. Altogether, our models serve as frameworks to simulate disease-specific scenarios. Subsequently, this can lead to early diagnosis, patient-specific treatments, and aid in drug discovery to identify novel inhibitory sites.

Consider a smooth connected closed two-dimensional Riemannian manifold
$\Sigma$ with positive Gauss curvature. If $u$ is a $C^2$ isometric
embedding of $\Sigma$, then $u (\Sigma)$ is convex. In the fifties Nash
and Kuiper showed, astonishingly, that this is not necessarily true when
the map is $C^1$. It is expected that the threshold at which isometric
embeddings "change nature" is the $\frac{1}{2}$-Hoelder continuity of
their derivatives, a conjecture which shares a striking similarity with
a (recently solved) problem in the theory of fully developed turbulence.

In my talk I will review several plausible reasons for the threshold
and a very recent work, joint with Dominik Inauen, which indeed shows a
suitably weakened form of the conjecture.

We will define the p-adic numbers and discuss some basic properties,
and in the process try to give the audience a sense of how analysis over
the p-adics compares to (and differs from) analysis over the real or
complex numbers. In particular, we hope to convey why a nice theory of
analytic geometry over the p-adics might be hard to come by, and then
discuss some of the more sophisticated methods created in order to develop
such a theory.

From number theory to knots, Hecke algebras have applications within
many areas of mathematics.

In this talk we describe a pictorial calculus for computing convolution
products in affine Hecke algebras over fields of characteristic zero.

Convolution products of this type have been understood since the work of
Iwahori and Matsumoto [1965].

However, using results of Parkinson, Ram and Schwer [2006], we can now
draw pictures illustrating the rich combinatorial nature of these products.

We describe this pictorial calculus in the example of
$\mathrm{SL}_3(\mathbb{Q}_p)$. Its applicability is limited to
characteristic zero.

Let $W^{(n)}$ be the $n$-letter word obtained by repeating a fixed word $W$, and let $R_n$ be a random $n$-letter word over the same alphabet. We show several results about the length of the longest common subsequence (LCS) between $W^{(n)}$ and $R_n$; in particular, we show that its expectation is $\gamma_W n-O(\sqrt{n})$ for an efficiently-computable constant $\gamma_W$.\\

This is done by relating the problem to a new interacting particle system, which we dub ``frog dynamics''. In this system, the particles (`frogs') hop over one another in the order given by their labels. Stripped of the labeling, the frog dynamics reduces to a variant of the PushASEP.\\

In the special case when all symbols of $W$ are distinct, we obtain an explicit formula for the constant $\gamma_W$ and a closed-form expression for the stationary distribution of the associated frog dynamics.\\

Froggies on a pond\\
They get scared and hop along\\
Scaring others too.\\

Their erratic gait\\
Gives us tools to calculate\\
LCS of words.\\

(Joint work with Boris Bukh)

The (Euclidean) isoperimetric inequality says that any set has larger perimeter than a ball with the same area. The quantitative isoperimetric inequality says that the difference in perimeters is bounded from below by the square of the distance from our set E to the ``closest'' ball of the same area. In this talk, we will discuss an extension of this result to closed Riemannian manifolds with analytic metrics. In particular, we show that a similar inequality holds but with the distance raised to a power that depends on the geometry. We also have examples which show that a greater power than two is sometimes necessary and that the analyticity condition is necessary. This is joint work with O. Chodosh (Stanford) and L. Spolaor (UCSD).

Grothendieck's inequality guarantees that a certain discrete optimization problem-optimizing a bilinear form
over +1, -1 inputsis equivalent up to a constant, called Grothendieck's constant, to a continuous optimization
problem -- optimizing the same bilinear form over unit vectors in a Hilbert space. This is important for
convex relaxation, because the former contains NP-hard problems such as max-cut, whereas the latter is
a semidefinite program and can be solved in polynomial time. A world apart from convex relaxation is
algebraic computational complexity, where it is well-known that the exponent of matrix multiplication is
exactly the sharp lower bound for the (log of) tensor rank of the Strassen matrix multiplication tensor.
We show that Grothendieck's constant is the sharp upper bound on a tensor norm of Strassen matrix
multiplication tensor. Hence these two important quantities from disparate areas of theoretical computer
science Strassen's exponent of matrix multiplication and Grothendieck's constantare just different measures
of the same underlying tensor. This allows us to rewrite Grothendieck's inequality as a norm inequality for
a 3-tensor, which in turn gives a family of natural generalized inequalities. We show that these are locally
sharp and prove that Grothendieck's inequality is unique.

A fundamental question in modern cell biology is how cellular and subcellular structures are formed and maintained given their particular molecular components. How are the different shapes, sizes, and functions of cellular organelles determined, and why are specific structures formed at particular locations and stages of the life cycle of a cell? In order to address these questions, it is necessary to consider the theory of self-organizing non-equilibrium systems. We are particularly interested in identifying and analyzing novel mechanisms for pattern formation that go beyond the standard Turing mechanism and diffusion-based mechanisms of protein gradient formation. In this talk we present three examples of non-classical biological pattern formation: (i) Space-dependent switching diffusivities and cytoplasmic protein gradients in the C. elegans zygote (ii) Transport models of cytoneme-based morphogenesis. (iii) Hybrid Turing mechanism for the homeostatic control of synaptogenesis in C. elegans.

Free probability is a subfield of mathematics at the intersection of
operator algebras, complex analysis, probability, and combinatorics. It is
used, among other things, to study the ``$n=\infty$'' case of various $n
\times n$ random matrix models. A concept of central importance in free
probability is \textit{free independence}, the ``noncommutative analogue''
of independence (of random variables) from classical probability. The goal
of this talk is to develop a rigorous understanding of the throw-away
clause in the previous sentence with an interesting mix of analysis and
algebra. Time permitting, we may also discuss why classical independence
and free independence are in a precise sense the only ``reasonable''
notions on independence.

Khovanov homology is a knot invariant, which generalizes and more specifically categorifies the Jones
polynomial of knots. However, in the construction of Khovanov homology we seem to make several
choices in the Frobenius algebra that we consider. In this talk, we will discuss a distinct alternative to
Khovanov homology known as Lee theory, in which we make different choices on our Frobenius algebra.
These different choices break the q-grading, but allow us to create a filtration, and use this to define a
Rasmussen's s-invariant. We will describe several properties and applications of this invariant.

The notion of an elliptic partial differential equation (PDE) goes back at least to 1908, when it appeared in a paper J. Hadamard. In this talk we present a recently discovered structural condition, called $p$-ellipticity, which generalizes classical ellipticity. It was co-discovered independently by Carbonaro and Dragicevic on one hand, and Pipher and myself on the other, and plays a fundamental role in many seemingly unrelated aspects of the $L^p$ theory of elliptic complex-valued PDE. So far, $p$-ellipticity has proven to be the key condition for:

(i) convexity of power functions (Bellman functions)
(ii) dimension-free bilinear embeddings,
(iii) $L^p$-contractivity and boundedness of semigroups $(P_t^A)_{t>0}$
associated with elliptic operators,
(iv) holomorphic functional calculus,
(v) multilinear analysis,
(vi) regularity theory of elliptic PDE with complex coefficients.

During the talk, I will describe my contribution to this development, in particular to (vi).

Given an affine Coxeter group W, the corresponding Shi arrangement is a refinement of the corresponding Coxeter hyperplane arrangements which was introduced by Shi to study the Kazhdan-Lusztig cells for W. Shi showed, in particular, that each region of a Shi arrangement contains exactly one element in W of minimal length. Garside shadows in W were introduced to study the word problem of the corresponding Artin-Tits (braid) group and turns out to produce automata to study the combinatorics of reduced words in W.

In this talk, we will discuss the following conjecture: the set of minimal length elements of the regions in a Shi arrangement is a Garside Shadow. The talk will be illustrated by the example of the affine symmetric group.

We first talk about Gage's area-preserving flow (GAPF) for simple, smooth and star-shaped curves. It is shown that GAPF drives a centrosymmetric initial curve into a circle as time tends to infinity. Then we generalize GAPF to deform one convex curve to another. This work gives a partial answer to Yau's problem on realizing Whitney-Graustein Theorem via curvature flows.

This talk will be a short review of the construction of the stochastic integral and some basic properties. The purpose is to remind the group of how the integral is defined, the importance of the Ito Isometry, and the Ito Formula.

The growth in scope and complexity of modern data sets presents the field of statistics and data science with numerous inferential and computational challenges, among them how to deal with various forms of heterogeneity. Latent variable models provide a principled approach to modeling heterogeneous collections of data. However, due to the over-parameterization, it has been observed that parameter estimation and latent structures of these models have non-standard statistical and computational behaviors. In this talk, we provide new insights into these behaviors under mixture models, a building block of latent variable models.

From the statistical viewpoint, we propose a general framework for studying the convergence rates of parameter estimation in mixture models based on Wasserstein distance. Our study makes explicit the links between model singularities, parameter estimation convergence rates, and the algebraic geometry of the parameter space for mixtures of continuous distributions.

From the computational side, we study the non-asymptotic behavior of the EM algorithm under the over-specified settings of mixture models in which the likelihood need not be strongly concave, or, equivalently, the Fisher information matrix might be singular. Focusing on the simple setting of a two-component mixture fit with equal mixture weights to a multivariate Gaussian distribution, we demonstrate that EM updates converge to a fixed point at Euclidean distance $\mathcal{O}((d/n)^{1/4})$ from the true parameter after $\mathcal{O}((n/d)^{1/2})$ steps where $d$ is the dimension.

From the methodological standpoint, we develop computationally efficient optimization-based methods for the multilevel clustering problem based on Wasserstein distance. Experimental results with large-scale real-world datasets demonstrate the flexibility and scalability of our approaches. If time allows, we further discuss a novel post-processing procedure, named Merge-Truncate-Merge algorithm, to determine the true number of components in a wide class of latent variable models.

Low-rank matrix recovery from structured measurements has been a topic of intense study in the last decade and many important problems such as blind deconvolution and matrix completion have been formulated in this framework. An important benchmark method to solve these problems is to minimize the nuclear norm, a convex proxy for the rank. A common approach to establish recovery guarantees for this convex program relies on the construction of a so-called approximate dual certificate. However, this approach provides only limited insight in various respects. Most prominently, the noise bounds exhibit seemingly suboptimal dimension factors. In this talk, we will discuss a more geometric viewpoint to analyze the blind deconvolution scenario. We find that for both these applications the dimension factors in the noise bounds are not an artifact of the proof, but the problems are intrinsically badly conditioned. We show, however, that bad conditioning only arises for very small noise levels: Under mild assumptions that include many realistic noise levels we derive near-optimal error estimates for blind deconvolution under adversarial noise. At the end, we will briefly discuss how the results can be extended to the scenario of matrix completion.

In this talk, we will focus on how one can deduce some
geometric invariants of an abelian variety or a K3 surface by
studying their Frobenius polynomials. In the case of an abelian variety,
we show how to obtain the decomposition of the endomorphism algebra, the
corresponding dimensions, and centers. Similarly, by studying the
variation of the geometric Picard rank, we obtain a sufficient criterion
for the existence of infinitely many rational curves on a K3 surface of
even geometric Picard rank.

We will discuss stabilization of the Betti numbers of the moduli space of
sheaves on surfaces, and in the special case of the projective plane we
will produce lower bounds on the second Chern class of the sheaves such
that the Betti numbers of their moduli space stabilizes.

Can an integer $n$ be represented as a sum of two squares $n=x^2+y^2$? If so, how many different representations are there? We begin with the answers to these classical questions due to Fermat and Jacobi. We then illustrate Hurwitz's class number formula for binary quadratic forms, and put all these classical formulas under the modern perspective of the Siegel-Weil formula. We explain how the latter perspective led Gross-Keating to discover a new type of identity between arithmetic intersection numbers on modular surfaces and derivatives of certain Eisenstein series. After outlining the influential program of Kudla and Rapoport for generalization to higher dimensions, we report a recent proof (joint with W. Zhang) of the Kudla-Rapoport conjecture and hint at the usage of the uncertainty principle in the proof.

Dear Math Enthusiast, have you ever been afraid that mathematics
contains nothing of substance? Are you afraid of realizing that, maybe,
all the proofs near and dear to you are actually easy corollaries of some
deeper, horrendous technicality with no interesting geometric,
number-theoretic, algebraic, combinatorial, or analytic interpretation?
Well, you are in luck! We will be providing the simulation of such an
experience. In this week's Food for Thought talk, we will present
interesting problems, such as a solution to a version of Fermat's Last
Theorem in a field with large enough characteristic, only to realize that
these problems boil down to dry logical tricks!

In this talk, we will introduce a new bootstrap algorithm for estimating linear combinations of coefficients in high dimensional threshold ridge regression model. In addition, we will also introduce a new definition about prediction and demonstrate how bootstrap calibration algorithm achieves this goal.

Cut and project sets are well-studied models of almost-periodic discrete subsets of $R^d$. In 2014 Marklof and Strombergsson introduced a natural class of random processes which generate cut and project sets in a way which is invariant under the group $ASL(d,R)$. Using Ratner's theorem and the theory of algebraic groups we classify all these measures. Using the classification we obtain results analogous to those of Siegel, Rogers, and Schmidt in geometry of numbers: summation formulas and counting points in large sets for typical cut and project sets. Joint work with Rene Ruehr and Yotam Smilansky.

Researchers around the globe are gathering biomedical information at a massive scale. We develop algorithms to compress this data that enable computation on the reduced representation. In this talk, I will discuss how we can leverage the low-dimensional true structure of biological data manifolds in order to build useable compact geometric summaries of this data. I will highlight our latest work on single-cell transcriptomic datasets, that enables an unprecedented scale of data to be effectively pooled from individuals and institutions across nations to enable novel life-saving discoveries.

In this talk, I will show that, given two distinct strictly stable self-expanders (of mean curvature flow) that are asymptotic to the same cone, there is a new self-expander trapped between these two. This is achieved by developing a min-max theory for the relative expander entropy. This is joint with Jacob Bernstein.

How much time does Brownian motion spends at 0? Even though it spends Lebesgue 0 amount of time there, this is not the end of the story. The Local Time for Brownian motion let us go deeper into this question, creating an object that keeps track of how much time Brownian motion spends at 0. Once established, we will discuss applications of Local Time such as Tanaka's Formula, change of variables formula, Engelbert-Schimdt 0-1 Law and Reflected Diffusions.

We revisit the tears of wine problem for thin films in water-ethanol mixtures and present a new model for the climbing dynamics. The new formulation includes a Marangoni stress balanced by both the normal and tangential components of gravity as well as surface tension which lead to distinctly different behavior. The combined physics can be modeled mathematically by a scalar conservation law with a nonconvex flux and a fourth order regularization due to the bulk surface tension. Without the fourth order term, shock solutions must sastify an entropy condition - in which characteristics impinge on the shock from both sides. However, in the case of a nonconvex flux, the fourth order term is a singular perturbation that allows for the possibility of undercompressive shocks in which characteristics travel through the shock. We present computational and experimental evidence that such shocks can happen in the tears of wine problem, with a protocol for how to observe this in a real life setting.

An independent set of size $k$ in a finite undirected graph is a set of $k$ vertices of the graph, no two of which are connected by an edge. The structure of independent sets of size $k$ as $k$ varies is of interest in probability, statistical physics, combinatorics, and computer science. In 1987, Alavi, Malde, Schwenk and Erdos conjectured that the number of independent sets of size $k$ in a tree is a unimodal sequence (this number goes up and then it goes down), and this problem is still open. A variation on this question is: do the number of independent sets of size $k$ form a unimodal sequence for Erdos-Renyi random graphs, or random trees? By adapting an argument of Coja-Oghlan and Efthymiou, we show unimodality for Erdos-Renyi random graphs, random bipartite graphs and random regular graphs (with high probability as the number of vertices in the graph goes to infinity, when the expected degree of a single vertex is large). The case of random trees remains open, as we can only show weak partial results there.

