The traveling salesman problem is one of the best-known examples of an algorithmically hard problem, but what does that mean formally? It turns out that a solution to this problem would immediately give a solution to any other NP problem, and in this sense we say it is "NP-complete." In this talk, we will give a more formal definition of what it means for a problem to be NP-complete, develop the machinery needed to prove NP-completeness, and then use this machinery to prove that the traveling salesman problem (and a few others) is indeed NP-complete.

The Springer resolution of the nilpotent cone of a semisimple Lie algebra has important

applications in representation theory, and in particular was used by Springer to give a geometric construction of the irreducible representations of Weyl groups.  This talk concerns a generalization of the Springer resolution constructed with the use of toric varieties.  We will discuss how this is connected in type A with Lusztig's generalized Springer correspondence, as well as an analogue of an affine paving of the fibers.  Part of this talk is joint work with Martha Precup and Amber Russell.

Markoff triples were introduced in 1879 and have a rich history spanning many branches of mathematics. In 2016, Bourgain, Gamburd, and Sarnak answered a long standing question by showing there exist infinitely many composite Markoff numbers. Their proof relied on showing the connectivity for an infinite family of graphs associated to Markoff triples modulo p for infinitely many primes p. In this talk we discuss what happens for the projective analogue of Markoff triples, that is surfaces W in $P^1$x$P^1$x$P^1$ cut out by the vanishing of a (2,2,2)-form that admit three non-commuting involutions and are fixed under coordinate permutations and double sign changes. Inspired by the work of B-G-S we investigate such surfaces over finite fields, specifically their orbit structure under their automorphism group. For a specific one-parameter subfamily $W_k$ of such surfaces, we construct finite orbits in $W_k(C)$ by studying small orbits that appear in  $W_k$($F_p$) for many values of p and k. This talk is based on joint work with E. Fuchs, J. Silverman, and A. Tran.

We motivate the study of coactions, developing our intuition by taking a tour through topological dynamics. We reinforce this intuition by exploring the particular example of skew product topological quivers - a subject of recent study by the speaker.

The self-dual Chern-Simons-Schrödinger model is a gauged cubic NLS on the plane with self-duality, i.e., energy minimizers are given by a first-order Cauchy-Riemann-type equation, rather than a second-order elliptic equation. While this equation shares all formal symmetries with the usual cubic NLS on the plane, the structure of solitary waves is quite different due to self-duality and nonlocality (which stems from the gauge structure). In accordance, this model possesses blow-up and global dynamics that are quite different from that of the usual cubic NLS. The goal of this talk is to present some recent results concerning the blow-up and global dynamics of this model, with emphasis on a few surprising features of this model such as the impossibility of a "bubble-tree'' blow-up and a nonlinear rotational instability of pseudoconformal blow-ups. This talk is based on joint work with Kihyun Kim (IHES) and Soonsik Kwon (KAIST).

In the late 1980’s, Krasauskas and Fiedorowicz-Loday independently developed the theory of crossed simplicial groups, which generalize Connes’ cyclic category. Of particular interest is the Dihedral category, which has recently been used to develop the theory of Real topological Hochschild homology, a first approximation to Grothendieck-Witt groups.

In the first part of my talk, I will discuss ongoing joint work with Mona Merling and Maximilien Péroux on a topological analogue of the homology of crossed simplicial groups. As a special case, we recover the theory of Real topological Hochschild homology.

In the second part of my talk, I will discuss joint work with Teena Gerhardt and Mike Hill. We provide a norm model for Real topological Hochschild homology, prove a multiplicative double coset formula for Real topological Hochschild homology, and we construct the Real Witt vectors of rings with anti-involution.

In this talk, we will explore several combinatorial objects whose existence depends on some (relatively) straightforward divisibility conditions. In each of these case, perhaps somewhat surprisingly, these necessary divisibility conditions are in fact sufficient. The talk will conclude by mentioned what a complete resolution is and will not require any background knowledge in combinatorics.

Convergence analysis of accelerated first-order methods for convex optimization problems are presented from the point of view of ordinary differential equation (ODE) solvers. We first take another look at the acceleration phenomenon via A-stability theory for ODE solvers and present a revealing spectrum analysis for quadratic programming. After that, we present the Lyapunov framework for dynamical system and introduce the strong Lyapunov condition. Many existing continuous convex optimization models, such as gradient flow, heavy ball system, Nesterov accelerated gradient flow, and dynamical inertial Newton system etc, are addressed and analyzed in this framework. Then we present convergence analyses of optimization algorithms obtained from implicit or explicit methods of underlying dynamical systems.

Modern learning algorithms such as deep neural networks operate in regimes that defy the traditional statistical learning theory. Neural networks architectures often contain more parameters than training samples. Despite their huge complexity, the generalization error achieved on real data is small. In this talk, we aim to study the generalization properties of algorithms in high dimensions. Interestingly, we show that algorithms in high dimensions require a small bias for good generalization. We show that this is indeed the case for deep neural networks in the over-parametrized regime. In addition, we provide lower bounds on the generalization error in various settings for any algorithm. We calculate such bounds using random matrix theory (RMT). We will review the connection between deep neural networks and RMT and existing results. These bounds are particularly useful when the analytic evaluation of standard performance bounds is not possible due to the complexity and nonlinearity of the model. The bounds can serve as a benchmark for testing performance and optimizing the design of actual learning algorithms. (Joint work with Ofer Zeitouni)

We study minimizers of a family of functionals arising in combustion theory, which converge, for infinitesimal values of the parameter, to minimizers of the one-phase free boundary problem. We prove a $C^{1,\alpha}$ estimate for the "interfaces'' of critical points (i.e. the level sets separating the burnt and unburnt regions). As a byproduct, we obtain the one-dimensional symmetry of minimizers in the whole $\mathbb{R}^N$ for $N \le 4$, answering positively a conjecture of Fernández-Real and Ros-Oton. Our results are to the one-phase free boundary problem what Savin's results for the Allen-Cahn equation are to minimal surfaces. This is a joint work with J. Serra (ETHZ).

The theory of unipotent flows plays a central role in homogeneous dynamics. Time-changes are a simple perturbation of a given flow. In this talk, we shall discuss the rigidity of time-changes of unipotent flows. More precisely, we shall see how to utilize the branching theory of the complementary series, combining it with the works of Ratner and Flaminio-Forni to get to our purpose.

In this talk, I will explain a theorem of Bhatt-Scholze: the equivalence between prismatic $F$-crystal and $\mathbb Z_p$-lattices inside crystalline representation, and how to extend this theorem to allow more general types of base ring like Tate algebra ${\mathbb Z}_p\langle t^{\pm 1}\rangle$.  This is a joint work with Heng Du, Yong-Suk Moon and Koji Shimizu.

This is a talk in integral $p$-adic Hodge theory.  So in the pre-talk, I will explain the motivations and base ideas in integral $p$-adic Hodge theory.
 

Stochastic reaction network models are a popular tool for studying the effects of dynamical randomness in biological systems. Such models are typically analysed by estimating the solution of Kolmogorov's forward equation, called the chemical master equation (CME), which describes the evolution of the probability distribution of the random state-vector representing molecular counts of the reacting species. The size of the CME system is typically very large or even infinite, and due to this high-dimensional nature, accurate numerical solutions of the CME are very difficult to obtain. In this talk we will present a novel deep learning approach for computing solution statistics of high-dimensional CMEs by reformulating the stochastic dynamics using Kolmogorov's backward equation. The proposed method leverages superior approximation properties of Deep Neural Networks (DNNs) to reliably estimate expectations under the CME solution for several user-defined functions of the state-vector. Our method only requires a handful of stochastic simulations and it allows not just the numerical approximation of various expectations for the CME solution but also of its sensitivities with respect to all the reaction network parameters (e.g. rate constants). We illustrate the method with a number of examples and discuss possible extensions and improvements. 