Classically, the transcendence (and even the irrationality) of
odd zeta values is widely conjectured, but yet unproven. However, for
zeta values defined over the rational function field, Jing Yu succeeded
in proving their transcendence in 1991, and many other transcendence
results (including algebraic independence) followed in the intervening
years. In this work (joint with T. Ngo Dac), we prove the algebraic
independence of zeta values defined over the function field of an
elliptic curve. The main technique we use is to construct a Tannakian
category of t-motives whose associated periods contain these zeta values
- thus we may exploit the existence of a motivic Galois group to study
the transcendence degree. We also discuss the difficulties and pathway
to proving algebraic independence for zeta values of function fields of
arbitrary curves.

Option contracts are a type of financial derivative that allow investors to hedge risk and speculate on the variation of an asset's future market price. In short, an option has a particular payout that is based on the market price for an asset on a given date in the future. In 1973, Black and Scholes proposed a valuation model for options that essentially estimates the tail risk of the asset price under the assumption that the price will fluctuate according to geometric Brownian motion. More recently, DeMarzo et al., among others, have proposed more robust valuation schemes, where we can even assume an adversary chooses the price fluctuations. This framework can be considered as a sequential two-player zero-sum game between the investor and Nature. We analyze the value of this game in the limit, where the investor can trade at smaller and smaller time intervals. Under weak assumptions on the actions of Nature (an adversary), we show that the minimax option price asymptotically approaches exactly the Black- Scholes valuation. The key piece of our analysis is showing that Nature's minimax optimal dual strategy converges to geometric Brownian motion in the limit.

I will introduce a program, begun by Dan-Cohen and Wewers, to compute
Minhyong Kim's Selmer varieties using mixed Tate motives. The idea is as
follows. We care about the Galois action on the unipotent fundamental
group of $\mathbf{P}^1 \setminus \{0,1,\infty\}$. This Galois action
lives in a certain category of $p$-adic Galois representations known as
mixed Tate representations. We will see that this category is Tannakian
and has a fairly simple description, in terms of its Ext groups, which
are just Bloch-Kato Selmer groups. The Bloch-Kato Selmer groups are
$p$-adic vector spaces, but they also have a rational structure coming
from algebraic K-theory. The category of mixed Tate motives gives us a
$\mathbb{Q}$-linear Tannakian category that underlies the
$\mathbb{Q}_p$-linear Tannakian category of Galois representations. This
in turn allows us to define a Selmer variety over $\mathbb{Q}$ (rather
than $\mathbb{Q}_p$), and a more explicit understanding of the category
of mixed Tate motives allows us to compute this variety explicitly.

In this talk I'll teach the basics of making a website and how to connect with the ucsd server. If you want to have a working website by the end of the week, it will help to (1) talk to Saul about getting your login information for the ucsd math servers, (2) have a text editor to make the website (I like notepad++), and (3) have a way to SSH to the math server (I like WinSCP). To stall for time I'll also talk about other resources that might be of use to a math phd student. Other people are more than welcome to contribute ideas in this direction!

When $X$ is a Grassmannian, Marian-Oprea-Pandharipande and Toda constructed alternate compactifications of spaces of maps from curves to $X$. The construction has been generalized to a large class of GIT quotients $X=W//G$ by Ciocan-Fontanine-Kim-Maulik and many others. It is called the theory of $\epsilon$-stable quasimaps. In this talk, we will introduce permutation-equivariant K-theoretic epsilon-stable quasimap invariants and prove their wall-crossing formulae for all targets in all genera. The wall-crossing formulae generalize Givental's K-theoretic toric mirror theorem in genus zero. In physics literature, these K-theoretic invariants are related to the $3d N = 2$ supersymmetric gauge theories studied by Jockers-Mayr, and the wall-crossing formulae can be interpreted as relations between invariants in the UV and the IR phases of the $3d$ gauge theory. It is based on joint work with Yang Zhou.

In this defense I will talk about two lines of research during my PhD. I will develop some scan statistics and derive their asymptotic distributions, using moderate deviation, large deviation, Kolmogorov's theorem and a Poisson approximation. I will briefly mention their applications in global testing and multiple testing, resulting from the work before advancement. Next I will talk about another project on causal inference, where the causal effects of etanercept on birth defects are investigated, in the presence of missing outcomes not at random, left truncation, observational nature, and rare events.

I will give a light introduction to the theory of quantum graphs. Quantum graphs are non-commutative generalizations of finite graphs that arise naturally in many areas, including the study of zero-error capacities of quantum channels, quantum symmetry groups of graphs, and in the theory of non-local games. I will give an overview of some of these connections and explain some ongoing joint work with Kari Eifler, Christian Voigt and Moritz Weber, where we generalize the well-known graph $C^\ast$-algebra construction to the setting of quantum graphs.

In this talk we will introduce a generalization of Babuska-Lax-Milgram theorem that is particularly well-suited for the study of global existence of solutions to certain parabolic PDEs on compact manifolds.

As part of a general theory for the isomorphism problem for actions of amenable groups, Ornstein and Weiss proved that any two Poisson point processes are isomorphic as measure-preserving actions. We give an elementary construction of an isomorphism between Poisson point processes that is finitary. This is joint work with Terry Soo.

The Laplacian is a canonical second order elliptic operator defined on any Riemannian manifold. The study of upper bounds for its eigenvalues under the volume constraint is a classical problem of spectral geometry going back to J. Hersch, P. Li and S.-T. Yau. The particular interest to this problem stems from a surprising connection to the theory of harmonic maps to spheres. In the present talk we will survey some recent results in the area with an emphasis on the role played by the energy index of harmonic maps. In particular, we will discuss some recent applications, including a min-max proof of the existence of optimisers and an optimal upper bound for Laplacian eigenvalues on the projective plane.

We show how to express the solution to an ordinary differential
equation as a sum over rooted trees, discover a group structure on the set
of rooted trees, extend it to a Hopf algebra related to quantum
renormalization, and finally bring ourselves back down to Earth and derive
the coefficients used in Runge-Kutta integrators.

In the first part of the talk (based on joint work with L.Coburn, J. Sj"ostrand, and F. White) we discuss continuity conditions for Toeplitz operators acting on spaces of entire functions with quadratic exponential weights (Bargmann spaces), in connection with a conjecture by C. Berger and L. Coburn, relating Toeplitz and Weyl quantizations. In the second part of the talk (based on joint work in progress with J. Sj"ostrand), we discuss elements of a semiglobal approach to analytic second microlocalization with respect to a hypersurface, in the semiclassical case, based on the study of the heat evolution semigroup for large times. We describe properties of the associated exponentially weighted spaces of holomorphic functions with
($h$--dependent) plurisubharmonic exponents and construct asymptotic Bergman projections in such spaces.

In his work on modularity theorems, Wiles proved a numerical criterion for a map of
rings R $\rightarrow$ T to be an isomorphism of complete intersections. He used this to show
that certain deformation rings and Hecke algebras associated to a mod p Galois
representation at non-minimal level were isomorphic and complete intersections,
provided the same was true at minimal level. In addition to proving modularity
theorems, this numerical criterion also implies a connection between the order of a
certain Selmer group and a special value of an L-function. In this talk I will
consider the case of a Hecke algebra acting on the cohomology a Shimura curve
associated to a quaternion algebra. In this case, one has an analogous map of ring
R $\rightarrow$ T which is known to be an isomorphism, but in many cases the rings R and T
fail to be complete intersections. This means that Wiles' numerical criterion will fail
to hold. I will describe a method for precisely computing the extent to which the
numerical criterion fails (i.e. the ``Wiles defect''), which will turn out to be
determined entirely by local information at the primes dividing the discriminant of
the quaternion algebra. This is joint work with Gebhard Bockle and Chandrashekhar
Khare.

Modeling and tractable computation form two fundamental but competing pillars of data science; indeed, fitting good models to data is often computationally challenging in modern applications. Focusing on the canonical tasks of ranking and regression, I introduce problems where this tension is immediately apparent, and present methodological solutions that are both statistically sound and computationally tractable.

I begin by describing a class of ``permutation-based'' models as a flexible alternative to parametric modeling in a host of inference problems including ranking from ordinal data. I introduce procedures that narrow a conjectured statistical-computational gap, demonstrating that carefully chosen non-parametric structure can significantly improve robustness to mis-specification while maintaining interpretability. Next, I address a complementary question in the context of convex regression, where I show that the curse of dimensionality inherent to non-parametric modeling can be mitigated via parametric approximation. Our provably optimal methodology demonstrates that it is often possible to enhance the interpretability of non-parametric models while maintaining important aspects of their flexibility.

We present one aspect of Bergman kernels, which includes the
following topics:
Fourier integral operators with complex phase and projections for certain scalar equations,
Bergman and Szeg"o kernels for strictly pseudoconvex domains,
Projections for exponentially weighted spaces and powers of complex line bundles.

Oprea and Pandharipande studied the virtual Euler characteristic of Quot schemes on surfaces. Based on the calculations in several cases, they conjectured the rationality of the generating series of the virtual invariants. In this talk, I will explain the virtual $\chi_{-y}$-genera of Quot schemes, thus refining the work of [OP]. The main result expresses the Quot scheme invariants universally in terms of Seiberg-Witten invariants of D"urr, Kabanov, and Okonek. Based on the calculations in several cases and the blow up formula, we resolve the $\chi_{-y}$-genus analogue of the rationality conjecture of [OP], for all surfaces with $p_g>0$. In addition, the reduced Quot scheme invariants for $K3$ surfaces will be discussed with connections to the Kawai-Yoshioka formula.

Building on previous work of Rozansky and Willis, we generalise Rasmussen's s-invariant to links in connected sums of $S^1 \times S^2$. Such an invariant can be computed by approximating the Khovanov-Lee complex of a link in $\#^r S^1 \times S^2$ with that of appropriate links in $S^3$. We use the approximation result to compute the s-invariant of a family of links in $S^3$ which seems otherwise inaccessible, and use this computation to deduce an adjunction inequality for null-homologous surfaces in a (punctured) connected sum of $\bar{CP^2}$. This inequality has several consequences: first, the s-invariant of a knot in the three-sphere does not increase under the operation of adding a null-homologous full twist. Second, the s-invariant cannot be used to distinguish $S^4$ from homotopy 4-spheres obtained by Gluck twist on $S^4$. We also prove a connected sum formula for the s-invariant, improving a previous result of Beliakova and Wehrli. We define two s-invariants for links in $\#^r S^1 \times S^2$. One of them gives a lower bound to the slice genus in $\natural^r S^1 \times B^3$ and the other one to the slice genus in $\natural^r D^2 \times S^2$ . Lastly, we give a combinatorial proof of the slice Bennequin inequality in $\#^r S^1 \times S^2$.

In this talk, we describe self-generating pointwise lower bounds for solutions of the non-cutoff Boltzmann equation, which models the evolution of the particle density of a diffuse gas. These lower bounds imply that vacuum regions in the initial data are filled instantaneously, and also lead to key coercivity estimates for the collision operator. As an application, we can remove the assumptions of mass bounded below and entropy bounded above, from the known criteria for smoothness and continuation of solutions. The proof strategy also applies to the Landau equation, and we will compare this (deterministic) proof with our prior (probabilistic) proof of lower bounds for the Landau equation. This talk is based on joint work with Chris Henderson and Andrei Tarfulea.

Complex networks are said to exhibit four key properties: large scale, evolving over time, small world properties, and power law degree distribution. The Preferential Attachment Model (Barab\'asi--Albert, 1999) and the ACL Preferential Attachment Model (Aiello, Chung, Lu, 2001) for random networks, evolve over time and rely on the structure of the graph at the previous time step. Further models of complex networks include: the Iterated Local Transitivity Model (Bonato, Hadi, Horn, Pralat, Wang, 2011) and the Iterated Local Anti-Transitivity Model (Bonato, Infeld, Pokhrel, Pralat, 2017). In this talk, we will define and discuss the Iterated Local Model. This is a generalization of the ILT and ILAT models, where at each time step edges are added deterministically according to the structure of the graph at the previous time step.

In this talk, we will first introduce several versions of existence and uniqueness theorems for stochastic differential equations(SDEs). Then we will focus on a special type of SDEs, Ito diffusions. We will give a detailed proof of the strong Markov property of Ito diffusions. This talk can be viewed as an extension of math 286 and material is drawn mostly from Chapter 10 of Chung and Williams' $Introduction$ $to$ $Stochastic$ $Integration$.

Though promised to solve intractable problems, quantum computers are much noisier than classical computers. To make quantum computers practical, efficient error correction schemes have to be developed. In the first half of the talk, I will first introduce the theory of Quantum Error Correction (QEC) and current proposals of using QEC to achieve fault-tolerant quantum computing, then I will talk about a new scheme that can greatly lower the cost of QEC. In the second half, I will talk about a promising near-term qubit encoding called the GKP qubit that has error-correcting abilities and a protocol to prepare such qubits.

The Gaussian multiplicative chaos is a relatively new universal object in probability that has many interesting geometric properties.The characteristic polynomial of many classes of random matrices is, in many cases conjecturally, one class of finite approximation to these random measures. Great progress has been made on showing the random matrices from specific ``circular ensembles" converge to the GMC.
Likewise, some progress has been made for unitarily-invariant random matrices. We show some new partial progress in showing the ``Gaussian-beta ensemble'' has a GMC limit. This we do by using the representation of its characteristic polynomial as an entry in a product of independent random two-by-two matrices. For a point $z$ in the complex plane, at which the transfer matrix is to be evaluated, this product of transfer matrices splits into three independent factors, each of which can be understood as a different dynamical system in the complex plane.
This is joint work with Gaultier Lambert.

Central leaves in the special fiber of Shimura varieties are the loci where the isomorphism class of the universal $p$-divisible group remains constant. The Hecke orbit conjecture asserts that every prime-to-$p$ Hecke orbit in a PEL type Shimura variety is dense in the central leaf containing it. This conjecture is proved for Hilbert modular varieties by C.-F. Yu, and for Siegel modular varieties by Chai and Oort. In this talk I give an overview of the conjecture and present my work that generalizes Chai and Oort's strategy to irreducible components of certain Newton strata on Shimura varieties of PEL type.

A smooth manifold is a topological structure that admits some standard
calculus constructions. A vector bundle over a manifold is a smooth choice
of a vector space at every point of the manifold. One motivation for such
an object is that the differential of a map between two manifolds is a map
between their tangent bundles. At first glance, vector bundles seem as
though they should be Cartesian products of the manifold with a vector
space. In general, this is false, and certain ``characteristic classes''
are the explicit obstructions. In this talk, we will construct these
classes and discuss some theorems explaining why they are of interest to
many mathematicians.

Quantum computing is receiving increasing attention as
it promises to revolutionize the current computational landscape. The
advent of quantum computers will be particularly disruptive in the
field of cryptography, where characteristically quantum properties
like entanglement and the ``no-cloning theorem'' open up a plethora of
novel opportunities. In this talk, I will describe two broad, and
connected, research questions. The first is: if a quantum device is
meant to perform tasks that are beyond the reach of classical
computers, how can a classical user trust that her quantum device is
behaving as intended? I will focus particularly on the connections of
this question with foundational questions in quantum information. The
second is: once we trust our quantum devices, what kinds of
cryptographic tasks can we realize that are beyond the reach of
classical computers? I will describe some concrete examples.

In my defense, I will talk about three relative topics relative to sparse recovery and representation of signals. I will start with the topic of recovering the low-rank matrix from incomplete measurements with prior information. Signal recovery assumes that we know the sensing matrix i.e. the linear transformation. But sometimes, we want the sparse representation of signals without knowing the transformation between the signal and its representation. Thus in the second topic, I'll talk about a novel algorithm that allows us to learn the linear transformation as well as the sparse representation and admits the transformation in complexity $O(n\log n)$ for a $n$ dimensional input signal. In the third topic, I'll introduce the usage of representation learning assuming that the transformation is a more complex function. I'll propose a deep neural network structure that can be used in image generation and introduce a specific application about it raised in computer game industry.