This is joint work with Prof. Christoph Schwab (Seminar for Applied Mathematics, ETH Zürich) and Prof. Mustafa Khammash (Department of Biosystems Science and Engineering, ETH Zürich)

Reference:  Gupta A, Schwab C, Khammash M (2021) DeepCME: A deep learning framework for computing solution statistics of the chemical master equation. PLoS Comput Biol 17(12): e1009623. https://doi.org/10.1371/journal.pcbi.1009623

The Christoffel-Darboux kernel is the reproducing kernel associated to the Hilbert space containing all polynomials up to a given degree. It can be naturally written in terms of any complete set of orthonormal polynomials. In classical analysis the Christoffel-Darboux kernel is useful for studying properties of the underlying measure with respect to which the Hilbert space of polynomials is defined. In this talk, we present the version of the Christoffel-Darboux kernel for $L^2$ spaces of tracial states on noncommutative polynomials. We view this kernel as a noncommutative function, and identify its values as maxima of certain sets of  non-negative matrices/operators.

In numerous cases, the classical version of the Christoffel-Darboux kernel can be used (after renormalization) to recover the measure to which it is associated as a weak derivative. This is done with the aid of the theory of plurisubharmonic functions. We use this same theory in order to introduce several noncommutative versions of the Siciak extremal function. We use the Siciak functions to prove that, in several cases of interest, the (properly normalized) limit of the evaluations of the Christoffel-Darboux kernel on matrix sets exists as a well-defined, quasi-everywhere finite plurisubharmonic function. Time permitting, we conclude with some conjectures regarding these objects. This is based on joint work with Victor Magron (LAAS) and Victor Vinnikov (Ben Gurion
University).

This talk describes work in progress computing the $H\underline{\mathbb{F}}_2$ homology of the $C_2$-equivariant Eilenberg-Maclane spaces associated to the constant Mackey functor $\underline{\mathbb{F}}_2$. We extend a Hopf ring argument of Ravenel-Wilson computing the mod p homology of non-equivariant Eilenberg-Maclane spaces to the $RO(C_2)$-graded setting. An important tool that arises in this equivariant context is the twisted bar spectral sequence which is quite complicated, lacking an explicit $E^2$ page and having arbitrarily long equivariant degree shifting differentials. We avoid working directly with these differentials and instead use a computational lemma of Behrens-Wilson along with norm and restriction maps to complete the computation.

Finding canonical metrics on compact Kähler varieties has been an intense topic of research for decades. A famous result of Yau says that every compact Kähler manifold with non-positive first Chern class admits a Kähler-Einstein metric (when the Chern class is negative this was also independently proved by Aubin). In this talk, I’ll present some recent joint works with Hamid Abban, Yuchen Liu and Chenyang Xu on the existence of Kähler-Einstein metrics when the first Chern class is positive and the variety is possibly singular (such varieties are called Fano varieties). I’ll focus on two particular aspects: the solution of the YauTian-Donaldson conjecture, which predicts that the existence of Kähler-Einstein metrics on Fano varieties is equivalent to an algebro-geometric stability condition called K-polystability, and a systematic approach (using birational geometry) to decide whether Kähler-Einstein metrics exist on explicit Fano varieties.

Ever find yourself lost in the woods? In this talk, we will speak for the trees with an overview of decision tree models and ensemble methods, including (but not limited to) random forests and XGBoost. We'll also discuss considerations of building such models and some applications. This talk does not require any background knowledge in machine learning.

We study exactness condition for Lasserre’s relaxation method for polynomial optimization problem with n variables under equality constraints defined by n polynomials. Under the assumption that the quotient ring has dimension equal to the product of the degrees of the n equality defining polynomials, we obtain an explicit bound on the order of Lasserre’s relaxation which guarantees exactness. When the common zero locus are real and all of multiplicity one, we describe the exact region as the convex hull of the moment map image of a vector subspace. For the relaxation of order equal to the explicit bound minus one, the convex hull coincides with the moment map image, and is diffeomorphic to its amoeba. Based on the theory of amoeba, we obtain an explicit description of the exact region, from which we further derive error estimations for relaxation of this specific order.

We present the probabilistic numeric solver BayesCG, for solving linear systems with real symmetric positive definite coefficient matrices. BayesCG is an uncertainty aware extension of the conjugate gradient (CG) method that performs solution-based inference with Gaussian distributions to capture the uncertainty in the solution due to early termination. Under a structure exploiting 'Krylov' prior, BayesCG produces the same iterates as CG. The Krylov posterior covariances have low rank, and are maintained in factored form to preserve symmetry and positive semi-definiteness. This allows efficient generation of accurate samples to probe uncertainty in subsequent computations.

We overview the recently developed level set approach to the existence theory of minimal submanifolds and present some joint work with A. Pigati and D. Stern. The underlying idea is to construct minimal hypersurfaces as limits of nodal sets of critical points of functionals. After starting with a general overview of the codimension one theory, we will move to the higher codimension setting, and introduce the self-dual Yang-Mills-Higgs functionals. These are a natural family of energies associated to sections and metric connections of Hermitian line bundles, whose critical points have long been studied in gauge theory. We will explain to what extent the variational theory of these energies is related to the one of the (n - 2)-area functional and how one can interpret the former as a relaxation/regularization of the latter. We will mention some elements of the proof, with special emphasis on the role played by the gradient flow.

Recent advances in the theory of smooth mod $p$ representations of a $p$-adic reductive group $G$ involve more and more derived methods.  It becomes increasingly clear that the proper framework to study smooth mod $p$ representations is the derived category $D(G)$.

I will talk about smooth mod $p$ representations and highlight their shortcomings compared to, say, smooth complex representations of $G$.  After explaining how the situation improves in the derived category, I will spend the remaining time on the left adjoint of the derived parabolic induction functor.
 

Let ${\mathbb{H}^n}$ be the hyperbolic 𝑛-space and Γ be a geometrically finite discrete subgroup in Isom$_+$(${\mathbb{H}^n}$) with parabolic elements. We investigate whether the geodesic flow (resp. the frame flow) over the unit tangent bundle T$^1$ (Γ \ ${\mathbb{H}^n}$) (resp. the frame bundle F(Γ \ ${\mathbb{H}^n}$)) mixes exponentially. This result has many applications, including spectral theory, prime geodesic theorems, orbit counting, equidistribution, etc.

I will start with a survey of the past results, methods, and related problems on this topic. Along the way, I will present the joint work with Jialun Li, Pratyush Sarkar.

For complex smooth threefolds, there are enumerative theories of curves defined using sheaves, such as Donaldson-Thomas (DT) theory using ideal sheaves and Pandharipande-Thomas (PT) theory using stable pairs. These theories are conjecturally related among themselves and conjecturally related to other enumerative theories of curves, such as Gromov-Witten theory. The conjectural relation between DT and PT theories is known only for Calabi-Yau threefolds by work of Bridgeland, Toda, where one can use the powerful machinery of motivic Hall algebras due to Joyce and his collaborators. Bryan-Steinberg (BS) defined enumerative invariants for Calabi-Yau threefolds $Y$ with certain contraction maps $Y\rightarrow X$. I plan to explain how to extend their definition beyond the Calabi-Yau case and what is the conjectural relation to the other enumerative theories. This conjectural relation is known in the Calabi-Yau case by work of Bryan-Steinberg using the motivic Hall algebra. In contrast to the DT/ PT correspondence, we manage to establish the BS/ PT correspondence in some non-Calabi-Yau situations.

I will introduce some recent progress towards understanding the scalability of Markov chain Monte Carlo (MCMC) methods and their comparative advantage with respect to variational inference. I will discuss an optimization perspective on the infinite dimensional probability space, where MCMC leverages stochastic sample paths while variational inference projects the probabilities onto a finite dimensional parameter space. Three ingredients will be the focus of this discussion: non-convexity, acceleration, and stochasticity. This line of work is motivated by epidemic prediction, where we need uncertainty quantification for credible predictions and informed decision making with complex models and evolving data.

In the 1980s, Mahowald and Kane used Brown-Gitler spectra to construct splittings of $bo \wedge bo$ and $l \wedge l$.These splittings helped make it feasible to do computations using the $bo$- and $l$-based Adams spectral sequences.I will discuss progress towards an analogous splitting for $BP\langle 2 \rangle \wedge BP \langle 2 \rangle$ at odd primes.

Capillary surfaces model interfaces between incompressible immiscible fluids. The Euler-Lagrange equations for the capillary energy functional reveals that such surfaces are solutions of the prescribed mean curvature equation, with prescribed contact angle where the interface meets the container of the fluids. Min-max methods have been used with great success to construct unstable critical points of various energy functionals, particularly for the special case of closed minimal surfaces. We will discuss the development of min-max methods to construct general capillary surfaces.