Quantum computing has notable impacts on computer
science in recent years. While quantum computers are about to achieve
so-called ``quantum supremacy'' (i.e., solving some
classically-intractable computational tasks), it is the right time to
understand the capabilities and limits of quantum computers. In this
talk, I will address the following two questions: 1) What is the power
of near-term quantum computers? 2) What speedup can general quantum
computers achieve for problems in machine learning and data analysis?
We will first see that a general quantum computer is strictly more
powerful than small-depth quantum computers in the presence of
classical computers. Then, I will show quantum-inspired classical
algorithms for problems including SVM, low-rank linear system,
low-rank SDP, and more. Our algorithms running in time polylog(n) are
asymptotically as good as existing quantum ones. This result also
implies that existing quantum machine learning algorithms have not
achieved exponential quantum speedups. Finally, I will discuss
polynomial quantum speedups for fundamental problems in data analysis
and their limits under plausible assumptions in complexity theory.

The position states of the harmonic oscillator describe
the location of a particle moving on the real line. Similarly, the
phase difference between two superconductors on either side of a
Josephson junction takes values in the configuration space of a
particle on a circle. More generally, many physical systems can be
described by a basis of ``position states,'' describing a particle
moving on a more general configuration or state space. Most of this
space is usually ignored due to the energy cost required to pin a
particle to a precise ``position''. However, as our control over quantum
systems improves, utilizing more of this higher-energy space harbors
benefits for protecting quantum information and probing quantum
matter. I will discuss quantum applications taking advantage of state
spaces associated with the harmonic oscillator, as well as molecular
rotational and nuclear states.\\

Bio: Victor V. Albert received his Bachelor's degree in physics
and mathematics from the University of Florida in 2010 and his
Ph.D. in physics from Yale University in 2017. He is currently a Lee
A. DuBridge Postdoctoral Scholar in Physics at the California
Institute of Technology. He pursues an interdisciplinary line of
research in quantum science, including open quantum systems,
error-correction, experimental realizations, and topological band
theory.

This is the first talk in a series on ``Sato-Tate
distributions''. See https://sites.google.com/view/vantageseminar for a
complete listing.

We will start by giving a brief introduction to representation stability
and combinatorial categories. Then we will introduce Fulton and
MacPherson's ``wonderful compactifications'' of configuration spaces,
and describe how they may be studied through the lens of representation
stability. In particular, we show that under a mild hypothesis we can
approach the representation theory of these spaces using the
combinatorial category $FS^{op}$. We end by discussing an attempt at
showing that these spaces do exhibit representation stability, although
to date the results of this approach have been fairly underwhelming.
This talk can be accessed at https://ucsd.zoom.us/j/309940113 .

Entanglement is an essential component of quantum
computation, information, and communication. Its applications range
from quantum key distribution and secret sharing to quantum
sensing. These applications drive the increasing need for a quantum
switching network that can supply end-to-end entangled states to
groups of endpoints that request them. To this end, I study a quantum
switch that distributes entanglement to users in a star topology. I
will present models for variants of this system and derive expressions
for switch capacity and the expected number of qubits stored in memory
at the switch.

Much of this work focuses on bipartite entanglement switching. For
this case, I will discuss how performance metrics are affected by
decoherence and link heterogeneity. In this talk, I will also discuss
a work wherein we explore a set of switching policies for a switch
that can serve both bipartite and tripartite entangled states. I will
conclude the talk with a discussion of future research directions and
a long-term vision of leveraging tools from performance evaluation to
analyze and help guide the design of future quantum networks.\\

Bio: Gayane Vardoyan is a PhD candidate in the College of
Information & Computer Sciences, at the University of Massachusetts,
Amherst. She is advised by Prof. Don Towsley. She holds a B.S. in
Electrical Engineering and Computer Sciences from the University of
California, Berkeley. Her research interests include the performance
evaluation of classical and quantum communication systems.

She interned at Inria, Sophia Antipolis during the summer of
2017. Previously, she worked at the Argonne National Lab and the
Computation Institute at the University of Chicago. She was a
participant of the Rising Stars 2019 academic career workshop for
women in EECS.

We shall discuss a classification result of gradient shrinking solitons.

The first demonstration of quantum supremacy in October
2019 was a major achievement of experimental physics. At the same
time, it relied on important developments in theoretical computer
science. In this talk I will describe my recent work laying the
complexity-theoretic foundations for Google/UCSB's quantum supremacy
experiment, providing evidence that their device is exponentially
difficult to simulate with a classical computer. This crossroad
between complexity theory and quantum physics also offers new insights
into both disciplines. For example, I will describe how techniques
from quantum complexity theory can be used to settle purely classical
problems.

Specifically, I will give a quantum argument which nearly resolves the
approximate degree composition conjecture, generalizing nearly 20
years of prior work. In a different direction, I will show that the
notion of computational pseudorandomness from complexity-based
cryptography has fundamental implications for black hole physics and
the theory of quantum gravity.\\

Bio: Adam Bouland is a postdoctoral researcher in computer science
at the University of California, Berkeley. He completed his Ph.D. in
computer science at MIT in 2017, where he was advised by Scott
Aaronson. Prior to that he completed master's degrees in computer
science and mathematics at the University of Cambridge, as well as a
B.S. in computer science/mathematics and physics at Yale. His research
focuses on the theory of quantum computation and its connections with
classical complexity theory and physics.

Grothendieck's section conjecture predicts that over arithmetically interesting fields (e.g. number fields or p-adic fields), rational points on a smooth projective curve X of genus at least 2 can be detected via the arithmetic of the etale fundamental group of X. We construct infinitely many curves of each genus satisfying the section conjecture in interesting ways, building on work of Stix, Harari, and Szamuely. The main input to our result is an analysis of the degeneration of certain torsion cohomology classes on the moduli space of curves at various boundary components. This is (preliminary) joint work with Padmavathi Srinivasan, Wanlin Li, and Nick Salter.

Biharmonic maps are maps between Riemannian manifolds which are critical points of the bi-energy. They are solutions of a system of 4thorder PDEs and they include harmonic maps and biharmonic functions as special cases. Biharmonic submanifolds (which include minimal submanifolds as special cases) are the images of biharmonic isometric immersions. The talk will review some problems, including classification of biharmonic submanifolds in space forms, biharmonic maps into spheres, biharmonic conformal maps, and unique continuation theorems, studied in this field and their progress since 2000. The talk also presents some recent work on equivariant biharmonic maps and the stability and index of biharmonic hypersurfaces in space forms.

For a moduli space $M$ of stable sheaves over a K3 surface $X$, we propose a series of conjectural identities in the Chow rings $CH_\ast (M\times X^l)$, $l\geq 1$, generalizing the classic Beauville-Voisin identity for a K3 surface. We emphasize consequences of the conjecture for the structure of the tautological subring $R_\ast (M)\subset CH_\ast (M)$. We prove the proposed identities when $M$ is the Hilbert scheme of points on a K3 surface. This is joint work with L. Flapan, A. Marian and R. Silversmith.

I will illustrate a recent result obtained in collaboration with L. Spolaor and B. Velichkov concerning the regularity of the free boundaries in the two phase Bernoulli problems. The new main point is the analysis of the free boundary close to branch points, where we show that it is given by the union of two $C^1$ graphs. This complete the analysis started by Alt Caffarelli Friedman in the '80s.

The P=W conjecture by de Cataldo, Hausel, and Migliorini suggests a surprising connection between the topology of Hitchin systems and Hodge theory of character varieties. In this talk, we will focus on interactions between compact and noncompact hyper-Kaehler geometries. Such connections, together with symmetries coming from the moduli of compact hyper-Kaehler manifolds, lead to new progress on the P=W conjecture. I will discuss the results we obtained and the difficulties we met. Based on joint work with Mark de Cataldo and Davesh Maulik.

Recently Okounkov proved Nekrasovs conjecture expressing the
partition function of K-theoretic DT invariants of the Hilbert scheme of
points Hilb($\mathbb C^3$, points) on affine 3-space as an explicit
plethystic exponential. We generalise Nekrasovs formula to higher rank,
where the Quot scheme of finite length quotients of the trivial rank $r$
bundle replaces Hilb($\mathbb C^3$,points). This proves a conjecture of
Awata-Kanno. Specialising to cohomological invariants, we obtain the
statement of Szabos conjecture. We discuss some further applications if
time permits. This is joint work with Nadir Fasola and Sergej Monavari.

The Iwasawa Theory of Selmer groups provides a natural way for p-adic approach to the celebrated Birch and Swinnerton Dyer conjecture. Over a non-commutative p-adic Lie extension, the (dual) Selmer group becomes a module over a non-commutative Iwasawa algebra and its structure can be understood by analyzing ``(left) reflexive ideals'' in the Iwasawa algebra. In this talk, we will start by recalling several existing conjectures in Iwasawa Theory and then we will give an explicit ring-theoretic presentation, by generators and relations, of such Iwasawa algebras and sketch its implications in understanding the (two-sides) reflexive ideals. Generalizing Clozel's work for $SL(2)$, we will also show that such an explicit presentation of Iwasawa algebras can be obtained for a much wider class of p-adic Lie groups viz. uniform pro-p groups and the pro-p Iwahori of $GL(n,Z_p)$. Further, if time permits, I will also sketch some of my recent Iwasawa theoretic results joint with Sujatha Ramdorai.

Using ideas from geometric group theory we provide a novel approach to Green's Conjecture on syzygies of canonical curves. Via a strong vanishing result for Koszul modules we deduce that a general canonical curve of genus $g$ satisfies Green's Conjecture when the characteristic is zero or at least $(g+2)/2$. Our results are new in positive characteristic (and answer positively a conjecture of Eisenbud and Schreyer), whereas in characteristic zero they provide a different proof for theorems first obtained in two landmark papers by Voisin. Joint work with Aprodu, Papadima, Raicu and Weyman.

Yang-Mills flow is one of the classical geometric evolution equations, having much in common both with harmonic map flow and with Ricci flow. I will describe its key properties, focusing on the refined energy identities that have led to new results about the flow in dimension four and on special-holonomy manifolds.

The integral geometric Satake equivalence in mixed characteristic
Abstract: The geometric Satake equivalence establishes a link between
two monoidal categories: the category of perverse sheaves on the local
Hecke stack and the category of finitely generated representations of
the Langlands dual group. It has many important applications in the
study of the geometric Langlands program and number theory. In this
talk, I will discuss the integral coefficient geometric Satake
equivalence in the mixed characteristic setting. It generalizes the
previous results of Lusztig, Ginzburg, Mirkovic-Vilonen, and Zhu. Time
permitting, I will discuss an application of this result in constructing
a Jacquet-Langlands transfer.

Stillman's conjecture asserts the existence of a uniform bound
on the projective dimension of an ideal in a polynomial ring generated
by a fixed number of polynomials of fixed degrees. Ananyan and Hochster
gave a proof of Stillman's conjecture by proving the existence of ``small
subalgebras''. I'll describe a simplification of their approach using
ultraproducts (and in particular, explain what ultraproducts are and all
of the terms mentioned above). This is based on joint work with Daniel
Erman and Andrew Snowden.

On a complete Riemannian manifold with Ricci curvature bounded below, Yau proved a sharp gradient estimate for positive harmonic functions. One comparison space is the hyperbolic space (but it is not unique). This talk will report on the corresponding question for complete noncompact Kaehler manifolds.

Given a semiabelian variety over a number field and a closed
subvariety thereof, it is a theorem (of various people) that the torsion
points on the subvariety have Zariski closure equal to some finite union
of torsion cosets of semiabelian subvarieties. We will focus on this
question for (split) tori, where it manifests as the more concrete
problem of finding solutions to a system of polynomial equations valued
in roots of unity. We compare two effective methods for solving this
problem: a combinatorial approach introduced by Conway-Jones, and a
commutative algebra approach introduced by Beukers-Smyth.

Hironaka's theorem states that every algebraic variety over a field of characteristic zero admits a resolution, however the algorithm to do so is very complex. It was conjectured by Nash that repeated Nash blowups (which replace each singular point by limiting positions of tangent spaces at nonsingular points) might provide a simple way to resolve. I plan to investigate the case of toric varieties.

In his landmark paper ``Modular forms and the Eisenstein
ideal,'' Mazur studied congruences modulo a prime p between the Hecke
eigenvalues of an Eisenstein series and the Hecke eigenvalues of cusp
forms, assuming these modular forms have weight 2 and prime level N. He
asked about generalizations to squarefree levels N. I will present some
work on such generalizations, which is joint with Preston Wake and
Catherine Hsu.

In the 1980s, Richard Hamilton invented the Ricci flow and initiated the geometric-analytic approach towards the Poincar\'e conjecture, which was eventually solved by Grisha Perelman. In this talk, we will briefly discuss the formation of singularities in the Ricci flow and the relations with Ricci solitons. We will introduce some recent advances and main conjectures in Ricci solitons, especially in steady gradient Ricci solitons. We end the talk by discussing some recent results (joint work with Professor Chow and Yuxing Deng) about asymptotic behaviors and a rigidity theorem of steady solitons.

A Calabi-Yau manifold is a Ricci-flat Kahler manifold. It is a longstanding question to understand how the Ricci-flat metrics develop singularities when the complex structure degenerates. An especially intriguing phenomenon is that these metrics can collapse to lower dimensions and exhibit very non-algebraic features, and it is challenging to describe the corresponding geometric behavior. In this talk I will discuss an answer to this question for a special class of degenerations. Based on joint works with H. Hein-J. Viaclovsky-R. Zhang 2018 and with R. Zhang 2019.

In this talk we will describe some recent works on Drinfeld modular forms with values in Tate algebras (in `equal positive characteristic'). In particular, we will discuss some remarkable identities (proved or conjectural) for Eisenstein and Poincar\'e series, and the problem of analytically interpolate families of Drinfeld modular forms for congruence subgroups at the infinity place.

Over the complex numbers, a famous theorem of Siu states that the plurigenera $P_m$ of projective manifolds are invariant under deformations. We give examples of families of elliptic surfaces over a DVR of positive or mixed characteristic such that $P_m$ fails to be constant for all sufficiently divisible $m\geq 0$. Time permitting, we will show that (asymptotic) invariance of plurigenera holds for families of quasi-elliptic surfaces.

With the motivation of generalizing the corresponding
geometrization of Tamagawa-Iwasawa theory after Kedlaya-Pottharst, and
with motivation of establishing the corresponding equivariant version of
the relative p-adic Hodge theory after Kedlaya-Liu aiming at the
deformation of representations of profinite fundamental groups and the
family of \'etale local systems, we initiate the corresponding
Hodge-Iwasawa theory with deep point of view and philosophy in mind from
early work of Kato and Fukaya-Kato. In this talk, we are going to
discuss some foundational results on the Hodge-Iwasawa modules and
Hodge-Iwasawa sheaves, as well as some interesting investigation towards
the goal in our mind, which are taken from our first paper in this
series project.

This talk will be about refined curve counting on local $P^2$, the noncompact Calabi-Yau 3-fold total space of the canonical line bundle of the projective plane. I will explain how to construct quasimodular forms starting from Betti numbers of moduli spaces of one-dimensional coherent sheaves on $P^2$. This gives a proof of some stringy predictions about the refined topological string theory of local $P^2$ in the Nekrasov-Shatashvili limit. This work is in part joint with Honglu Fan, Shuai Guo, and Longting Wu.

We show that certain singularities do not arise as blow-ups of mean curvature flow starting at a generic hypersurface. This is joint work with Kyeongsu Choi, Christos Mantoulidis, and Felix Schulze.