Do you ever hear your number theorist friends say the word "perfectoid"? Does it make you feel confused? Afraid? If you have some vague idea that perfectoid spaces are an important concept, but have no idea what purpose they serve, then this talk is for you. We will attempt to describe how the theory of perfectoid spaces can be used to prove a simple statement in ring theory, which a priori has nothing to do with perfectoid spaces. We will assume familiarity with some basic notions regarding rings and modules at the level of Math 200, but we will do our best to strip away all unnecessary jargon and communicate plainly the role of perfectoid geometry in the proof.

A group is said to have the infinite conjugacy class (ICC) property if every non-identity element has an infinite conjugacy class. In this talk I will survey some ideas in geometric group theory, harmonic functions on groups, and topological dynamics and show how the ICC property sheds light on these three seemingly distinct areas. In particular I will discuss when a group has only constant bounded harmonic functions, when every proximal dynamical system has a fixed point, and what this all has to do with the growth of a group. No prior knowledge of harmonic functions on groups or Topological dynamics will be assumed.

This talk will include joint work with Anna Erschler, Yair Hartman, Omer Tamuz, and Pooya Vahidi Ferdowsi.

We say that a finitely generated group is self-simulable if every action of the group on a zero-dimensional space which is effectively closed (this means it can be described by a Turing machine in a specific way) is the topological factor of a subshift of finite type on said group. Even though this seems like a property which is very hard to satisfy, we will show that these groups do exist and that their class is stable under commensurability and quasi-isometries of finitely presented groups. We shall present several examples of well-known groups which are self-simulable, such as Thompson's V and higher-dimensional general linear groups. We shall also show that Thompson's group F satisfies the property if and only if it is non-amenable, therefore giving a computability characterization of this well-known open problem. Joint work with Mathieu Sablik and Ville Salo.

We use ideas from the theory of absolute concentration robustness to control a species of interest in a given chemical reaction network. The results are based on the network topology and the deficiency of the system, independent of reaction parameter values. The control holds in the stochastic regime and the quasistationary distribution of the controlled species is shown to be approximately Poisson under a specific scaling limit.

https://mathweb.ucsd.edu/~bli/research/mathbiosci/MBBseminar/

The theory of local theta correspondence is built up from two main ingredients: a reductive dual pair inside a symplectic group, and a Weil representation of its metaplectic cover. Exceptional correspondences arise similarly: dual pairs inside exceptional groups can be constructed using so-called Freudenthal Jordan algebras, while the minimal representation provides a suitable replacement for the Weil representation. The talk will begin by recalling these constructions. Focusing on a particular dual pair, we will explain how one obtains Howe duality for the correspondence in question. Finally, we will discuss applications of these results. The new work in this talk is joint with Gordan Savin.

In both normal tissue and disease states, cells interact with one another, and other tissue components. These interactions are fundamental in determining tissue fates, and the outcomes of normal development, and cancer metastasis. I am interested in collective cell behaviours, which I view as swarms with a twist: (1) cells are not simply point-like particles but have spatial extent, (2) interactions between cells go beyond simple attraction-repulsion, and (3) cells “live” in a regime where friction dominates over inertia. Examples include: wound healing, embryogenesis, the immune response, and cancer metastasis. In this seminar, I will give an overview of my computational, modelling, and theoretical contributions to tissue modelling at the sub-cellular, cellular, and population level.

In the first part, I focus on the nonlocal “Armstrong adhesion model” (Armstrong et al. 2006) for adhering tissue (an example of an aggregation-diffusion equation). Since its introduction, this approach has proven popular in applications to embyonic development and cancer modeling. However many mathematical questions remain. Combining global bifurcation results pioneered by Rabinowitz, equivariant bifurcation theory, and the mathematical properties of the non-local term, we prove a global bifurcation result for the non-trivial solution branches of the scalar Armstrong adhesion model. I will demonstrate how we used the equation’s symmetries to classify the solution branches by the nodal properties of the solution’s derivative.

In the second part, I focus on agent-based modelling of cell migration. Small GTPases, such as Rac and Rho, are well known central regulators of cell morphology and motility, whose dynamics play a role in coordinating collective cell migration. Experiments have shown GTPase dynamics to be affected by both spatio-temporally heterogeneous chemical and mechanical cues. While progress on understanding GTPase dynamics in single cells has been made, a major remaining challenge is to understand the role of GTPase heterogeneity in collective cell migration. Motivated by recent one-dimensional experiments (e.g. microchannels) we introduce a one-dimensional modelling framework allowing us to integrate cell bio-mechanics, changes in cell size, and detailed intra-cellular signalling circuits (reaction-diffusion equations). We use numerical simulations, and analysis tools, such as bifurcation analysis, to provide insights into the regulatory mechanisms coordinating collective cell migration.

Hilbert schemes of points on a smooth projective curve are simply symmetric powers of the curve itself; they are smooth and we know essentially everything about them. We propose a variation by studying double nested Hilbert schemes of points, which parametrize flags of 0-dimensional subschemes satisfying certain nesting conditions dictated by Young diagrams. These moduli spaces are almost never smooth but admit a virtual structure à la Behrend-Fantechi. We explain how this virtual structure plays a key role in (re)proving the correspondence between Gromov-Witten invariants and stable pair invariants for local curves, and say something on their K-theoretic refinement.

As the continuous limit of many discretized algorithms, PDEs can provide a qualitative description of algorithm’s behavior and give principled theoretical insight into many mysteries in machine learning. In this talk, I will give a theoretical interpretation of several machine learning algorithms using Fokker-Planck (FP) equations. In the first one, we provide a mathematically rigorous explanation of why resampling outperforms reweighting in correcting biased data when stochastic gradient-type algorithms are used in training. In the second one, we propose a new method to alleviate the double sampling problem in model-free reinforcement learning, where the FP equation is used to do error analysis for the algorithm. In the last one, inspired by an interactive particle system whose mean-field limit is a non-linear FP equation, we develop an efficient gradient-free method that finds the global minimum exponentially fast.

I will present joint work with Jacek Jendrej (CRNS, Sorbonne Paris Nord) on equivariant wave maps with values in the two-sphere. We prove that every finite energy equivariant wave map resolves, as time passes, into a superposition of decoupled harmonic maps and radiation, settling the soliton resolution conjecture for this equation.  It was proved in works of Côte, and Jia and Kenig, that such a decomposition holds along a sequence of times. We show the resolution holds continuously-in-time via a “no-return” lemma based on the virial identity. The proof combines a collision analysis of solutions near a multi-soliton configuration with concentration compactness techniques. As a byproduct of our analysis we also prove that there are no elastic collisions between pure multi-solitons. 

Let $C_2$ denote the cyclic group of order 2. In this talk, we’ll explore some recent computations done in $RO(C_2)$-graded cohomology with constant integral coefficients for $C_2$-surfaces. We’ll also explore some interesting patterns in these computations, and discuss how these might generalize to $C_2$-manifolds of higher dimension.

The ability of cells to exert forces and move about in their environment is essential to the survival of single-celled and multicellular organisms. Cell movement requires the coordination a number of sub-processes involving biochemical signaling and mechanical force generation. How such coordination can occur is a major area of study. In this talk I will discuss two lines of research that address different aspects of cell motion and biological transport.

In the first part, I will discuss a model and numerical simulation method to study how cell membranes change shape in response to forces that arise from the cell cortex. The cell cortex is a thin layer of cytoskeletal material that lies beneath the cell membrane in many cells. While models of biological membranes have existed for some time, the cell cortex is much more complicated, and detailed models do not yet exist. Thus, as a first step toward understanding the effect of the cell cortex, I will discuss how forces that mimic those generated by the cell cortex affect cell shape, leading to biologically realistic results.

In the second part, I will discuss how cell-level behaviors impact tissue and organ-scale properties through the use of multi-state continuous-time random walk models. These models can be posed in a general framework that includes details such as spatially heterogeneous binding, stochastic internal state changes, and various modes of spatial transport. Macro-scale equations with coefficients that depend on the local details are then obtained to describe transport on a tissue or organ-scale. Both lines of research have extensions that will be discussed throughout the talk.

Approximately an eternity ago, I gave a talk about Fréchet derivatives of maps between normed vector spaces and an infinite-dimensional Taylor's Theorem. I also promised this talk would be part of a series of infinite-dimensional calculus talks. I shall finally partially deliver on this promise by discussing "vector-valued integrals": what they are, when they exist, and -- time permitting -- some applications.

We study the problem of decision-making in the setting of a scarcity of shared resources when the preferences of agents are unknown a priori and must be learned from data. Taking the two-sided matching market as a running example, we focus on the decentralized setting, where agents do not share their learned preferences with a central authority. Our approach is based on the representation of preferences in a reproducing kernel Hilbert space, and a learning algorithm for preferences that accounts for uncertainty due to the competition among the agents in the market. Under regularity conditions, we show that our estimator of preferences converges at a minimax optimal rate. Given this result, we derive optimal strategies that maximize agents’ expected payoffs and we calibrate the uncertain state by taking opportunity costs into account. We also derive an incentive-compatibility property and show that the outcome from the learned strategies has a stability property. Finally, we prove a fairness property that asserts that there exists no justified envy according to the learned strategies.

This is a joint work with Michael I. Jordan.

For more than a century and a half it has been widely-believed that the physics of diffraction imposes certain fundamental limits on the resolution of an optical system. However our understanding of what exactly can and cannot be resolved has never risen above heuristic arguments which, even worse, appear contradictory.

In this work we remedy this gap by studying the diffraction limit as a statistical inverse problem and, based on connections to provable algorithms for learning mixture models, we rigorously prove upper and lower bounds on how many photons we need (and how precisely we need to record their locations) to resolve closely-spaced point sources. Moreover we show the emergence of a phase transition, which helps explain why the diffraction limit can be broken in some domains but not in others.

This is based on joint work with Sitan Chen.

In 1986, Nishikawa conjectured that a closed Riemannian manifold with positive curvature operator of the second kind is diffeomorphic to a spherical space form and a closed Riemannian manifold with nonnegative curvature operator of the second kind is diffeomorphic to a Riemannian locally symmetric space. Recently, the positive case of Nishikawa's conjecture was proved by Cao-Gursky-Tran and the nonnegative case was settled by myself. In this talk, I will first talk about curvature operators of the second kind and then present a proof of Nishikawa's conjecture under weaker assumptions.

A famous theorem of Giordano, Putnam and Skau (1995) states that two minimal homeomorphisms of a Cantor space X are orbit equivalent (i.e, the equivalence relations induced by the two associated actions are isomorphic) as soon as they have the same invariant Borel probability measures. I will explain a new "elementary" approach to prove this theorem, based on a strengthening of a result of Krieger (1980). I will not assume prior familiarity with Cantor dynamics. This is joint work with S. Robert (Lyon).

Fluctuations of synaptic-weights, among many other physical, biological and ecological quantities, are driven by coincident events originating from two 'parent' processes. We propose a multiplicative shot-noise model that can capture the behavior of a broad range of such natural phenomena, and analytically derive an approximation that accurately predicts its statistics. We apply our results to study the effects of a multiplicative synaptic plasticity rule that was recently extracted from measurements in physiological conditions. Using mean-field theory analysis and network simulations we investigate how this rule shapes the connectivity and dynamics of recurrent spiking neural networks. We show that the multiplicative plasticity rule, without fine-tuning, gives a stable, unimodal synaptic-weight distribution with a large fraction of strong synapses. The strong synapses remain stable over long times but do not `run away'. Our results suggest that the multiplicative plasticity rule offers a new route to understand the tradeoff between flexibility and stability in neural circuits and other dynamic networks. Joint work with Bin Wang.

https://mathweb.ucsd.edu/~bli/research/mathbiosci/MBBseminar/

Take $E$ to be an elliptic curve over a number field whose four torsion obeys certain technical conditions. In this talk, we will outline a proof that 100% of the quadratic twists of $E$ have rank at most one. To do this, we will find the distribution of $2^k$-Selmer ranks in this family for every positive $k$. We will also show how are techniques may be applied to find the distribution of $2^k$-class groups of quadratic fields.

The pre-talk will focus on the definition of Selmer groups. We will also give some context for the study of the arithmetic statistics of these groups.

That ellipsoidal shells do not exert gravitational force inside the cavity of the shell was known to Newton, Laplace, and Ivory.

In early 30’s P. Dive proved the inverse of this theorem. In this talk, I shall recall the (partially geometric) proof of this fact and then extend this result to unbounded domains.

Since ellipsoids, and any limit of a sequence of ellipsoids, are the so-called coincidence sets for the obstacle problem, there is a close link between the ellipsoidal potential theory and global solutions to the obstacle problem.

In this talk we present a complete classification (in terms of limit domains of ellipsoids) for global solutions to the obstacle problem in dimensions greater than five. The interesting ramification of this result is a new interpretation of the structure of the regular free boundary close to singular points.

This is a joint work with S. Eberle, and G.S. Weiss.

For further details and references see: https://www.scilag.net/problem/P-200218.1

Past work has shown that discretizing dynamical systems in a structure-preserving way can improve upon the performance of numerical methods constructed using more traditional approaches.  We aim to show that the Maxwell-Vlasov system of equations, which models plasma dynamics, is amenable to such a structure-preserving approach.  In particular, we will appeal to results obtained for compressible fluids and electromagnetic fields in our treatment of Maxwell-Vlasov, while discussing obstacles unique to that system.

The Newell-Littlewood numbers are tensor product multiplicities of Weyl modules for the classical groups in the stable range. Littlewood-Richardson coefficients form a special case. A. Klyachko connected eigenvalues of sums of Hermitian matrices to the saturated LR-cone and established defining linear inequalities. We prove analogues for the saturated NL-cone: an eigenvalue interpretation and a previously conjectured description by Extended Horn inequalities. This is joint work with S. Gao, N. Ressayre, and A. Yong.

Many natural and social phenomena involve individual agents coming together to create group dynamics, whether the agents are drivers in a traffic jam, cells in a developing tissue, or locusts in a swarm. Here I will focus on the specific example of pattern formation in zebrafish, which are named for the dark and light stripes that appear on their bodies and fins. Mutant zebrafish, on the other hand, feature different skin patterns, including spots and labyrinth curves. All of these patterns form as the fish grow due to the interactions of tens of thousands of pigment cells. The longterm motivation for my work is to better link genes, cell behavior, and visible animal characteristics — I seek to identify the specific alterations to cell interactions that lead to different mutant patterns. Toward this goal, I develop agent-based models to simulate pattern formation and make experimentally testable predictions. In this talk, I will overview my models and highlight several future directions. Because agent-based models are not analytically tractable using traditional techniques, I will also discuss the topological methods that we have developed to quantitatively describe cell-based patterns, as well as the associated nonlocal continuum limits of my models.

We use addition of two two-digit numbers as a motivating example to introduce addition with carrying. Prerequisites for the talk include familiarity with the place value system and single-digit addition in base ten. Some knowledge of multi-digit addition, in particular addition with carrying, is recommended but not required.

The Knaster continuum, also known as the buckethandle, or the Brouwer–Janiszewski–Knaster continuum can be viewed as an inverse limit of 2-tent maps on the interval. However, there is a whole class (with continuum many non-homeomorphic members) of Knaster continua, each viewed as an inverse limit of p-tent maps, where p is a sequence of primes. In this talk, for each Knaster continuum K, we will give a projective Fraïssé class of finite objects that approximate K (up to homeomorphism) and examine the combinatorial properties of that the class (namely whether the class is Ramsey or if it has a Ramsey extension). We will give an extremely amenable subgroup of the homeomorphism group of the universal Knaster continuum.

The Tower variant of the Number Field Sieve (TNFS) is known to be asymptotically the most efficient algorithm to solve the discrete logarithm problem in finite fields of medium characteristics, when the extension degree is composite. A major obstacle to an efficient implementation of TNFS is the collection of algebraic relations, as it happens in dimensions greater than 2. This requires the construction of new sieving algorithms which remain efficient as the dimension grows.

In this talk,  I will present how we overcome this difficulty by considering a lattice enumeration algorithm which we adapt to this specific context. We also consider a new sieving area, a high-dimensional sphere, whereas previous sieving algorithms for the classical NFS considered an orthotope. Our new sieving technique leads to a much smaller running time, despite the larger dimension of the search space, and even when considering a larger target, as demonstrated by a record computation we performed in a 521-bit finite field GF($p^6$). The target finite field is of the same form as finite fields used in recent zero-knowledge proofs in some blockchains. This is the first reported implementation of TNFS.