In 1998, P. Biane and R. Speicher developed a theory of stochastic integration against free additive self-adjoint Brownian motion, the large-$n$ limit of $n \times n$ Hermitian Brownian motion. This development included a kind of It\={o} Formula for certain functions of free It\={o} processes. In this talk, we discuss -- for motivation -- the finite-$n$ version of this It\={o} Formula and use objects called \textit{double operator integrals} to give a new analytic interpretation of the quantities therein, which to date have been understood and interpreted mostly in combinatorial ways. This analytic interpretation yields an extension, which we shall also discuss, of Biane and Speicher's original formula.

Langlands's functoriality conjectures predict the existence of
``liftings'' of automorphic representations along morphisms of L-groups. A
basic case of interest comes from the irreducible algebraic
representations of GL(2), thought of as the L-group of the reductive
group GL(2) over Q. I will discuss the proof, joint with James Newton,
of the existence of the corresponding functorial liftings for a broad
class of holomorphic modular forms, including Ramanujan's Delta function.

In applications to number theory, it is often desirable to
find effective dynamical results. We want to provide effective
information about averages of orbits of the horospherical subgroup
acting on a hyperbolic manifold of infinite volume. We will start by
presenting the setting and results for manifolds with finite volume.
Then, we will discuss the difficulties that arise when studying the
infinite volume setting, and the measures that play a crucial role in
it. This is joint work with Jacqueline Warren.

How many smooth structures are there on a sphere? For dimensions at least 5, Kervaire--Milnor solved this problem in terms of another problem in algebraic topology: The computations of stable homotopy groups of spheres. In this talk, I will discuss recent progress on this problem in algebraic topology and its applications on smooth structures, which includes the following result with Guozhen Wang, building up on Moise, Kervaire--Milnor, Browder, Hill--Hopkins--Ravenel: Among all odd dimensions, the n-sphere has a unique smooth structure if and only if n = 1, 3, 5, 61. I will also discuss some recent progress towards the Kervaire invariant problem in dimension 126.

Based on the Lagarias-Odlyzko effectivization of the
Chebotarev density theorem, Kumar Murty gave aneffective version of the
Sato-Tate conjecture for an elliptic curve conditional on the analytic
continuationand the Riemann hypothesis for all the symmetric power
L-functions. Using similar techniques, Kedlaya and I obtained a
similar conditional effectivization of the generalized Sato-Tate
conjecture for an arbitrary motive. As an application, we obtained a
conditional upper bound of the form $O((\log N)^2(\log \log N)^2)$ for
the smallest prime at which two given rational elliptic curves with
conductor at most $N$ have Frobenius traces of opposite sign. In this
talk, I will discuss how to improve this bound to the best possible in
terms of Nand under slightly weaker assumptions. Our new approach
extends to abelian varieties. This is joint work with Kiran Kedlaya and
Francesc Fit\'e.

Circle packings in which all circles have integer curvature,
particularly Apollonian circle packings, have in the last decade become
objects of great interest in number theory. In this talk, we explore
some of their most fascinating arithmetic features, from local to global
properties to prime components in the packings, going from theorems, to
widely believed conjectures, to wild guesses as to what might be true.

Deligne's weight-monodromy conjecture gives control over the zeros of local factors of L-functions of varieties at places of bad reduction. His proof in characteristic p was a step in his proof of the generalized Weil conjectures. Scholze developed the theory of perfectoid spaces to transfer Deligne's proof to characteristic 0, proving the conjecture for complete intersections in toric varieties. Building on Scholze's techniques, we prove the weight-monodromy conjecture for complete intersections in abelian varieties.

Fix a smooth projective variety in projective space and project it to a linear subspace of the same dimension. The ramification divisor of this projection is classically known as the ``apparent boundary''. How does the apparent boundary move when we move the center of projection? I will discuss the geometry arising from this natural question. I will explain why the situation is most interesting for varieties of minimal degree, and how it is related to limit linear series for vector bundles.

A complete Kahler metric $g$ on a Kahler manifold $M$ is a ``steady gradient Kahler-Ricci soliton'' if there exists a smooth real-valued function $f:M \rightarrow R$ with $\nabla^{g}f$ holomorphic such that $Ric(g)=Hess(f)$. I will present a theorem of existence and uniqueness for such metrics. This is joint work with Alix Deruelle (Sorbonne).

When studying the cohomology of Shimura varieties and arithmetic manifolds, one encounters the following phenomenon: the same Hecke eigensystem shows up in multiple degrees around the middle dimension, and its multiplicities in these degrees resembles that of an exterior algebra.\\

In a series of recent papers, Venkatesh and his collaborators provide an explanation: they construct graded objects having a graded action on the cohomology, and show that under good circumstances this action factors through that of an explicit exterior algebra, which in turn acts faithfully and generate the entire Hecke eigenspace.\\

In this talk, we discuss joint work with Khare where we investigate the p=p situation (as opposed to the l $\neq$ p situation, which is the main object of study of Venkatesh's Derived Hecke Algebra paper): we construct a degree-raising action on the cohomology of the ordinary Hida tower and show that (under some technical assumptions), this action generates the full Hecke eigenspace under its lowest nonzero degree. Then, we bring Galois representations into the picture, and show that the derived Hecke action constructed before is in fact related to the action of a certain dual Selmer group.

A classical conjecture of Shafarevich, solved by Parshin and Arakelov, predicts that any smooth projective family of high genus curves over the complex line minus a point or an elliptic curve is isotrivial (has zero variation in its algebraic structure). A natural question then arises as to what other families of manifolds and base spaces might behave in a similar way. Kebekus and Kov\'acs conjecture that families of manifolds with good minimal models form the most natural category where Shafarevich-type conjecturesshould hold. For example, analogous to the original setting of Shafarevich Conjecture, they expect that over a base space of Kodaira dimension zero such families are always (birationally) isotrivial. In this talk I will discuss a solution to Kebekus-Kov\'acs Conjecture.

In this talk we discuss a classification of ribbon categories with the tensor product rules of the finite-dimensional complex representations of $SO(N)$, for $N \geq 5$ and $N=3$. The equivalence class of a category with $SO(N)$ fusion rules depends only on the eigenvalues of the braid operator on $X \otimes X$, where $X$ corresponds to the defining representation. The classification applies both to generic $SO(N)$ tensor product rules, and to certain fusion rings having only finitely many simple objects.

In recent years, K-stability of Fano varieties has been proved to be a rich topic for higher dimensional geometers. The transition of knowledge is mutual. In one direction, we use the powerful machinery from higher dimensional geometry, especially the minimal model program, to have a better understanding of various concepts in K-stability. On the other direction, K-stability provides the right subclass to construct moduli spaces of Fano varieties, which had been once considered beyond reach by algebraic geometers. In the first half hour, I will explain how people change their viewpoint on the definition of K-stability. Then in the main talk, I will focus on the moduli of Fano varieties.

I will outline the proof of the classification of two-dimensional shrinking gradient Kahler-Ricci solitons with scalar curvature tending to zero at infinity. This is joint work with Alix Deruelle and Song Sun.

http://www.math.ucsd.edu/\~{}benchow/cc-seminar.html.

http://www.math.ucsd.edu/\~{}benchow/cc-seminar.html

\textbf{August 18-21}\\
\\
\textbf{Organizers}\\
Shiferaw Berhanu (Temple University)\\
Ming Xiao (University of California, San Diego)\\
\\
\textbf{Speakers}\\
Paulo Cordaro (Universidade de Sao Paulo)\\
Jean-Pierre Demailly (Universit\'e Grenoble Alpes)\\
Tien-Cuong Dinh (National University of Singapore)\\
Peter Ebenfelt (University of California, San Diego)\\
John Erik Fornaess (Norwegian University of Science and Technology)\\
Xianghong Gong (University of Wisconsin-Madison)\\
Xiaojun Huang (Rutgers University-New Brunswick)\\
Bernhard Lamel (Texas A&M University at Qatar)\\
Min Ru (University of Houston)\\
Berit Stensones (Norwegian University of Science and Technology)\\
Laurent Stolovitch (University of Nice Sophia Antipolis)\\
Emil Straube (Texas A&M University)\\
Andrew Zimmer (University of Wisconsin-Madison)\\

Dropout and its extensions (e.g. DropBlock and DropConnect) are popular heuristics for training neural networks, which have been shown to improve generalization performance in practice. However, a theoretical understanding of their optimization and regularization
properties remains elusive. This talk will present a theoretical analysis of several dropout-like regularization strategies, all of which can be understood as stochastic gradient descent methods for minimizing a certain regularized loss. In the case of single
hidden-layer linear networks, we will show that Dropout and DropBlock induce nuclear norm and spectral k-support norm regularization, respectively, which promote solutions that are low-rank and balanced (i.e. have factors with equal norm). We will also show
that the global minimizer for Dropout and DropBlock can be computed in closed form, and that DropConnect is equivalent to Dropout. We will then show that some of these results can be extended to a general class of Dropout-strategies, and, with some assumptions,
to deep non-linear networks when Dropout is applied to the last layer.

In addition to frequent single-nucleotide mutations, mammalian and many
other genomes undergo rare and dramatic changes called genome
rearrangements. These include inversions, fissions, fusions, and
translocations. Although analysis of genome rearrangements was pioneered
by Dobzhansky and Sturtevant in 1938, we still know very little about the
rearrangement events that produced the existing varieties of genomic
architectures. Recovery of mammalian rearrangement history is a difficult
combinatorial problem that I will cover in this talk. Our data sets have
included sequenced genomes (human, mouse, rat, and others), as well as
radiation hybrid maps of additional mammals.

We present a new compactness theory of Ricci flows. This theory states that any sequence of Ricci flows that is pointed in an appropriate sense, subsequentially converges to a synthetic flow. Under a natural non-collapsing condition, this limiting flow is smooth on the complement of a singular set of parabolic codimension at least 4. We furthermore obtain a stratification of the singular set with optimal dimensional bounds depending on the symmetries of the tangent flows. Our methods also imply the corresponding quantitative stratification result and the expected $L^p$-curvature bounds.
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As an application we obtain a description of the singularity formation at the first singular time and a long-time characterization of immortal flows, which generalizes the thick-thin decomposition in dimension 3. We also obtain a backwards pseudolocality theorem and discuss several other applications.
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The schedule of the lecture series will be approximately as follows:
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1. Heat Kernel and entropy estimates on Ricci flow backgrounds and related geometric bounds.
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2. Continuation of Lecture 1 + Synthetic definition of Ricci flows (metric flows) and basic properties
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3. Convergence and compactness theory of metric flows
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4. Partial regularity of limits of Ricci flows

In this talk I'll introduce the Zoom for Thought seminar and then
teach the basics of making a website and how to connect with the UCSD
server. If you want to have a working website by the end of the week, it
will help to (1) talk to Saul about getting your login information for the
UCSD math servers, (2) have a text editor to make the website (I like
notepad++), and (3) have a way to SSH to the math server (I like WinSCP).
To stall for time I'll also talk about other resources that might be of
use to a math PhD student. Other people are more than welcome to
contribute ideas in this direction.

Given an $r$-uniform hypergraph $H$ and an $r$-uniform hypergraph $F$, define the relative Tur\'an number $ex(H,F)$ to be the maximum number of edges in an $F$-free subgraph of $H$. In this talk we discuss bounds for $ex(H,F)$ when $F$ is a loose cycle, Berge cycle, and a complete $r$-partite $r$-graph when the host $H$ is either arbitrary or when it is the random hypergraph $H_{n,p}^r$.
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This is joint work with Jiaxi Nie and Jacques Verstraete.

We present a new compactness theory of Ricci flows. This theory states that any sequence of Ricci flows that is pointed in an appropriate sense, subsequentially converges to a synthetic flow. Under a natural non-collapsing condition, this limiting flow is smooth on the complement of a singular set of parabolic codimension at least 4. We furthermore obtain a stratification of the singular set with optimal dimensional bounds depending on the symmetries of the tangent flows. Our methods also imply the corresponding quantitative stratification result and the expected $L^p$-curvature bounds.
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As an application we obtain a description of the singularity formation at the first singular time and a long-time characterization of immortal flows, which generalizes the thick-thin decomposition in dimension 3. We also obtain a backwards pseudolocality theorem and discuss several other applications.
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The schedule of the lecture series will be approximately as follows:
\\
1. Heat Kernel and entropy estimates on Ricci flow backgrounds and related geometric bounds.
\\
2. Continuation of Lecture 1 + Synthetic definition of Ricci flows (metric flows) and basic properties
\\
3. Convergence and compactness theory of metric flows
\\
4. Partial regularity of limits of Ricci flows

Brakke flow is a measure-theoretic generalization of the mean curvature flow which exploits the flexibility of geometric measure theory in order to describe the evolution by (generalized) mean curvature of surfaces exhibiting singularities, such as, for instance, a planar network with multiple junctions. In the first part of this talk, I will discuss the proof of the following result: given any $n$-dimensional rectifiable subset $\Gamma_0$ of a strictly convex bounded domain $U \subset \mathbb{R}^{n+1}$ such that $U \setminus \Gamma_0$ is not connected, there exists a Brakke flow of surfaces (possibly weighted with integer multiplicities) starting from $\Gamma_0$ and with the additional property that their boundary coincides with $\partial \Gamma_0$ at all times. Furthermore, the flow subconverges, as $t \to \infty$ and in the sense of varifolds, to a (generalized) minimal surface in $U$ with the prescribed boundary $\partial \Gamma_0$, thus providing a dynamical solution to Plateau's problem. In the second part, I will discuss recent developments concerning the relationship between the singularities of a stationary initial surface $\Gamma_0$ and the (non) uniqueness of the flow. This investigation leads to the definition of a class of \emph{dynamically stable} stationary varifolds, which seems worthy of further study. Based on joint works with Yoshihiro Tonegawa (Tokyo Institute of Technology).

Abstract: Consider an $n\times n$ deterministic matrix $A$ and a random matrix $M$ with independent standard Gaussian entries. In this talk I will discuss recent results that state that, if $||A||\leq 1$, for any $\delta \textgreater 0$, with high probability $A+\delta M$ has eigenvector condition number of order poly$(n/\delta)$ and eigenvalue gaps of order poly$(\delta/n)$, which implies that the randomly perturbed matrix has a stable spectrum. This has useful applications to numerical analysis and was used to obtain the fastest known provable algorithm for diagonalizing an arbitrary matrix.

This is joint work with Jess Banks, Archit Kulkarni and Nikhil Srivastava.

In this talk I will discuss the optimal transport problem with ``storage fees.'' This is a variant of the semi-discrete optimal transport (a.k.a. Monge-Kantorovich) problem, where instead of transporting an absolutely continuous measure to a fixed discrete
measure and minimizing the transport cost, one must choose the weights of the target measure, and minimize the sum of the transport cost and some given ``storage fee function'' of the target weights. This problem arises in queue penalization and semi-supervised
data clustering. I will discuss some basic theoretical questions, such as existence, uniqueness, a dual problem, and characterization of solutions. Then, I will present a numerical algorithm which has global linear and local superlinear convergence for a subcase
of storage fee functions.
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All work in this talk is joint with M. Bansil (UCLA).

The exceptional group $E_{7,3}$ has a symmetric space with
Hermitian tube structure. On it, Henry Kim wrote down low weight
holomorphic modular forms that are ``singular'' in the sense that their
Fourier expansion has many terms equal to zero. The symmetric space
associated to the exceptional group $E_{8,4}$ does not have a Hermitian
structure, but it has what might be the next best thing: a quaternionic
structure and associated ``modular forms''. I will explain the
construction of singular modular forms on $E_{8,4}$, and the proof that
these special modular forms have rational Fourier expansions, in a
precise sense. This builds off of work of Wee Teck Gan and uses key
input from Gordan Savin.