In the pre-talk, I will briefly present the core ideas of the quadratic sieve algorithm and its evolution to the Number Field Sieve algorithm.

Evolutionary dynamics permeates life and life-like systems. Mathematical methods can be used to study evolutionary processes, such as selection, mutation, and drift, and to make sense of many phenomena in life sciences. I will present two very general types of evolutionary patterns, loss-of-function and gain-of-function mutations, and discuss scenarios of population dynamics  -- including stochastic tunneling and calculating the rate of evolution. I will also talk about evolution in random environments.  The presence of temporal or spatial randomness significantly affects the competition dynamics in populations and gives rise to some counterintuitive observations. Applications to biomedical problems will be discussed.

During this talk I will introduce the notion of a k-AOU space, which we may think of as a matricial Archimedean order unit space. I will then describe the relationship between the category of k-AOU spaces and k-positive maps, and the category of operator systems and completely positive maps. After demonstrating the existence of injective envelopes and C*-envelopes in the category of k-AOU spaces, I will describe a connection with quantum correlations. Combined with previous work, this yields a reformulation of Tsirelson's conjecture. 

I will describe some recent progress in periodic homogenization of Hamilton-Jacobi equations. First, we show that the optimal rate of convergence is $O(\varepsilon)$ in the convex setting. We then give a minimalistic explanation that the class of centrally symmetric polygons with rational vertices and nonempty interior is admissible as effective fronts in two dimensions. Joint works with Wenjia Jing and Yifeng Yu.

Most biochemical reactions in living cells are open system interacting with environment through chemostats. At a mesoscopic scale, the number of  each species in those biochemical reactions can be modeled by the random time-changed Poisson processes. To characterize the macroscopic behaviors in the large volume limit, the law of large number in path space determines a mean-field limit nonlinear Kurtz ODE, while the WKB expansion yields a Hamilton-Jacobi equation and the corresponding Lagrangian gives the good rate function in the large deviation principle. A parametric variation principle can be formulated to compute the reaction paths. We propose a gauge-symmetry criteria for a class of non-equilibrium chemical reactions including  enzyme reactions, which identifies a new concept of balance within the same reaction vector due to flux grouping degeneracy. With this criteria,  we (i) formulate an Onsager-type gradient flow structure in terms of the energy landscape given by a steady solution to the Hamilton-Jacobi equation;  (ii) find transition paths between multiple non-equilibrium steady states (rare events in biochemical reactions).  We illustrate this idea through a bistable catalysis  reaction. In contrast to the standard diffusion approximations via Kramers-Moyal expansion, a new drift-diffusion approximation sharing the same gauge-symmetry is constructed based on the Onsager-type gradient flow formulation to compute the correct energy barrier.

We construct ramified families of curves to explicitly model the Lubin-Tate action, the action of a formal group law on its deformation space, for a maximal finite subgroup $G$. We will see that as a $G$-representation, this deformation space is a quotient of a regular representation of a finite cyclic group! This allows us to partially compute the $E_2$ page of the homotopy fixed point spectral sequence of the $K(h, p)$-local homotopy groups of spheres for height $h=p^{k-1}(p-1)$, for all such $h$ and $p$ simultaneously. Thus, we resolve a 40 year old computational stalemate.

In this talk, I will introduce two modeling approaches for biomedical diseases, one is pathophysiology-driven modeling, the other one is data-driven modeling. The former one is used when the pathophysiology of such a disease is well known. As an example, a mathematical model of atherosclerosis, based on this modeling approach, provides a personalized cardiovascular risk by solving a free boundary problem. Some interesting mathematical problems are also introduced by this new model to help us understand cardiovascular risk. The second modeling approach is used to learn the mathematical model based on clinical data when the pathophysiology of a particular disease is not well understood. I will use Alzheimer's disease as an example to illustrate the idea of this modeling approach and apply it to personalized treatment studies of aducanumab, a recently FDA-approved Alzheimer's medication.
 

The point of topology is to study shapes--and these shapes tend to have points. However, points aren't actually that cool. We will develop a theory of shapes called locales, which is 100% point-free. That is, completely pointless. Then we'll say a few words about the Banach-Tarski Paradox.

We study the decomposition methods for solving a class of nonconvex and nonsmooth two-stage stochastic programs, where both the objective and constraints of the second-stage problem are nonlinearly parameterized by the first-stage variable.  Due to the failure of the Clarke-regularity of the resulting nonconvex recourse function, classical decomposition approaches such as Benders decomposition and (augmented) Lagrangian-based algorithms cannot be directly generalized to solve such models. By exploring an implicitly convex-concave structure of the recourse function, we introduce a novel surrogate decomposition framework based on the so-called partial Moreau envelope. Convergence for both fixed scenarios and interior sampling strategy is established. Numerical experiments are conducted to demonstrate the effectiveness of the proposed algorithm.

Detecting differences and building classifiers between distributions, given only finite samples, are important tasks in a number of scientific fields. 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 can be slow, and it often fails to exploit reduced complexity in case the family of distributions is generated by simple group actions.  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. The proposed framework significantly reduces both the computational effort and the required training data in supervised settings. We demonstrate the benefits in pattern recognition tasks in imaging and provide some medical applications.

This talk is based on joint work with Alex Cloninger, Harish Kannan, Varun Khurana, and Jinjie Zhang.

I will report on joint work with Spencer Leslie where we define an analogue of the Cohen-Zagier Eisenstein series to the exceptional group $G_2$. Recall that the Cohen-Zagier Eisenstein series is a weight $3/2$ modular form whose Fourier coefficients see the class numbers of imaginary quadratic fields. We define a particular modular form of weight $1/2$ on $G_2$, and prove that its Fourier coefficients see (certain torsors for) the 2-torsion in the narrow class groups of totally real cubic fields. In particular:

1) we define a notion of modular forms of half-integral weight on certain exceptional groups,
2) we prove that these modular forms have a nice theory of Fourier coefficients, and
3) we partially compute the Fourier coefficients of a particular nice example on $G_2$.

The RAS protein network presents a unique situation in biology where all of the critical reactions are very well characterized both for the wild-type versions of the RAS proteins and for the cancer causing mutant forms of RAS.  We have previously shown that mathematical models that build up from reaction mechanisms can be used to make non-obvious and novel predictions about the behaviors of RAS mutants.  Recently, we have used these mathematical models to understand why some RAS mutations respond to drugs, when others do not.

The fermionic diagonal coinvariant ring was introduced by Rhoades and Jongwon Kim and is a quotient of a polynomial ring in two sets of $n$ anticommuting variables modulo $\mathfrak{S}_n$ invariant polynomials with no constant term, where the action of $\mathfrak{S}_n$ permutes both sets of variables simultaneously. In this talk, we will introduce a basis of this ring for which the action of $\mathfrak{S}_{n-1} \subset \mathfrak{S}_n$ can be interpreted combinatorially and use this basis to determine the isomorphism type of the ring. We will also relate our basis to a cyclic sieving result by Thiel.

Stoichiometric theory includes multiple biological scales from elements to ecosystems, and allows the construction of robust mechanistic, predictive, and empirically testable models via rigorous chemical and physical laws. Experimental and fundamental evidence motivates the application of this microscopic approach to understand macroscopic phenomena. I will introduce a series of stoichiometric models and their novel dynamics that resolve some biological paradoxes and lead to new insights. Selected new mathematical development will be briefly described. “True” model validation will be presented in contrast to conventional methods with many freedoms. I will briefly mention my recent expansion on a new graduate program and research of data science and machine learning.

We discuss results generalizing a result of Cordero-Erausquin and Klartag involving transport of log-concave measures to the free probabilistic setting. We also discuss open problems in extending it further.

We develop and analyze a numerical method proposed for solving high-dimensional Fokker-Planck equations by leveraging the generative models from deep learning. Our starting point is a formulation of the Fokker-Planck equation as a system of ordinary differential equations (ODEs) on finite-dimensional parameter space with the parameters inherited from generative models such as normalizing flows. We call such ODEs "neural parametric Fokker-Planck equations". The fact that the Fokker-Planck equation can be viewed as the 2-Wasserstein gradient flow of the relative entropy (also known as KL divergence) allows us to derive the ODE as the 2-Wasserstein gradient flow of the relative entropy constrained on the manifold of probability densities generated by neural networks. For numerical computation, we design a bi-level minimization scheme for the time discretization of the proposed ODE. Such an algorithm is sampling-based, which can readily handle computations in higher-dimensional space. Moreover, we establish bounds for the asymptotic convergence analysis as well as the error analysis for both the continuous and discrete schemes of the neural parametric Fokker-Planck equation. Several numerical examples are provided to illustrate the performance of the proposed algorithms and analysis.

Chromatic homotopy theory is a powerful tool to study periodic phenomena in the stable homotopy groups of spheres. Under this framework, the homotopy groups of spheres can be built from the fixed points of Lubin--Tate theories $E_h$.  These fixed points are computed via homotopy fixed points spectral sequences.  In this talk, we prove that at the prime 2, for all heights $h$ and all finite subgroups $G$ of the Morava stabilizer group, the $G$-homotopy fixed point spectral sequence of $E_h$ collapses after the $N(h,G)$-page and admits a horizontal vanishing line of filtration $N(h,G)$.

This vanishing result has proven to be computationally powerful, as demonstrated by Hill--Shi--Wang--Xu’s recent computation of $E_4^{hC_4}$.  Our proof uses new equivariant techniques developed by Hill--Hopkins--Ravenel in their solution of the Kervaire invariant one problem.  As an application, we extend Kitchloo--Wilson’s $E_n^{hC_2}$-orientation results to all $E_n^{hG}$-orientations at the prime 2. This is joint work with Zhipeng Duan and XiaoLin Danny Shi.
 

Key polynomials were introduced by Demazure for Weyl groups. They are non-symmetric generalizations of Schur polynomials, which are important in representation theory and geometry. Lascoux polynomials are K-theoretic analogues of key polynomials. In this talk, we describe several rules to compute key polynomials and Lascoux polynomials using tableaux. 

We discuss Kurt Heegner's work on the "class number 1 problem", and other fun stories like why ${e^{\pi\sqrt{163}}}$ is pretty much an integer

The concept of period growth was defined by Grigorchuk in the 80s, but still there are only a few examples of groups where we can estimate this invariant. We will sketch a connection to the Burnside problems and introduce a family of groups with very small period growth, answering a question by Bradford.

The defining feature of biological systems is their ability to replicate, which has at its foundation the process of cell division.  We are focused on understanding the inherent "two-ness" of cells and how accuracy and optimality are ensured during the trilliions of cell divisions that are needed to build and maintain multicellular organisms.  Our recent work highlights temporal optimization during cell division that is frequently disrupted in human cancers, highlighting a new type of tumor suppressor mechanism.

https://mathweb.ucsd.edu/~bli/research/mathbiosci/MBBseminar/

Branching Brownian motion (BBM) is a random particle system which incorporates both the tree-like structure and the diffusion process. BBM has a natural interpretation as a population model. In this talk, we will see how one variant model of BBM, BBM with an inhomogeneous branching rate can be used to study the evolution of populations undergoing selection. We will provide a mathematically rigorous justification for the biological observation that the distribution of the fitness levels of individuals in a population evolves over time like a traveling wave with a profile defined by the Airy function. This talk is based on joint work with Jason Schweinsberg.

An important area of study in the classification of II_1 factors is to investigate the dependence of a group von Neumann algebra L(G) relative to the group G. Popa’s deformation/rigidity theory has provided novel insights into this question over the past 20 years. 

In this talk, we demonstrate how one can import group cohomological information into the von Neumann algebra framework to unravel the structure of a large family of von Neumann algebras.

The truncated Brown-Peterson spectra admit actions by the cyclic group of order 2 via complex conjugation. Their fixed point spectra are higher height analogues of real K-theory. We describe how to use Tate square methods along with the slice spectral sequence to compute their mod 2 homology. This is joint work in progress with Mike Hill and Doug Ravenel.

The balancing of items — or discrepancy — arises naturally in Computer Science and Discrete Mathematics. Here we consider n vectors in n-space, all coordinate +1 or −1. We create a signed sum of the vectors, with the goal that this signed sum be as small as possible, Here we use the max (or ${L^∞}$) norm, though many variants are possible.

We create a game with Paul (Erdos) selecting the vectors and Carole (find the anagram!) choosing to add or subtract. This becomes four (two TIMES two) different problems. The vectors (Paul) can be chosen randomly or adversarially, equivalently average case and worst case analysis for Carole. The choice of signed sum (Carole) can be done on-line or off-line.

All four variants are interesting and are at least partially solved. We emphasize the random (Paul) on-line (Carole) case, joint work with Nikhil Bansal.

Since there is NOTHING special about March 2nd, in particular, it is no famous person's birthday, we will get back on track with serious, graduate-level mathematics. I will present the definition of a von Neumann algebra which is the main object I study. We will go through some common constructions and see their relationships to concepts in group theory. The talk will DEFINITELY NOT incorporate anything frivolous such as rhyme, meter, or visual media.
 

Stochastic majorization-minimization (SMM) is an online extension of the classical principle of majorization-minimization, which consists of sampling i.i.d. data points from a fixed data distribution and minimizing a recursively defined majorizing surrogate of an objective function. In this paper, we introduce stochastic block majorization-minimization, where the surrogates can now be only block multi-convex and a single block is optimized at a time within a diminishing radius. Relaxing the standard strong convexity requirements for surrogates in SMM, our framework gives wider applicability including online CANDECOMP/PARAFAC (CP) dictionary learning and yields greater computational efficiency especially when the problem dimension is large. We provide an extensive convergence analysis on the proposed algorithm, which we derive under possibly dependent data streams, relaxing the standard i.i.d. assumption on data samples. We show that the proposed algorithm converges almost surely to the set of stationary points of a nonconvex objective under constraints at a rate $O(  \frac{  (\log n)^{1+\epsilon}  }{  n^{1/2}  }  )$ for the empirical loss function and $O(  \frac{  (\log n)^{1+\epsilon}  }{  n^{1/4}  }  )$ for the expected loss function, where n denotes the number of data samples processed. Under some additional assumption, the latter convergence rate can be improved to $O(  \frac{  (\log n)^{1+\epsilon}  }{  n^{1/2}  }  )$. Our results provide first convergence rate bounds for various online matrix and tensor decomposition algorithms under a general Markovian data setting.

Introduced by Lehmer in 1933, the classical Mahler measure of a complex rational function $P$ in one or more variables is given by integrating $\log|P(x_1, \ldots, x_n)|$ over the unit torus. Lehmer asked whether the Mahler measures of integer polynomials, when nonzero, must be bounded away from zero, a question that remains open to this day. In this talk we generalize Mahler measure by associating it with a discrete dynamical system $f: \mathbb{C} \to \mathbb{C}$, replacing the unit torus by the $n$-fold Cartesian product of the Julia set of $f$ and integrating with respect to the equilibrium measure on the Julia set. We then characterize those two-variable integer polynomials with dynamical Mahler measure zero, conditional on a dynamical version of Lehmer's conjecture.
 

Fully connected networks are roughly described by two structural parameters: a depth L and a width n. It is well known that, with some important caveats on the scale at initialization, in the regime of fixed L and the limit of infinite n, neural networks at the start of training are a free (i.e. Gaussian) field and that network optimization is kernel regression for the so-called neural tangent kernel (NTK). This is a striking and insightful simplification of infinitely overparameterized networks. However, in this particular infinite width limit neural networks cannot learn data-dependent features, which is perhaps their most important empirical feature. To understand feature learning one must therefore study networks at finite width. In this talk I will do just that. I will report on recent work joint with Dan Roberts and Sho Yaida (done at a physics level of rigor) and some more mathematical ongoing work which allows one to compute, perturbatively in 1/n and recursively in L, all correlation functions of the neural network function (and its derivatives) at initialization. An important upshot is the emergence of L/n, instead of simply L, as the effective network depth. This cut-off parameter provably measures the extent of feature learning and the distance at initialization to the large n free theory.

I will outline a proof that the Ising model has a continuous phase transition on any nonamenable Cayley graph. This will involve some neat probabilistic applications of ergodic-theoretic machinery such as factors of IID and the spectral theory of group actions. I will aim to make the talk accessible to a broad community.