How does one measure area? As an example, how can one determine
the area of a region on a map for the purpose of real estate appraisal?
Wouldn't it be great if there were an instrument that would measure the
area of a region by simply tracing its boundary? It turns out that there
is such an instrument: it is called a planimeter. In this talk we will
discuss a particular type of planimeter called the compensating polar
planimeter. There will be a little bit of history and some analysis
involving line integrals and Green's theorem. Finally, there will be a
chance to (virtually) see actual examples of these fascinating instruments
from the speaker's collection.

We present a new compactness theory of Ricci flows. This theory states that any sequence of Ricci flows that is pointed in an appropriate sense, subsequentially converges to a synthetic flow. Under a natural non-collapsing condition, this limiting flow is smooth on the complement of a singular set of parabolic codimension at least 4. We furthermore obtain a stratification of the singular set with optimal dimensional bounds depending on the symmetries of the tangent flows. Our methods also imply the corresponding quantitative stratification result and the expected $L^p$-curvature bounds.
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As an application we obtain a description of the singularity formation at the first singular time and a long-time characterization of immortal flows, which generalizes the thick-thin decomposition in dimension 3. We also obtain a backwards pseudolocality theorem and discuss several other applications.
\\
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The schedule of the lecture series will be approximately as follows:
1. Heat Kernel and entropy estimates on Ricci flow backgrounds and related geometric bounds.
2. Continuation of Lecture 1 + Synthetic definition of Ricci flows (metric flows) and basic properties
3. Convergence and compactness theory of metric flows
4. Partial regularity of limits of Ricci flows

Polynomial optimization is a class of optimization programs whose objective and constraining functions are polynomials. The core task in polynomial optimization is to compute global optimizers when the optimization is nonconvex. Tensor computation is about optimization and decompositions of tensors, such as tensor norms, tensor eigenvalues and tensor decompositions. All these problems are connected to each other by the theory nonnegative polynomials and moment problems. This talk will give an introduction about classical backgrounds, currently existing results and remaining challenges for the research of these topics.

The minimal genus problem is a fundamental question in smooth 4-manifold topology. Every 2-dimensional homology class can be represented by a surface. But how small can this surface be? A generation ago, techniques from gauge theory were used to solve this in a large class of 4-manifolds with extra geometric structure, namely symplectic 4-manifolds. Recent work on trisections if 4-manifolds has revealed a deep connection with symplectic geometry and gives a new perspective on this result.

If $\operatorname{M}_n(\mathbb{C})$ is the set of $n \times n$ complex
matrices and $A \in \operatorname{M}_n(\mathbb{C})$, then we write
$\sigma(A) \subseteq \mathbb{C}$ for the set of eigenvalues of $A$. If $A$
is diagonalizable and $f \colon \sigma(A) \to \mathbb{C}$ is any function,
then one can define $f(A) \in \operatorname{M}_n(\mathbb{C})$ in a
reasonable way. Now, let
$\operatorname{M}_n(\mathbb{C})_{\operatorname{sa}}$ be the set of $n
\times n$ Hermitian matrices, which are unitarily diagonalizable and have
real eigenvalues. If $f \colon \mathbb{R} \to \mathbb{C}$ is a continuous
function, then one can fairly easily show that the map $\tilde{f} \colon
\operatorname{M}_n(\mathbb{C})_{\operatorname{sa}} \to
\operatorname{M}_n(\mathbb{C})$ defined by $A \mapsto f(A)$ is also
continuous. In this talk, we shall discuss the less elementary fact that
if $f$ is $k$-times continuously differentiable, then so is $\tilde{f}$.
Time permitting, we shall also discuss the much more complicated
infinite-dimensional case -- where instead of matrices, one considers
linear operators on a Hilbert space -- which is still an active area of
research.

This is an example-based introduction to the characteristic zero representation theory of p-adic GL(n). Highlights include Reeder's construction of the ``simple supercuspidal representations'', the proof that these are irreducible, and a discussion of their Langlands parameters. We assume a few basic definitions from representation theory and p-adic arithmetic.

We present a new compactness theory of Ricci flows. This theory states that any sequence of Ricci flows that is pointed in an appropriate sense, subsequentially converges to a synthetic flow. Under a natural non-collapsing condition, this limiting flow is smooth on the complement of a singular set of parabolic codimension at least 4. We furthermore obtain a stratification of the singular set with optimal dimensional bounds depending on the symmetries of the tangent flows. Our methods also imply the corresponding quantitative stratification result and the expected $L^p$-curvature bounds.
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As an application we obtain a description of the singularity formation at the first singular time and a long-time characterization of immortal flows, which generalizes the thick-thin decomposition in dimension 3. We also obtain a backwards pseudolocality theorem and discuss several other applications.
\\
\\
The schedule of the lecture series will be approximately as follows:
1. Heat Kernel and entropy estimates on Ricci flow backgrounds and related geometric bounds.
2. Continuation of Lecture 1 + Synthetic definition of Ricci flows (metric flows) and basic properties
3. Convergence and compactness theory of metric flows
4. Partial regularity of limits of Ricci flows

The theory of RCD spaces has seen a huge development in the last teen years. They are metric measure structures satisfying a synthetic notion of Ricci bounded below. This class includes several spaces with boundary, such as Gromov-Hausdorff limits of manifolds with convex boundary and Ricci bounded below in the interior. In this talk we will present new stability and regularity results for boundaries of RCD spaces. We will focus mostly on a new epsilon-regularity theorem which is new even in the setting of smooth Riemannian manifolds. It is based on a work in progress joint with Aaron Naber and Daniele Semola.

Convolutional Neural Networks are highly successful tools for image recovery and restoration. A major contributing factor to this success is that convolutional networks impose
strong prior assumptions about natural images - so strong that they enable image recovery without any training data. A surprising observation that highlights those prior assumptions is that one can remove noise from a corrupted natural image by simply fitting
(via gradient descent) a randomly initialized, over-parameterized convolutional generator to the noisy image.
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In this talk, we discuss a simple un-trained convolutional network, called the deep decoder, that provably enables image denoising and regularization of inverse problems such as compressive sensing with excellent performance. We formally characterize the dynamics
of fitting this convolutional network to a noisy signal and to an under-sampled signal, and show that in both cases early-stopped gradient descent provably recovers the clean signal.
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Finally, we discuss our own numerical results and numerical results from another group demonstrating that un-trained convolutional networks enable magnetic resonance imaging from highly under-sampled measurements, achieving results surprisingly close to trained
networks, and outperforming classical untrained methods.

I will present a vast generalization of Taelman's 2012
celebrated class-number formula for Drinfeld modules to the setting of
(rigid analytic) L-functions of Drinfeld module motives with Galois
equivariant coefficients. I will discuss applications and potential
extensions of this formula to the category of t-modules and t-motives.
This is based on joint work with Ferrara, Green and Higgins, and a
result of meetings in the UCSD Drinfeld Module Seminar.

Recently, Marian-Oprea-Pandharipande proved Lehn's conjecture for
Segre numbers associated to Hilbert schemes of points on surfaces.
They also provided a conjectural correspondence between Segre and
Verlinde numbers. For surfaces with holomorphic 2-form, we propose
conjectural generalizations of their results to moduli spaces of
stable sheaves of any rank. We provide several verifications by using
Mochizuki's formula. Joint work with Gottsche.

Percy MacMahon introduced the ``major index'' of a permutation a century
ago and proved that it has the same distribution as the ``inversion
number''. Such statistics are now called ``Mahonian''. Baxter and
Zeilberger considered the joint distribution of inversions and the major
index on permutations and showed they are jointly independently
asymptotically normally distributed. However, Baxter-Zeilberger's argument
has no hope of giving a more precise ``local limit theorem'' and
Zeilberger subsequently offered a \$300 reward for a different proof. This
is the story of that reward.

In these two lectures we plan to survey some known results about complete noncompact shrinking Ricci solitons in dimension four. The main goal is to understand the geometry at infinity of these manifolds, and for that we address the following basic questions:
*Estimate the number of ends of 4d shrinkers.
*Understand the asymptotic geometry along each end.
*Uniqueness results.
*Open problems/conjectures.

Holonomy vectors of translation surfaces provide a geometric generalization for higher genus surfaces of (primitive) integer lattice points. The counting and distribution properties of holonomy vectors on translation surfaces have been studied extensively. A natural question to ask is: How random are the holonomy vectors of a translation surface? We motivate the gap distribution of slopes of holonomy vectors as a measure of randomness and compute the gap distribution for the class of translation surfaces given by gluing two identical tori along a slit. No prior background on translation surfaces or gap distributions will be assumed.

This talk concerns a classical problem in computational wave propagations: How does one truncate an infinite domain to a finite size without introducing reflection waves from the artificial boundaries? The state-of-the-art approach is attaching to those boundaries a perfectly matched layer (PML). In the continuous theory, PMLs are subject to an analytically continued wave equation that damps all incident waves without creating any interfacial reflection. However, it is believed that ``numerical reflections'' are unavoidable after discretization. In this talk, I will demonstrate a truly reflectionless discrete PML. The key is to uncover the geometric mechanism hidden in the differential calculus formalism; in discretizing the theory, approximations are the best one can hope for the latter, while the former often admits exact discretization.

I will talk about why some classical theorems (e.g. Waldhausen, Laudenbach-Poenaru) can be modified to hold for non-orientable 3- and 4-manifolds. As a consequence, we can draw Kirby diagrams and trisection diagrams and conclude the usual main trisection theorems for non-orientable 4-manifolds. This is joint work with Patrick Naylor (University of Waterloo).

Dear ZFT attendee, has your research ever come down to proving a polynomial inequality? If so, would you want to use boring old analysis to come up with a five line proof? Or would you rather spend two days creating an algorithm to find the exact factorization of the polynomial? The latter is the obvious choice. Come on over to ZFT this week and see how Clarkson's inequality can be proved in a special case by using linear recursion and binomial coefficients! I guarantee that you will think of the polynomial $y^2-2y+1$ differently for the rest of your life!

Roth famously used fourier analysis to upper bound the size of a set of integers without 3-term arithmetic progressions. One might hope that similar techniques can be used for 4 or more term progressions, and some simple examples demonstrate otherwise. However, Gowers (2001) introduced a ``higher order'' fourier analysis which generalizes Roth's proof to longer arithmetic progressions. In this talk, we will give a combinatorial sketch of the methods of higher order fourier analysis culminating in the key problem of the field, the inverse conjecture for the gowers norms.

An expander graph is a type of graph that is sparse yet highly-connected. They have found uses in many subjects including computer science, topology, algebra, number theory and algebraic geometry. We will talk about the basics of expander graphs and how they can be used to give quantitative versions of algebraic statements regarding interesting groups

Ricci soliton is a self-similar solution to the Ricci flow and arises naturally in the singularity analysis of the flow. Steady Ricci soliton is a kind of soliton whose associated Ricci flow evolves by reparametrizing a fixed metric. It is closely related to the Type II limit solution to the Ricci flow. Steady Ricci soliton with integrable scalar curvature was studied by Deruelle in 2012, later by Catino-Mastrolia-Monticelli in 2016, Munteanu-Sung-Wang in 2019, Deng-Zhu in 2020. In this talk, we shall discuss a classification result on steady Kaehler Ricci soliton with nonnegative Ricci curvature and integrable scalar curvature. We then apply the result to study the steady Kaehler Ricci soliton with subquadratic volume growth or fast curvature decay.

In these two lectures we plan to survey some known results about complete noncompact shrinking Ricci solitons in dimension four. The main goal is to understand the geometry at infinity of these manifolds, and for that we address the following basic questions:
*Estimate the number of ends of 4d shrinkers.
*Understand the asymptotic geometry along each end.
*Uniqueness results.
*Open problems/conjectures.

The Multiple Access Channel (MAC) is one of the most well studied and understood network information theoretic models, describing a scenario where K users wish to deliver their information message to one receiver, sharing the same transmission channel. Beyond the multitude of practical and somewhat heuristic MAC protocols (e.g., TDMA, FDMA, CDMA, CSMA, Aloha, and variations thereof), the information theoretic capacity region is well understood under many situations of interest, and in particular in the Gaussian case, modeling the uplink of a wireless system with one access point or base station (receiver) and several users, sharing the same frequency band.
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More recently, a variant of this model has been proposed for a situation where a very large (virtually unlimited) number of users wish to communicate only very sporadically, such that at any point in time only a finite and relatively small number of users are ``active''. This scenario is appropriate for machine-type communications and Internet of Things, where a multitude of sensors and objects have only rather sporadic data to send, but they need to send them when they are created, at random times.
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The identification of the active user set, or the active message set (the list of messages transmitted, irrespectively of who is transmitting them) has some points in common with a compressed sensing problem, where the activity vector (entry 1 if a user/message is active and 0 otherwise) is the key object to be estimated at the receiver side.
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In this talk we review the basic MAC model and results, a variant for massive random access called ``unsourced random access'' where all users use the same codebook, and related recent results and algorithms. including some capacity scaling for this model, which remains quite open as far as a full information theoretic characterization is concerned.

This will be a mainly expository talk about some recent applications of number theory to topology. The crux of these applications is the construction of a cohomology theory called topological modular forms (TMF) out of the moduli of elliptic curves. I'll explain what TMF is, what we have been doing with it, and what we'd still like to know; I'll also discuss more recent attempts to extend the theory using level structures, higher-dimensional abelian varieties, and K3 surfaces. Time permitting, I'll talk about my work with Dominic Culver on some partial number-theoretic interpretations of TMF co-operations.

Morphogen gradients play a vital role in developmental biology by enabling embryonic cells to infer their spatial location and determine their developmental fate accordingly. The standard mechanism for generating a morphogen gradient involves a morphogen being produced from a localized source and subsequently degrading. While this mechanism is effective over the length and time scales of tissue development, it fails over typical subcellular length scales due to the rapid dissipation of spatial asymmetries. Single-particle tracking experiments have recently found that C. elegans zygotes rely on space-dependent switching diffusivities to form intracellular gradients during cell polarization. We analyze a model of switching diffusivities to determine its role in protein concentration gradient formation. In particular, we determine how the presence of switching diffusivities modifies the standard theory and show that space-dependent switching diffusivities can yield a gradient in the absence of a localized source. Our mathematical analysis yields explicit formulas for the intracellular concentration gradient which closely match the results of previous experiments and numerical simulations. We further consider how this mechanism of switching diffusive states interacts with a locally varying periodic microstructure in the cell, and use homogenization theory to show that at the typical cellular scales involved, such a microstructure does not necessarily need to be resolved in fine detail in order to accurately capture the dynamics of the system.

This will be a mainly expository talk about some recent
applications of number theory to topology. The crux of these
applications is the construction of a cohomology theory called
topological modular forms (TMF) out of the moduli of elliptic curves.
I'll explain what TMF is, what we have been doing with it, and what we'd
still like to know; I'll also discuss more recent attempts to extend the
theory using level structures, higher-dimensional abelian varieties, and
K3 surfaces. Time permitting, I'll talk about my work with Dominic
Culver on some partial number-theoretic interpretations of TMF
co-operations.

Many familiar objects in algebra, geometry and topology belong to
2-Calabi-Yau categories, e.g. coherent sheaves on a K3 or Abelian
surface, Higgs bundles on a smooth and proper curve, local systems on a
Riemann surface, and representations of a fixed preprojective algebra.
In each case the Borel-Moore homology of the stack of objects carries a
Hall algebra structure, which in the preprojective algebra case contains
all of the raising operators on the cohomology of Nakajima quiver
varieties, and in other cases produces a kind of nonlinear/global
analogue of this theory. By reducing everything locally to the case of
representations of a preprojective algebra, I'll show that in each case
the Borel-Moore homology of the stack of objects carries a perverse
filtration. I'll explain how this filtration can be used to produce
generators for the Borel-Moore homology of the stack of objects coming
from intersection cohomology of the coarse moduli space, along with
tautological classes. In joint work with Sjoerd Beentjes, we show that
in all but one of the examples in the first sentence of this abstract
this Borel-Moore homology is pure. In the K3 case this proves a
conjecture of Halpern-Leistner.

Given a class of finite structures, is there a way to assemble them into a
countably infinite limit object in a canonical way? For example, consider
the collection of finite linear orders. Clearly the limit object should be
some infinite linearly ordered set, but there are lots to choose from.
Should we use the order of the naturals? The integers? The rationals?
Similarly, consider the class of finite graphs. There are lots of
countably infinite graphs which embed every finite graph. Which one is
best? Fraisse theory allows us to answer this question; for certain
collections of finite structures, we can consider the ``generic''
countably infinite structure which embeds every member of the finite
class.

Isotropic Quot scheme is a projective scheme that compactifies the moduli space of maps (of a given degree) from a smooth projective algebraic curve to an Isotropic Grassmannian. Isotropic quot schemes are often not smooth, but we construct a virtual fundamental class which makes it 'virtually' smooth. We use this to understand intersection theory of this space and find a Vafa-Intriligator type formula for the intersection of certain cohomology classes.

By Ratner's famous equidistribution theorem, we know that unipotent orbits in finite volume quotients of Lie groups equidistribute in their closures. Often, in applications, one needs to know more: specifically, at what rate does the orbit equidistribute? We call a statement that includes a quantitative error term effective. In this talk, I will present an effective equidistribution theorem, with polynomial rate, for horospherical orbits in the frame bundle of certain infinite volume hyperbolic manifolds. This is joint work with Nattalie Tamam.

In a series of papers, Voiculescu generalized the notions of entropy and Fisher information to the free probability setting. In particular, the notions of free entropy have several applications in the theory of von Neumann algebras and free probability such as demonstrating certain von Neumann algebras do not have property Gamma, demonstrating the absence of atoms in the distributions of polynomials of random matrices, and the construction of free monotone transport. With the recent bi-free extension of free probability being sufficiently developed, it is natural to ask whether there are bi-free extensions of Voiculescu's notions of free entropy. In this talk, we will provide an introduction to a few notions of bi-free entropy and discuss the difficulties and peculiarities that occur. This is joint work with Ian Charlesworth.

In 2010, Hedden showed that the Ozsvath-Szabo concordance invariant tau detects whether a fibered knot induces a tight contact structure on the three-sphere. In 2017, Ozsvath, Stipsicz, and Szabo constructed a one-parameter family of concordance invariants Upsilon, which recovers tau as a special case. I will discuss a sufficient condition using Upsilon for the monodromy of the open book decomposition of a fibered knot to be right-veering. I will also discuss a generalization of a result of Baker on ribbon concordances between fibered knots. This is joint work with Dongtai He and Diana Hubbard.

This talk discusses the saddle point problem of polynomials. We give an algorithm for computing saddle points, based on Lasserre's hierarchy of Moment-SOS relaxations. Under some genericity assumptions, we show that: i) if there exists a saddle point, the algorithm can get one by solving a finite number of relaxations; ii) if there is no saddle point, the algorithm can detect its nonexistence.

A mean field game is usually denoted as a game with the number of
players tending to infinity, which leads us to consider population
densities of players and their resulting ``Nash equilibrium'' strategies.
Mean field games have been used to model financial markets, population
dynamics, and more, but solving the mean field games ends up requiring us
to solve a high-dimensional PDE. To avoid the curse of dimensionality, we
can instead use the method of characteristics to turn one of the PDEs in
the mean field game into a system of ODEs and then solve the mean field
game by parameterizing the dynamics of the ODE with a neural network.

Theta functions are certain quasi-periodic functions in several complex variables. ``Non-abelian'' analogues of theta functions can be defined via moduli of stable bundles over a curve. There exist surprising symmetries in the dimension formulas, called Verlinde numbers, of spaces of non-abelian theta functions. We will talk about the strange duality which is a geometric explanation for the symmetry between spaces of non-abelian theta functions.

The fractional obstacle problem in $\mathbb R^n$ with obstacle $\varphi\in C^\infty(\mathbb{R}^n)$ can be written as \[ \min\{(-\Delta)^s u , u-\varphi\} = 0,\quad\textrm{in }\quad\mathbb{R}^n. \] The set $\{u = \varphi\} \subset \mathbb{R}^n$ is called the contact set, and its boundary is the free boundary, an unknown of the problem. The free boundary for the fractional obstacle problem can be divided between two subsets: regular points (around which the free boundary is smooth, and is $n-1$ dimensional) and degenerate points. The set of degenerate points, even for smooth obstacles, can be very large (for example, with infinite $\mathcal{H}^{n-1}$ measure). In a joint work with X. Ros-Oton we show, however, that generically solutions to the fractional obstacle problem have a lower dimensional degenerate set. That is, for almost every solution (in an appropriate sense), the set of degenerate points is lower dimensional.

I will present MCMC algorithms as optimization over the KL-divergence in the space of probabilities. By incorporating a momentum variable, I will discuss an algorithm which performs accelerated gradient descent over the KL-divergence. Using optimization-like ideas, a suitable Lyapunov function is constructed to prove that an accelerated convergence rate is obtained. I will then discuss how MCMC algorithms compare against variational inference methods in parameterizing the gradient flows in the space of probabilities and how it applies to sequential decision making problems.

Right and left eigenvectors of non-Hermitian matrices form a bi-orthogonal system to which one can associate homogeneous quantities known as overlaps. The matrix of overlaps quantifies the stability of the spectrum and characterizes the joint eigenvalues increments under Dyson-type dynamics. Overlaps first appeared in the physics literature: Chalker and Mehlig calculated their conditional expectation for complex Ginibre matrices (1998). For the same model, we extend their results by deriving the distribution of the overlaps and their correlations (joint work with P. Bourgade). Similar results have been obtained in other integrable settings, namely quaternionic Gaussian matrices, as well as matrices from the spherical and truncated unitary ensembles.

Modeling Malle's Conjecture with Random Groups
Abstract: We construct a random group with a local structure that models
the behavior of the absolute Galois group ${\rm Gal}(\overline{K}/K)$,
and prove that this random group satisfies Malle's conjecture for
counting number fields ordered by discriminant with probability 1. This
work is motivated by the use of random groups to model class group
statistics in families of number fields (and generalizations). We take
care to address the known counter-examples to Malle's conjecture and how
these may be incorporated into the random group.

Outside of laboratories, microbial communities (biofilms and other types)
often exist in relatively stable environments where, on average, resource quality and quantity
are predictable. In these conditions, these communities are able to organize
into tuned chemical factories, efficiently turning resources into biomass and
waste byproducts. To do so, physical, chemical, and biological constraints
must be accomodated. In this seminar, techniques to model this
organization will be discussed. In particular, the importance of coupling microscale
metabolic information to community scale transport processes will be emphasized.

One approach to computing integrals over Hilbert schemes of
points on surfaces (and other moduli spaces of sheaves on surfaces) is
to reduce to the special case when the surface in question is $C^2$.
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I'll explain how to use the (K-theoretic) Donaldson-Thomas theory of
threefolds to deduce identities for holomorphic Euler characteristics
of tautological bundles over the Hilbert scheme of points on $C^2$. I'll
also explain how these identities control the behavior of such Euler
characteristics over Hilbert schemes of points on general surfaces.

It is known both scientifically and anecdotally that typical large-scale
social networks exhibit short paths between pairs of nodes, hence
the common phrase ``six degrees of separation''. In this talk, mathematical
background for this phenomenon is given, together with a study of the
algorithmic question of how to search such large-scale networks using only
local information. In particular, one could imagine that each
person in the network is allowed to pass a message to one of their
acquaintances, until a particular target person in the network receives the
message. Remarkably, for many networks with $n$ nodes, there is a simple
algorithm which does this extremely efficiently -- a ``decentralized
search'' algorithm which runs in time polylogarithmic in the number of
nodes. The mathematics in this talk involves only elementary combinatorics
and probability.

The Mahowald invariant is a highly nontrivial map (with indeterminacy) from the homotopy groups of spheres to itself with deep connections to chromatic homotopy theory. In this talk I will discuss a variant of the Mahowald invariant that can be computed using knowledge of the R-motivic stable homotopy groups of spheres, and discuss its comparison to the classical Mahowald invariant. This is joint work with Dan Isaksen.

Consider the following experiment: a deck with $m$ copies of $n$ different card types is randomly shuffled, and a guesser attempts to guess the cards sequentially as they are drawn. Each time a guess is made, some amount of ``feedback'' is given. For example, one could tell the guesser the true identity of the card they just guessed (the complete feedback model) or they could be told nothing at all (the no feedback model).
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In this talk we explore a partial feedback model, where upon guessing a card, the guesser is only told whether or not their guess was correct. We show in this setting that, uniformly in $n$, at most $m+O(m^{3/4}\log m)$ cards can be guessed correctly in expectation, which is roughly the number of cards one gets by naively guessing the same card type $mn$ times. This is joint work with Persi Diaconis, Ron Graham, and Xiaoyu He.

The direct summand conjecture, part of the homological conjectures in commutative algebra, was first proposed by Mel Hochster in 1973. The conjecture is easily proven when the given ring contains a field, but the ``mixed characteristic'' case seemed, for the most part, impenetrable. In 2016 Yves Andre announced a proof of the conjecture which utilizes the theory of perfectoid spaces. We will attempt to give a (somewhat) detailed proof of the DSC in a simplified scenario, and then give a general idea of the methods used to prove the theorem in the general case.

In this talk, we will discuss the existence theory of Kahler-Ricci flow when the Kahler metric has unbounded curvature. We will discuss some application of the Kahler-Ricci flow in the study of uniformization and the regularity of Gromov-Hausdorff limit. This is joint work with L.-F. Tam.

I will talk about some recent work determining the archimedean
components of certain Eisenstein series and CAP forms induced from the
long root parabolic of $G_2$. I will also discuss how these results are
being used in some work in progress on producing nonzero classes in
symmetric cube Selmer groups.

At the microscopic level, the dynamics of arbitrary networks of chemical reactions can be modeled as jump Markov processes whose sample paths converge, in the limit of large number of molecules, to the solutions of a set of algebraic ordinary differential equations. Fluctuations around these asymptotic trajectories and the corresponding phase transitions can in principle be studied through large deviations theory in path space, also called Wentzell-Freidlin (W-F) theory. However, the specific form of the jump rates for this family of processes does not satisfy the standard regularity assumptions imposed by such theory, and weaker conditions need to be developed to deal with the framework at hand. Using tools of Lyapunov stability theory we design sufficient conditions for the applicability of large deviations estimates to the asymptotics of the Markov process at hand. We then translate such conditions in terms of the topological structure of the chemical reaction network. This ultimately allows to define a large class of chemical reaction systems to which the estimates of interest can automatically be applied.

We introduce projective space over a field, and we investigate its
geometry. Along the way we find connections to many other fascinating
geometries including M\``{o}bius strips, the Hopf fibration, and the Fano
plane. We also discuss the basics of algebraic geometry in projective
space.

We define a localised Euler class for isotropic
sections, and isotropic cones, in SO(N) bundles. We use this to give an
algebraic definition of Borisov-Joyce sheaf counting invariants on
Calabi-Yau 4-folds. When a torus acts, we prove a localisation result.
This talk is based on the joint work with Richard. P. Thomas.

A solution to a geometric flow is called ancient if it has a backhistory going back infinitely far in time. Ancient solutions of parabolic PDE are analogous to entire solutions of elliptic PDE. In particular, they play a fundamental role in understanding singularity formation.
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Perelman studied ancient solutions to the Ricci flow in dimension 3 which are kappa-noncollapsed, and proved a crucial structure theorem for these ancient kappa-solutions. Moreover, Perelman conjectured that, up to scaling, every noncompact ancient kappa-solution in dimension 3 is isometric to either the Bryant soliton or the standard cylinder (or a quotient thereof). In these lectures, I will discuss the proof of this conjecture.

For every infinite set X we define S(X) as the group of all permutations of X. On its subgroup consisting of all finitely supported permutations there exists a natural group homomorphism signature. However, thanks to an observation of Vitali in 1915, we know that this group homomorphism does not extend to S(X). In the talk we extend the signature on the subgroup of S(X) consisting of all piecewise isometric elements (strongly related to the Interval Exchange Transformation group). This allows us to list all of its normal subgroups and gives also informations about an element of the second cohomology group of some groups.

Recent work of both Gabai and Schneiderman-Teichner on the smooth isotopy of homotopic surfaces with a common dual has reinvigorated the study of concordance invariants defined by Freedman and Quinn in the 90's, along with homotopy theoretic invariants of Dax from the 70's obstructing isotopy of disks. Using the Dax invariant, we will give conditions under which pairs of homotopic properly embedded disks in a smooth 4-manifold with boundary with a common dual are isotopic.

In recent years, topological and geometric data analysis (TGDA) has emerged as a new and promising field for processing, analyzing and understanding complex data. Indeed, geometry and topology form natural platforms for data analysis, with geometry describing the ``shape'' behind data; and topology characterizing / summarizing both the domain where data are sampled from, as well as functions and maps associated to them.
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In this talk, I will show how topological (and geometric ideas) can be used to analyze graph data, which occurs ubiquitously across science and engineering. Graphs could be geometric in nature, such as road networks in GIS, or relational and abstract. I will particularly focus on the reconstruction of hidden geometric graphs from noisy data, as well as graph matching and classification. I will discuss the motivating applications, algorithm development, and theoretical guarantees for these methods. Through these topics, I aim to illustrate the important role that topological and geometric ideas can play in data analysis.

Modeling high dimensional data by latent tree graphical models is a common approach in multiple
machine learning applications. In these models, the key task is to infer the structure of the tree, given
only observations on its leaves. A canonical example of this setting is the tree of life, where the
evolutionary history of a set of organisms is inferred by their nucleotide or protein sequences.
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In this talk, we will show that the tree structure is strongly related to the spectral properties of a fully
connected graph, defined over the terminal nodes of the tree. This relation forms the theoretical basis
of two new methods to recover latent tree models. Comparing our approach to several competing
methods, we show that in many settings, spectral methods have stronger theoretical guarantees and
work better in practice.

I will discuss topics in Natural Language Processing

It has been proved by R. E. Greene and H. Wu that a simply-connected complete Kahler manifold negatively pinched sectional curvature possesses a complete Bergman metric. I will briefly review the history and present the estimates of the Bergman metric using the bounded geometry. This talk is based on the joint work with Yau.

A solution to a geometric flow is called ancient if it has a backhistory going back infinitely far in time. Ancient solutions of parabolic PDE are analogous to entire solutions of elliptic PDE. In particular, they play a fundamental role in understanding singularity formation.
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Perelman studied ancient solutions to the Ricci flow in dimension 3 which are kappa-noncollapsed, and proved a crucial structure theorem for these ancient kappa-solutions. Moreover, Perelman conjectured that, up to scaling, every noncompact ancient kappa-solution in dimension 3 is isometric to either the Bryant soliton or the standard cylinder (or a quotient thereof). In these lectures, I will discuss the proof of this conjecture.

A central problem of machine learning is the following. Given data of the form $\{(y_i, f(y_i)+\epsilon_i)_{i=1}^M\}$, where $y_i$'s are drawn randomly from an unknown (marginal) distribution $\mu^*$ and $\epsilon_i$ are random noise variables from another unknown distribution, find an approximation to the unknown function $f$, and estimate the error in terms of $M$.
The approximation is accomplished typically by neural/rbf/kernel networks, where the number of nonlinear units is determined on the basis of an estimate on the degree of approximation, but the actual approximation is computed using an optimization algorithm.
Although this paradigm is obviously extremely successful, we point out a number of perceived theoretical shortcomings of this paradigm, the perception reinforced by some recent observations about deep learning.
We describe our efforts to overcome these shortcomings and develop a more direct and elegant approach based on the principles of approximation theory and harmonic analysis.\\
%We demonstrate a duality between certain problems of function approximation and probability estimation in machine learning and problems of super-resolution in signal separation. In particular, we will explain how the same tools from harmonic analysis can be used for both purposes, leading to a unified theory. We will demonstrate our ideas with some numerical examples.