Inherent fluctuations may play an important role in biochemical and biophysical systems when the system involves some species with low copy numbers. This talk will present the recent work on the stochastic modeling of enzyme-catalyzed reactions in biology.

In the first part of the talk, I will introduce a multiscale approximation method that helps reduce network complexity using various scales in species numbers and reaction rate constants. I will apply the multiscale approximation method to simple enzyme kinetics and derive quasi-steady-state approximations. In the second part of the talk, I will show another example for glucose metabolism where we see different-sized enzyme complexes. We hypothesized that the size of multienzyme complexes is related to their functional roles. We will see two models: one using a system of ordinary differential equations and the other using the Langevin dynamics.

Often, a goal of mathematics is to classify objects of a particular type.  In algebraic geometry, the objects are usually of some geometric interest: manifolds, varieties, vector bundles, etc; and after fixing several discrete invariants (like the dimension of the object), we try to classify all the objects with those invariants. This leads to a notion of moduli space, i.e. a space parametrizing all of these objects. We will do several examples and mention both the usefulness and difficulty of these problems!  No background in algebraic geometry is required.

We apply results of Garsia-Haiman-Tesler on Macdonald polynomials to the problem of computation of integrals of tautological classes over the Hilbert schemes of surfaces, studied by Marian-Oprea-Pandharipande. Using localization, these results allow us to find new functional equations for the generating series of integrals. The MOP paper considers two kinds of integrals: the so-called Chern integrals resp. Verlinde integrals. The answer to the problem is encoded in series A1, A2, A3, A4, A5 resp. B1, B2, B3, B4. All the series except A4, A5, B3, B4 were computed in MOP and a conjecture motivated by mathematical physics was formulated relating A4 to B3 and A5 to B4. It was also conjectured that A4, A5, B3, B4 are algebraic functions. Solving our functional equations we prove the former conjecture and obtain explicit formulas for A4 and B3, thus proving a part of the latter conjecture. We also give a conjectural formula for A5 and B4. This is a joint work with Lothar Göttsche

We apply results of Garsia-Haiman-Tesler on Macdonald polynomials to the problem of computation of integrals of tautological classes over the Hilbert schemes of surfaces, studied by Marian-Oprea-Pandharipande. Using localization, these results allow us to find new functional equations for the generating series of integrals. MOP paper considers two kind of integrals: the so-called Chern integrals resp. Verlinde integrals. The answer to the problem is encoded in series A1, A2, A3, A4, A5 resp. B1, B2, B3, B4. All the series except A4, A5, B3, B4 were computed in MOP and a conjecture motivated by mathematical physics was formulated relating A4 to B3 and A5 to B4. It was also conjectured that A4, A5, B3, B4 are algebraic functions. Solving our functional equations we prove the former conjecture and obtain explicit formulas for A4 and B3, thus proving a part of the latter conjecture. We also give a conjectural formula for A5 and B4. This is a joint work with Lothar Göttsche.

As a noncommutative fractal geometer, I look for new expressions of the geometry of a fractal through the lens of noncommutative geometry.  At the quantum scale, the wave function of a particle, but not its path in space, can be studied.  Riemannian methods often rely on smooth paths to encode the geometry of a space.  Noncommutative geometry generalizes analysis on manifolds by replacing this requirement with operator algebraic data.  These same "point-free" techniques can also be used to study the geometry of spaces like fractals.  Recently, Michel Lapidus, Frédéric Latrémoliére, and I identified conditions under which differential structures defined on fractal curves can be realized as a metric limit of differential structures on their approximating finite graphs.  Currently, I am using some of the same tools from that project to understand noncommutative discrete structures.  Progress in noncommutative geometry has produced a rich dictionary of quantum analogues of classical spaces.  The addition of noncommutative discrete structure to this dictionary would enlarge its potential to yield insights about both noncommutative sets and classically pathological sets like fractals.  Time permitting, other works in progress, such as on classification of $C^*$-algebras on fractals, may be discussed. 

Interior-point methods are some of the most effective and widely used methods to finding local minimizers of large-scale non-convex optimization problems. In this talk, we introduce three different mechanisms for ensuring global convergence to second-order local minimizers from arbitrary feasible starting points by solving a sequence of trust-region subproblems defined by quadratic models of a shifted primal-dual penalty-barrier merit function. Each of these methods begins by solving the trust-region subproblem to form a new trial point, and proceeds to refine the trial iterate until a sufficient-decrease condition is met. We suggest two different definitions of the trust region, and provide numerical results comparing each of the different approaches.

In this talk, I will discuss characterizations of stationary solutions of the 2D Euler equations in two different directions under different assumptions: rigidity (is every stationary solution radial?) and flexibility (do there exist non-radial stationary solutions?). The proofs will have a calculus of variations' flavor, a new observation that finite energy, stationary solutions with simply-connected vorticity have compactly supported velocity, and an application of the Nash-Moser iteration procedure. Based on joint work with Jaemin Park, Jia Shi and Yao Yao.

Hilbert’s Nullstellensatz is a fundamental result in commutative algebra which is the starting point for classical algebraic geometry.

In this talk, I will discuss joint work with Robert Burklund and Tomer Schlank on a chromatic version of Hilbert’s Nullstellensatz in which Lubin-Tate theories play the role of algebraically closed fields.I will then sample some applications of our results to chromatic support, redshift, and orientation theory for $E_\infty$ rings.

Golden ratio primal-dual algorithm (GRPDA) is a new variant of the classical Arrow-Hurwicz method for solving structured convex-concave saddle point problem. In this talk, we present GRPDAs with adaptive linesearch, which potentially allows much larger stepsizes, and hence, could significantly accelerate the convergence speed. We show global iterate convergence as well as O($\frac{1}{N}$) ergodic convergence rate results, measured by the function value gap and constraint violations of an equivalent optimization problem. When one of the component functions is strongly convex, faster O($\frac{1}{N^2}$) ergodic convergence rate can be established. In addition, linear convergence can be established when subdifferential operators of the component functions are strongly metric subregular. Our preliminary numerical results show our algorithms perform much better than other state-of-art
 comparison algorithms. 

A powerful tool to study the geometry of Radon measures is the decomposability bundle, which I introduced with Alberti in [On the differentiability of Lipschitz functions with respect to measures in the Euclidean space, GAFA, 2016]. This is a map which, roughly speaking, captures at almost every point the tangential directions to the Lipschitz curves along which the measure can be disintegrated. In this talk I will discuss some recent applications of this flexible tool, including a characterization of rectifiable measures as those measures for which Lipschitz functions admit a Lusin type approximation with functions of class ${C^1}$, the converse of Pansu's theorem on the differentiability of Lipschitz functions between Carnot groups, and a characterization of Federer-Fleming flat chains with finite mass.

In this talk, we will consider the angular component of multipliers of repelling cycles of a hyperbolic rational map in one complex variable. Oh-Winter have shown that these angles of multipliers uniformly distribute in the circle (-$\pi$, $\pi$]. Motivated by the sector problem in number theory, we show that for a fixed $K \gg 1$, almost all intervals of length $\frac{2 \pi}{K}$ in (-$\pi$, $\pi$] contains a multiplier angle with the property that the norm of the multiplier is bounded above by a polynomial in K. This is joint work with Hongming Nie.

Given a smooth projective family $f : X \to S$ defined over the ring of integers of a number field, the Shafarevich problem is to describe those fibres of f which have everywhere good reduction. This can be interpreted as asking for the dimension of the Zariski closure of the set of integral points of $S$, and is ultimately a difficult diophantine question about which little is known as soon as the dimension of $S$ is at least 2. Recently, Lawrence and Venkatesh gave a general strategy for addressing such problems which requires as input lower bounds on the monodromy of f over essentially arbitrary closed subvarieties of $S$. In this talk we review their ideas, and describe recent work which gives a fully effective method for computing these lower bounds. This gives a fully effective strategy for solving Shafarevich-type problems for essentially arbitrary families $f$.

Using a combination of methods from applied mathematics and nonlinear dynamics, we present a constructive way to give a discrete time dynamical rule that accurately forecasts the voltage across a neuron cell membrane. This is the only quantity required to build a biological network of realistic neurons. The construction uses simulated 'data' or observed biophysical data alone to develop the dynamical map. We call this data driven forecasting (DDF). The method is described in detail at first using 'data' from simple neuron models and then using observed neurobiological data from laboratory experiments. It provides accurate forecasting of observed quantities in each setting. 