For most microbial communities, the environment and the microbial community structure and function are intimately connected. Most environments outside of the lab are physically and chemically heterogeneous, shaping and complicating the metabolisms of their resident microbial communities: spatial variations introduce physics such as diffusive and advective transport of nutrients and byproducts for example. Conversely, microbial metabolic activity can strongly affect the environment in which the community must function. Hence it is important to link metabolism at the cellular level to physics and chemistry at the community level.
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To introduce metabolism to community-scale population dynamics, many modeling methods rely on large numbers of reaction kinetics parameters that are unmeasured, also making detailed metabolic information mostly unusable. The bioengineering community has addressed this difficulty by moving to kinetics-free formulations at the cellular level, termed flux balance analysis. To combine and connect the two scales, we propose to replace classical kinetics functions in community scale models with cell-level metabolic models, and predict metabolism and how it is influenced by and influences the environment. Further, our methodology permits assimilation of many types of measurement data. We will discuss the background and motivation, model development, and some numerical simulation results.

Let $X/\mathbb{F}_q$ be a smooth affine curve over a finite
field of characteristic $p > 2$. In this talk we discuss the $p$-adic
variation of zeta functions $Z(X_n,s)$ in a pro-covering
$X_\infty:\cdots \to X_1 \to X_0 = X$ with total Galois group
$\mathbb{Z}_p$. For certain ``monodromy stable'' coverings over an
ordinary curve $X$, we prove that the $q$-adic Newton slopes of
$Z(X_n,s)/Z(X,s)$ approach a uniform distribution in the interval
$[0,1]$, confirming a conjecture of Daqing Wan. We also prove a
``Riemann hypothesis'' for a family of Galois representations associated
to $X_\infty/X$, analogous to the Riemann hypothesis for
equicharacteristic $L$-series as posed by David Goss. This is joint
work with Joe Kramer-Miller.

A foliation on an algebraic variety is a partition of the variety into ``parallel'' disjoint immersed complex submanifolds.
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This turns out to be a very useful notion and holomorphic foliations
have played a central role in several recent developments in the study
of the geometry of projective varieties.
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This is the first part of a two talks series (with Roberto Svaldi)
in which we will explain some recent work building towards the
birational classification of holomorphic foliations on projective
varieties 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.
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Features joint work with P. Cascini and R. Svaldi

Random matrices are an incredibly useful tool for modelling noise and understanding the behavior of large, chaotic systems. They can help us understand the ``average'' complexity of a linear algebra algorithm, model ``on the cheap'' the behavior of large data sets, serve as benchmarks for clustering algorithms, and so much more. And they can do all of this for a very good reason: random matrices are, well, not so random after all. Their spectral asymptotics are highly structured, and highly concentrated, and that is the source of their usefulness.
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This talk will be accessible to an audience familiar with basic linear algebra and probability.

I will briefly review the convergence theory for non-collapsed Einstein 4-manifolds developed by Anderson, Bando-Kasue-Nakajima and Tian around 1990. This was the main precursor for the more recent higher-dimensional theory of Cheeger-Colding-Naber. However, several difficult problems have remained open even in dimension 4. I will focus on the structure of the possible bubbles and bubble trees in the 4-dimensional theory. In particular, I will explain Kronheimer's classical work on gravitational instantons as well as a recent result of Biquard-H concerning the renormalized volume of a 4-dimensional Ricci-flat ALE space.

The f-invariant was introduced by Lewis Bowen in 2008 and is a real-valued isomorphism invariant that is defined for a large class of probability measure-preserving actions of finite-rank free groups. Most notably, the f-invariant provided the first classification up to isomorphism of Bernoulli shifts over finite-rank free groups. It is also quite useful for the study of finite state Markov chains with values indexed by a finite-rank free group. The f-invariant is conceptually similar to entropy, and it has a formal connection to sofic entropy. In this expository talk, I will introduce the f-invariant and discuss some of its basic properties.

Detecting differences and building classifiers between distributions, given only finite samples, is an important task in data science applications. Optimal transport (OT) has evolved as the most natural concept to measure the distance between distributions, and has gained significant importance in machine learning in recent years. There are some drawbacks to OT: Computing OT is usually slow, and it often fails to exploit reduced complexity in case the family of distributions is generated by simple group actions.
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In this talk, we discuss how optimal transport embeddings can be used to deal with these issues, both on a theoretical and a computational level. In particular, we'll show how to embed the space of distributions into an $L^2$-space via OT, and how linear techniques can be used to classify families of distributions generated by simple group actions in any dimension. We'll also show conditions under which the $L^2$ distance in the embedding space between two distributions in arbitrary dimension is nearly isometric to Wasserstein-2 distance between those distributions. This is of significant computational benefit, as one must only solve N optimal transport problems to define the $N^2$ pairwise distances between N distributions. We'll present some applications in image classification and supervised learning.
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This is joint work with Alex Cloninger.

In free probability theory, there is no direct analog of density for probability distributions, but there is something like a notion of ``log-density'' in the study of free Gibbs laws and free score functions. Non-commutative notions of smoothness are important for studying both these log-densities and the functions used for changes of variables (or transport of measure) in free probability. In the single-variable setting, we have a good understanding of the smoothness properties of a function $f: \mathbb{R} \to \mathbb{R}$ applied to self-adjoint operators thanks to the work of Peller, Aleksandrov, and Nazarov; see the recent paper of Evangelos Nikitopoulos. However, in the multivariable setting, much of the literature has used classes of functions that are either too restrictive (such as analytic functions) or very technical to define (such as Dabrowski, Guionnet, and Shlyakhtenko's smooth functions where Haagerup tensor norms were used for the derivatives). We will discuss a notion of tracial non-commutative smooth functions that is modeled on trace polynomials. These smooth functions have many desirable properties, such as a chain rule, good behavior under conditional expectations, and a natural way to incorporate the one-variable functional calculus. We will sketch current work about the smooth transport of measure for free Gibbs laws as well as future directions in relating these smooth functions to free SDE.

In this talk, I will introduce the Chow t-structure on the motivic stable homotopy category over a general base field. This t-structure is a generalization of the Chow-Novikov t-structure defined on a p-completed cellular motivic module category in work of Gheorghe--Wang--Xu.
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Moreover, we identify the heart of this t-structure with a purely algebraic category, and expand the results of Gheorghe-Wang-Xu to integral results on the entire motivic category over general base fields. This leads to computational applications on determining the Adams spectral sequences in the classical stable homotopy category, as well as that in the motivic stable homotopy category over C, R, and $F_p$. This is joint work with Tom Bachmann, Guozhen Wang and Zhouli Xu.

Once upon a time, in the land of Mathlandia, the people in the city of Algebraic Geometria had a brilliant vision: what if all math was algebra, every theorem was category theory, and every proof used cohomology? Armed with their categories and rings, they declared war on the other cities of Mathlandia. Number Theoryberg was the first to fall, succumbing to the glory of sheaves and schemes. Next was Representation Theory City, which fell soon after. But alas, despite their efforts the algebraists had not made any progress since... until now! Scholze and Clausen have started the Condensed Mathematics program, which vows to take over Functional Analysisberg and Differential Geometria. We will deal with the very first matter in this program, the story of locally compact abelian groups.

Hurwitz spaces parametrize finite covers of the projective line by curves of genus g. We show that the Chow rings of Hurwitz spaces of low degree covers exhibit interesting stabilization properties as g goes to infinity. We also consider applications to the moduli space of curves. This is joint work with Hannah Larson.

Often in Number Theory and Geometry, it is necessary to consider topological algebraic objects, like locally compact abelian groups, continuous group cohomology, and adic spaces. However, many of these topological structures don't have good categorical properties, at least when compared to their purely algebraic counterparts. We will explore Condensed Mathematics, a program of Clausen and Scholze to systematically add topological structure to algebra with nice categorical properties.

We will discuss classification of ancient solutions to geometric flows. We will focus especially on the Ricci flow.

Machine learning achieves tremendous success in image classification, speech recognition, and medical diagnosis. In the meantime, machine learning also brings some interesting mathematical questions: How to solve the resulting optimization more efficiently and robustly? How to apply the machine learning technique to tackle challenging problems in mathematics. In this talk, I will view the neural networks model from a nonlinear scientific computing point of view and present some recent work on developing a homotopy training algorithm to train neural networks ``layer-by-layer'' and ``node-by-node.'' I will also review the neural network from numerical algebraic geometry point view and provide a novel initialization for ReLU networks.

The classical convergence theory and singularity formation for non-collapsed Einstein 4-manifolds leaves some important issues open. Which singular Einstein metrics are actually limits of smooth ones? Can we describe the moduli space of Einstein metrics close to its boundary? We partially answer these by proving that any Einstein manifold sufficiently GH-close to an Einstein orbifold is the result of a gluing-perturbation procedure. This completely describes the singularity formation of compact Einstein metrics and moreover lets us show that the desingularization of Einstein metrics by Kronheimer's gravitational instantons is obstructed. The recent results of Biquard-Hein let us extend a weaker obstruction for general Ricci-flat ALE spaces.

The famous Shannon-Nyquist theorem has become a landmark in the development of digital signal and image processing. However, in many modern applications, the signal bandwidths have increased tremendously, while the acquisition capabilities have not scaled sufficiently fast. Consequently, conversion to digital has become a serious bottleneck. Furthermore, the resulting digital data requires storage, communication and processing at very high rates which is computationally expensive and requires large amounts of power. In the context of medical imaging sampling at high rates often translates to high radiation dosages, increased scanning times, bulky medical devices, and limited resolution.
In this talk, we present a framework for sampling and processing a large class of wideband analog signals at rates far below Nyquist in space, time and frequency, which allows to dramatically reduce the number of antennas, sampling rates and band occupancy. Our framework relies on exploiting signal structure and the processing task. We consider applications of these concepts to a variety of problems in communications, radar and ultrasound imaging and show several demos of real-time sub-Nyquist prototypes including a wireless ultrasound probe, sub-Nyquist MIMO radar, super-resolution in microscopy and ultrasound, cognitive radio, and joint radar and communication systems. We then discuss how the ideas of exploiting the task, structure and model can be used to develop interpretable model-based deep learning methods that can adapt to existing structure and are trained from small amounts of data. These networks achieve a more favorable trade-off between increase in parameters and data and improvement in performance, while remaining interpretable.

The 2-spin spherical Sherrington Kirkpatrick (SSK) model was introduced by Kosterlitz, Thouless and Jones as a simplification of the usual Sherrington-Kirkpatrick model with Ising spins. The SSK model admits tractable formulas for many of its observables, allowing for a detailed analysis of its fluctuations using techniques from random matrix theory. We discuss recent results on the fluctuations of the SSK model with a magnetic field as well as at critical temperature. Based on joint work with P. Sosoe.

In this talk, we study the Chow group of the motive associated
to a tempered global L-packet $\pi$ of unitary groups of even rank with
respect to a CM extension, whose global root number is -1. We show that,
under some restrictions on the ramification of $\pi$, if the central
derivative $L'(1/2,\pi)$ is nonvanishing, then the $\pi$-nearly isotypic
localization of the Chow group of a certain unitary Shimura variety over
its reflex field does not vanish. This proves part of the
Beilinson--Bloch conjecture for Chow groups and L-functions. Moreover,
assuming the modularity of Kudla's generating functions of special
cycles, we explicitly construct elements in a certain $\pi$-nearly
isotypic subspace of the Chow group by arithmetic theta lifting, and
compute their heights in terms of the central derivative $L'(1/2,\pi)$ and
local doubling zeta integrals. This is a joint work with Chao Li.

Kuznetsov conjectured the existence of a correspondence
between different types of Fano threefolds which identifies a
distinguished semiorthogonal component of the derived category on each
side. I will explain joint work with Arend Bayer which resolves one of
the outstanding cases of this conjecture. This relies on the study of
the Hodge theory of certain K3 categories associated to the
semiorthogonal components.

Arithmetic statistics is a subfield of number theory which studies arithmetic objects in the context of statistical properties like distribution, expected value, etc. Work in this field involves both an understanding of the underlying arithmetic structure and the application of strong analytic tools. We will first explore a few types of questions that are studied in this field today, then we discuss a major analytic tool used by researchers called a Tauberian theorem.

Quot schemes are fundamental objects in moduli theory of algebraic geometry. We show that the generating series of certain virtual invariants of Quot schemes of surfaces are expressed by the universal series and Seiberg-Witten invariants.
We apply this to several cases including homological and K-theoretic descendent series, reduced invariants of K3 surfaces, virtual Segre and Verlinde series. In particular, descendent series are shown to be rational functions whenever the curve class is of Seiberg-Witten length N.

Branching Brownian motion (BBM) is a random particle system where each particle diffuses as Brownian motion and branches into a random number of particles at a constant rate. In this talk, we will focus on two variant models of BBM, BBM with absorption and BBM with inhomogeneous breeding potential. In the first model, we derive the long run expected number of particles conditioned on survival in the near critical case. In the second model, we study the entire configuration of particles.

In this talk I will explain an interpretation (due to Lewis Bowen) of the f-invariant as a variant of sofic entropy: it is the exponential growth rate of the expected number of ``good models'' for an action over a random sofic approximation. I will then introduce the relative f-invariant and provide a similar interpretation of this quantity. This provides a formula for the growth rate of the expected number of good models over a type of stochastic block model.

I will talk about joint work with Siegfried Echterhoff and Rufus Willett in which we introduce and study amenable actions of locally compact groups on C*-algebras, building on previous similar notions by Anantharaman-Delaroche for actions of discrete groups. Among the new results we prove an extension of Matsumura's theorem giving a characterisation of the weak containment property (coincidence of full and reduced crossed products) for actions on commutative C*-algebras and give examples showing that this result does not extend to general noncommutative C*-algebras.

Corks are special contractible 4-manifolds that play a key role in the study of exotic smooth 4-manifolds. In this talk, I will describe applications of corks to the study of surfaces in 4-manifolds. I'll begin with some badly behaved 2-spheres in 4-space, based on joint work with Piccirillo. Then I'll use a twist on these ideas to construct smoothly (indeed, holomorphically) embedded disks in the 4-ball that are isotopic through ambient homeomorphisms but not through diffeomorphisms. There will be lots of cartoons.

In the 90s', Witten gave a physical derivation of an isomorphism
between the Verlinde algebra of GL(n) of level l and the quantum
cohomology ring of the Grassmannian Gr(n,n+l). In the joint work arXiv:1811.01377 with Yongbin Ruan, we proposed a K-theoretic generalization of Witten's work by relating the $GL_n$ Verlinde numbers to
the level l quantum K-invariants of the Grassmannian Gr(n,n+l), and refer
to it as the Verlinde/Grassmannian correspondence. The correspondence was formulated precisely in the aforementioned paper, and we proved the rank 2
case (n=2) there.
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In this talk, I will first explain the background of this correspondence
and its interpretation in physics. Then I will discuss the main idea of
the proof for arbitrary rank. A new technical ingredient is the virtual
nonabelian localization formula developed by Daniel Halpern-Leistner.

Curvature pinching theorems restrict the topology of smooth manifolds satisfying suitable curvature assumptions. In some situations the Ricci flow can transform initial integral curvature bounds into later pointwise bounds and thereby extend pointwise to integral pinching results. I will first review $L^p$ integral pinching theorems of Gursky, Hebey--Vaugon, Dai--Wei--Ye, and others, which all rely on supercritical powers p greater than n/2 or on Chern--Gauss--Bonnet in dimension four. Then I will discuss how stronger control of the Sobolev inequality obtained using Perelman's mu-functional can be used to address the critical case $p=n/2$, leading both to a generalization of previous results as well as to a separate pinching result in the asymptotically flat setting. Some of the work presented is joint with Guofang Wei and Rugang Ye.

In the 70s, Fred Cohen and Peter May gave a description of the mod $p$ homology of a free $E_n$-algebra in terms of certain homology operations, known as Dyer--Lashof operations, and the Browder bracket. These operations capture the failure of the $E_n$ multiplication to be strictly commutative, and they prove useful for computations. After reviewing the main ideas from May and Cohen's work, I will discuss a framework to generalize these operations to homology with certain twisted coefficient systems and give a complete classification of twisted operations for $E_{\infty}$-algebras. I will also explain computational results that show the existence of new operations for $E_2$-algebras.

The variational principle of Lagrangian PDE is encoded geometrically in a multisymplectic structure. This multisymplectic structure gives rise to a covariant formulation of Noether's theorem and conservation of multisymplecticity, a spacetime generalization of the symplecticity of Lagrangian mechanics. Multisymplecticity has many important physical implications, such as reciprocity in electromagnetism and conservation of wave action, a key ingredient in the stability analysis of wave propagation problems. Furthermore, upon introducing a foliation of spacetime, multisymplecticity can be reformulated as symplecticity on an infinite-dimensional phase space. Consequently, understanding how these multisymplectic structures are affected under discretization is an important aspect to the numerical integration of this class of PDE.
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In this talk, after discussing the multisymplectic formulation of Lagrangian PDE, we discuss how the multisymplectic structure is affected under discretization of the variational principle via the finite element method. We show how choices of discretization which preserve functional and geometric relationships of the underlying function spaces naturally give rise to discrete analogs of the continuum multisymplectic structures. In particular, we discuss how cochain projections and group-equivariant projections from the continuum function spaces to the finite element function spaces induce discrete analogs of the variational, multisymplectic, and Noether-theoretic structures.
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This is joint work with Melvin Leok.

Of all the polynomials over $\mathbb{Q}_p$, those of degree divisible by $p$ are most complicated. Similarly, the $p$-local subgroup structure of $\operatorname{GL}_n(\mathbb{C})$ is more complicated if $p$ divides $n$. The talk will be an example based introduction to these ideas, which fall under the ``wild aspect'' of the local Langlands correspondence. Along the way we will define $\mathbb{Q}_p$, some reductive groups of exceptional type, and exhibit wild $p$-structures in both of these examples.

We give a brief introduction to moduli of shtukas, with an eye towards its application in the Langlands correspondence for global function fields.

Leon Simon showed that if an area minimizing hypersurface admits a cylindrical tangent cone of the form C x R, then this tangent cone is unique for a large class of minimal cones C. One of the hypotheses in this result is that C x R is integrable and this excludes the case when C is the Simons cone over $S^3 x S^3$. The main result in this talk is that the uniqueness of the tangent cone holds in this case too. The new difficulty in this non-integrable situation is to develop a version of the Lojasiewicz-Simon inequality that can be used in the setting of tangent cones with non-isolated singularities.

In this talk, I will talk about my construction of relative moduli
spaces of stable sheaves over the stack of quasipolarized surfaces.
For this, I first retrace some of the classical results in the theory
of moduli spaces of sheaves on surfaces to make them work over the
nonample locus. Then I will recall the theory of good moduli spaces,
whose study was initiated by Alper and concerns an intrinsic (stacky)
reformulation of the notion of good quotients from GIT. Finally, I use
a criterion by Alper-Heinloth-Halpern-Leistner to prove existence of
the good moduli space. The application of the construction that I have
in mind is extending the Strange Duality results to degree two K3
surfaces - this part is still work in progress.

In this talk, we discuss a few simple and optimal methods for solving stochastic variational inequalities (VI). A prominent application of our algorithmic developments is the stochastic policy evaluation problem in reinforcement learning. Prior investigations in the literature focused on temporal difference (TD) learning by employing nonsmooth finite time analysis motivated by stochastic subgradient descent leading to certain limitations. These encompass the requirement of analyzing a modified TD algorithm that involves projection to an a-priori defined Euclidean ball, achieving a non-optimal convergence rate and no clear way of deriving the beneficial effects of parallel implementation. Our approach remedies these shortcomings in the broader context of stochastic VIs and in particular when it comes to stochastic policy evaluation. We developed a variety of simple TD learning type algorithms motivated by its original version that maintain its simplicity, while offering distinct advantages from a non-asymptotic analysis point of view. We first provide an improved analysis of the standard TD algorithm that can benefit from parallel implementation. Then we present versions of a conditional TD algorithm (CTD), that involves periodic updates of the stochastic iterates, which reduce the bias and therefore exhibit improved iteration complexity. This brings us to the fast TD (FTD) algorithm which combines elements of CTD and our newly developed stochastic operator extrapolation method. For a novel index resetting stepsize policy FTD exhibits the best known convergence rate. We also devised a robust version of the algorithm that is particularly suitable for discounting factors close to 1.

The theory of Drinfeld modules is remarkably similar to the
theory of abelian varieties, but their local monodromy behaves
differently and is poorly understood. In this talk I will present a
research program which aims to fully describe this monodromy. The
cornerstone of this program is a ``z-adic'' variant of Grothendieck's
l-adic monodromy theorem.
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The talk is aimed at a general audience of number theorists and
arithmetic geometers. No special knowledge of monodromy theory or
Drinfeld modules is assumed.

In an SRPT queue, the job with the shortest remaining processing time is served first, with preemption. The SRPT scheduling rule is of interest due to its optimality properties; it minimizes queue length (number of jobs in system). However, even with Markovian distributional assumptions on the processing times, an exact analysis is not possible. Hence, approximations in the form of a fluid (functional law of large numbers) limit or a diffusion (functional central limit theorem) limit can provide insights into system performance. It was shown by Gromoll, Kruk and Puha (2011) that, if the processing time distribution has unbounded support, then, under standard heavy traffic conditions, the diffusion limit of the queue length process is identically equal to zero. This exhibits the queue length minimization property of SRPT in sharpest relief. It also demonstrates that the SRPT queue length process is orders of magnitude smaller than the workload process in the diffusion limit. (The workload process tracks the time it will take the server to process the work associated with each job in system.) In this talk, we report on progress in characterizing this order of magnitude difference. We find that distribution dependent scaling must be used to obtain a nontrivial limit for the queue length and the associated measure valued state descriptor. The scaling captures the order of magnitude difference, and the nature of the limit is dependent on the tail decay of the processing time distribution.
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This work is joint with Sayan Banerjee (UNC) and Amarjit Budhiraja (UNC).

The spreading behavior of new or invasive species is a central topic in ecology. The modeling of free boundary problems is widely studied to better understand the nature of spreading behaviors of new species. From mathematical modeling point of view, it is a challenge to perform numerical simulations of the free boundary problems, due to the moving boundaries, the topological changes, and the stiffness of the system.
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Our work is concerned with numerical simulations of the long-term dynamical behavior of invasive species modeled by reaction-diffusion equations with free boundaries. We develop a front-tracking method to track the locations of the moving boundary explicitly in one dimension and higher dimensions with spherical symmetry. In two dimensional cases, we employ the level set method to handle topological bifurcations. For single invasive species, we numerically analyze the spreading-vanishing dichotomy in the diffusive logistic model. Various numerical experiments are presented in the two-dimensional spaces to show that the population range tend to be more and more spherical as time increases no matter what geometrical shape the initial population range has if the invasive species spreads successfully. For two invasive species in a weak-strong competition case, we examine how the long-time dynamics of the model changes as the initial functions are varies. Specifically, we simulate the ``chase-and-run coexistence'' phenomenon by choosing the initial function properly. The spreading behavior under time-periodic perturbation of the environment is also considered in our work.

Most of us learn about determinants in an algebra course, but they have important geometrical meanings. I'll explain a few instances of these in this talk.

Curvature pinching theorems restrict the topology of smooth manifolds satisfying suitable curvature assumptions. In some situations the Ricci flow can transform initial integral curvature bounds into later pointwise bounds and thereby extend pointwise to integral pinching results. I will first review $ L^p$ integral pinching theorems of Gursky, Hebey--Vaugon, Dai--Wei--Ye, and others, which all rely on supercritical powers p greater than n/2 or on Chern--Gauss--Bonnet in dimension four. Then I will discuss how stronger control of the Sobolev inequality obtained using Perelman's mu-functional can be used to address the critical case p=n/2, leading both to a generalization of previous results as well as to a separate pinching result in the asymptotically flat setting. Some of the work presented is joint with Guofang Wei and Rugang Ye.

Multiscale substitution tilings are a new family of tilings of Euclidean space that are generated by multiscale substitution rules. Unlike the standard setup of substitution tilings, which is a basic object of study within the aperiodic order community and includes examples such as the Penrose and the pinwheel tilings, multiple distinct scaling constants are allowed, and the defining process of inflation and subdivision is a continuous one. Under a certain irrationality assumption on the scaling constants, this construction gives rise to a new class of tilings, tiling spaces and tiling dynamical systems, which are intrinsically different from those that arise in the standard setup. In the talk I will describe these new objects and discuss various structural, geometrical, statistical and dynamical results. Based on joint work with Yaar Solomon.

In MIP*$=$RE the authors settle the CEP in the negative by showing that two computational complexity classes are equal. But the heart of their argument is the construction of a synchronous game with certain properties. In this talk we will describe the theory of synchronous games, and show how our construction of the algebra of a game leads more directly to the CEP. This approach to the CEP still depends on their construction of a synchronous game with particular properties, but stays within the context of operator algebras and games.

Given the ubiquity of vector bundles, it is perhaps surprising that there are so many open questions about them -- even on projective spaces. In this talk, I will outline some results about vector bundles on projective spaces, including my ongoing work on complex rank 3 topological vector bundles on $CP^5$. In particular, I will describe a classification of such bundles which involves a connection to topological modular forms; a concrete, rank-preserving additive structure which allows for the construction of new rank 3 bundles on $CP^5$ from ``simple'' ones; and future directions related to this project.

As an independent mathematician, you want to determine which of your axioms are internally consistent. Since you only have a finite amount of time, you may choose to come to this talk! I will be giving a countably long survey of results related to the Axiom of Choice. Well, hopefully you find it a maximally useful way to spend your Tuesday afternoon. (Or maybe you won't not find it useful, depending on what you decide.)

Let $V$ be a vector bundle over a smooth projective curve of rank $n$. We are interested in understanding the set of sub-bundles of rank $r$ with maximal degree. When we impose some numerical conditions, this set happens to be finite. In this talk, we will go over chronological developments in finding the number of maximal sub-bundles. Surprisingly, these numbers are related to the Gromov-Witten invariants of Grassmannian which are computed by the Vafa-Intriligator formula.

In this talk I will discuss certain singular (and degenerate) solutions to the Kaehler Ricci flow (KRF) on smooth compact complex manifolds. Algebraically this will correspond to solving the Kahler Ricci flow on a projective varieties with so called log canonical singularities. Analytically this will correspond to solving a complex parabolic Monge Ampere equation on a smooth manifild, with degeneracies and singularities in the equation and possibly the initial condition. Settings for this study include the analytic minimal model program via KRF, pluri-potential theory and KRF, the conical KRK, and the flow of complete unbounded curvature metrics. Our results will be discussed within each of these contexts.

This talk discusses multiple methods for clustering high-dimensional data, and explores the delicate balance between utilizing data density and data geometry. I will first present path-based spectral clustering, a novel approach which combines a density-based metric with graph-based clustering. This density-based path metric allows for fast algorithms and strong theoretical guarantees when clusters concentrate around low-dimensional sets. However, the method suffers from a loss of geometric information, information which is preserved by simple linear dimension reduction methods such as classic multidimensional scaling (CMDS). The second part of the talk will explore when CMDS followed by a simple clustering algorithm can exactly recover all cluster labels with high probability. However, scaling conditions become increasingly restrictive as the ambient dimension increases, and the method will fail for irregularly shaped clusters. Finally, I will discuss how a more general family of path metrics, combined with MDS, give low-dimensional embeddings which respect both data density and data geometry. This new method exhibits promising performance on single cell RNA sequence data and can be computed efficiently by restriction to a sparse graph.

The mod p cohomology of locally symmetric spaces for definite
unitary groups at infinite level is expected to realize the mod p local
Langlands correspondence for $GL_n$. In particular, one expects the
(component at p) of the associated Galois representation to be
determined by cohomology as a smooth representation. I will describe how
one can establish this expectation in many cases when the local Galois
representation is Fontaine-Laffaille.
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This is joint work with D. Le, S. Morra, C. Park and Z. Qian.

Deciding nonnegativity of real polynomials is a key
question in real algebraic geometry with crucial importance in
polynomial optimization. It is well-known that in general this problem
is NP-hard, therefore one is interested in finding sufficient conditions
(certificates) for nonnegativity, which are easier to check. Since the
19th century, sums of squares (SOS) are a standard certificate for
nonnegativity, which can be detected by using semidefinite programming
(SDP). This SOS/SDP approach, however, has some issues, especially in
practice if the problem has many variables or high degree.
In this talk I will introduce sums of nonnegative circuit polynomials
(SONC). SONC polynomials are certain sparse polynomials having a special
structure in terms of their Newton polytopes and supports and serve as a
nonnegativity certificate for real polynomials, which is independent of
sums of squares. I will present some structural results of SONC
polynomials and I will provide an overview about polynomial optimization
via SONC polynomials.

We consider the construction of a polygon $P$ with $n$ vertices whose turning angles at the vertices are given by a sequence $A=(\alpha_0,\ldots, \alpha_{n-1})$, $\alpha_i\in (-\pi,\pi)$,
for $i\in\{0,\ldots, n-1\}$.
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The problem of realizing $A$ by a polygon can be seen as that of constructing a straight-line drawing of a graph with prescribed angles at vertices, and hence, it is a special case of the well studied problem of constructing an \emph{angle graph}.
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In 2D, we characterize sequences $A$ for which every generic polygon $P\subset \mathbb{R}^2$ realizing $A$ has at least $c$ crossings, for every $c\in \mathbb{N}$, and describe an efficient algorithm that constructs, for a given sequence $A$, a generic polygon $P\subset \mathbb{R}^2$ that realizes $A$ with the minimum number of crossings.
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In 3D, we describe an efficient algorithm that tests whether a given sequence $A$ can be realized by a (not necessarily generic) polygon $P\subset \mathbb{R}^3$, and for every realizable sequence the algorithm finds a realization.

The Variational Implicit Solvent Model (VISM) is a theoretical and computational framework for studying bimolecular systems with complex topology. Central in VISM is an effective free-energy of all possible interfaces separating solutes (e.g., proteins) from solvent (e.g., water). Such a functional can be minimized numerically by a level-set method to determine the stable equilibrium conformations and solvation free energies. In this talk we introduce two numerical methods related to VISM. The first one is the Compact Coupling Interface Method (CCIM), which is a finite difference method for elliptic interface problems with accurate gradient approximation. The second one is the Binary Level Set-VISM, which is coupled with the Monte Carlo method to simulate the binding of biomolecules.

Please join the Division of Physical Sciences for a virtual public lecture presented by the 2019-20 Lattimer Fellows: Rommie Amaro (Chemistry and Biochemistry), Suckjoon Jun (Physics), and Bo Li (Mathematics).
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The George W. and Carol A. Lattimer Faculty Research Fellowship seeks to stimulate thought about avenues of productive scientific exploration that may result from unique fusions and collaborations and promotes cross-collateral work between scientists and non-scientists at UC San Diego.
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Learn more about this opportunity and hear from past fellows about their research made possible through this fellowship.

We give an overview of dimensionality reduction methods, or sketching, for a number of problems in optimization, first surveying work using these methods for classical problems, which gives near optimal algorithms for regression, low rank approximation, and natural variants. We then survey recent work applying sketching to column subset selection, kernel methods, sublinear algorithms for structured matrices, tensors, trace estimation, and so on. The focus in the talk will be on fast algorithms.