In an example where a detailed Hodgkin-Huxley (HH) model was developed using data assimilation for observed laboratory observations the DDF neuron runs an order of magnitude faster than the HH version in forecasting the important neuron voltage time course. As the computation required for a network of N nodes will be faster by about a factor of 10N using DDF neurons, this will permit building and analyzing the very large networks desired to address realistic biological questions using elements determined via the biophysics of the component neurons. 

If time permits, we will describe how one may use the DDF idea to substantially reduce the geophysical computations required for regional numerical weather forecasting.

The Tower variant of the Number Field Sieve (TNFS) is known to be asymptotically the most efficient algorithm to solve the discrete logarithm problem in finite fields of medium characteristics, when the extension degree is composite. A major obstacle to an efficient implementation of TNFS is the collection of algebraic relations, as it happens in dimensions greater than 2. This requires the construction of new sieving algorithms which remain efficient as the dimension grows. In the work I will present, we overcome this difficulty by considering a lattice enumeration algorithm which we adapt to this specific context. We also consider a new sieving area, a high-dimensional sphere, whereas previous sieving algorithms for the classical NFS considered an orthotope. Our new sieving technique leads to a much smaller running time, despite the larger dimension of the search space, and even when considering a larger target, as demonstrated by a record computation we performed in a 521-bit finite field $GF(p^6)$. The target finite field is of the same form as finite fields used in recent zero-knowledge proofs in some blockchains. This is the first reported implementation of TNFS.

As anyone at UCSD can tell you, I really like making dumb jokes. Unfortunately, I can end up spending so much time on my jokes that I don't end up doing any mathematics. My solution to this problem has been to write math papers which are based on jokes. Somehow I've managed to do this 3 times. In this talk I'll briefly discuss these joke papers. No prior knowledge of jokes or any sense of humor will be assumed. Current UCSD students, prospective students, and anyone else who isn't on my thesis committee is welcome to attend.

I will present conjectures and results concerning intersection theoretic invariants of parameter spaces of sheaves over low dimensional varieties, with emphasis on the Quot schemes of curves and surfaces.

A Schubert variety in the complete flag variety $GL_n/B$ is Levi-spherical if the action of a Borel subgroup in a Levi subgroup of a standard parabolic has an open dense orbit. I will present some recent work, joint with Alexander Yong and Yibo Gao, giving a classification of Levi-spherical Schubert varieties in terms of spherical elements of a symmetric group. I will also discuss a conjectural extension of the classification to other Lie types.

For property (T) II$_1$ factors, any inclusion into a tracial von Neumann algebra has spectral gap, and therefore weak spectral gap. I will discuss characterizations of property (T) for II$_1$ factors by weak spectral gap in inclusions. I will explain how this is related to the non-weakly-mixing property of the bimodules containing almost central vectors, from which we also obtain a characterization of property (T).

Typical comparison results in Riemannian geometry, such as for volume or for spectrum of the Laplacian, require Ricci curvature lower bounds. In dimension three, we can prove several sharp comparison estimates assuming only a scalar curvature bound. The talk will present these results, their applications and describe how dimension three is used in the proofs. Joint work with Jiaping Wang.

Decision tree-based models are a popular tool for use in prediction and regression machine learning problems. In this talk, we will provide an overview of decision tree models and ensemble methods, including (but not limited to) random forests and XGBoost. We'll also discuss considerations of building such models and some applications. This talk does not require any background knowledge in machine learning.

Groupoids give a very large class of examples of C*-algebras. For example, it is known that every classifiable C*-algebra arises as the reduced C*-algebra of some twisted groupoid.

In joint work with Matthew Kennedy, Se-Jin Kim, Xin Li, and Sven Raum, we fully characterize when the essential C*-algebra of an étale groupoid $\mathcal{G}$ with locally compact unit space has the ideal intersection property. This is done in terms of the dynamics of $\mathcal{G}$ on the space of subgroups of the isotropy groups of $\mathcal{G}$. The essential and reduced C*-algebras coincide in the case of Hausdorff groupoids, and the ideal intersection property is the same as simplicity in the case of minimal groupoids. This generalizes the case of the reduced crossed product $C(X) \rtimes_r G$ done by Kawabe, which in turn generalizes the case of the reduced C*-algebra $C^*_r(G)$ of a discrete group done by Breuillard, Kalantar, Kennedy, and Ozawa.

No prior knowledge of groupoids will be required for this talk.

I will communicate an amusing observation about the K theory of non-noetherian schemes in characteristic zero. The lowest K group in this setting can sometimes identify with the top cohomology of the structure sheaf. No knowledge of negative K theory (or even K theory!) will be assumed, with the hope that both topologists and algebraic geometers can learn something.

Stochastic iterative methods lie at the core of large-scale optimization and its modern applications to data science. Though such algorithms are routinely and successfully used in practice on highly irregular problems (e.g. deep neural networks), few performance guarantees are available outside of smooth or convex settings. In this talk, I will describe a framework for designing and analyzing stochastic gradient-type methods on a large class of nonsmooth and nonconvex problems. The problem class subsumes such important tasks as matrix completion, robust PCA, and minimization of risk measures, while the methods include stochastic subgradient, Gauss-Newton, and proximal point iterations. I will describe a number of results, including finite-time efficiency estimates, avoidance of extraneous saddle points, and asymptotic normality of averaged iterates.

Shrinking Ricci solitons are Ricci flow solutions that self-similarly shrink under the flow. Their significance comes from the fact that finite-time Ricci flow singularities are typically modeled on gradient shrinking Ricci solitons. Here, we shall address a certain converse question, namely, "Given a complete, noncompact gradient shrinking Ricci soliton, does there exist a Ricci flow on a closed manifold that forms a finite-time singularity modeled on the given soliton?"

We'll discuss recent work that shows the answer is yes when the soliton is asymptotically conical. No symmetry or Kahler assumption is required, and so the proof involves an analysis of the Ricci flow as a nonlinear degenerate parabolic PDE system in its full complexity. We'll also discuss applications to the (non-)uniqueness of weak Ricci flows through singularities.

In joint work with Lucas Mann, we establish several new properties of the p-adic Jacquet-Langlands functor defined by Scholze in terms of the cohomology of the Lubin-Tate tower. In particular, we prove a duality theorem, establish bounds on Gelfand-Kirillov dimension, prove some non-vanishing results, and show a kind of partial Künneth formula. The key new tool is the six functor formalism with solid almost $\mathcal{O}^+/p$-coefficients developed recently by Mann.

The aim of this talk is to introduce the audience to the Halpern-Lauchli theorem, which is a Ramsey-theoretic statement about products of trees. We will discuss several applications of the theorem and outline a number of different proofs. While the original proof was combinatorial in nature, there are now a number of proofs that interact with ideas from set-theoretic forcing. One of these proofs is new, and is joint work with Chris Lambie-Hanson.

In the last two decades great progress has been made in the study of dispersive and wave equations. Over the years the toolbox used in order to attack highly nontrivial problems related to these equations has developed to include a collection of techniques: Fourier and harmonic analysis, analytic number theory, math physics, dynamical systems, probability and symplectic geometry. In this talk I will introduce a variety of results using as model problem mainly the periodic 2D cubic nonlinear Schrödinger equation. I will start by giving a physical derivation of the equation from a quantum many-particles system, I will introduce periodic Strichartz estimates along with some remarkable connections to analytic number theory, I will move on to the concept of energy transfer and its connection to dynamical systems, and I will end with some results on the derivation of a wave kinetic equation.

We will construct a noncommutative polynomial, P, in 2n variables such that every 2n-tuple of nxn matrices vanish when plugged into P. The Cayley-Hamilton theorem will be the key ingredient.
 

Nash equilibrium problems (NEPs) are games for several players . A Nash Equilibrium (NE) is a tuple of strategies such that each player's benefits cannot be improved when the other players' strategies are fixed. For NEPs given by polynomial functions, we formulate efficient polynomial optimization problems for computing NEs. The Moment-SOS relaxations are used to solve them. Under genericity assumptions, the method can find a Nash equilibrium if there is one; it can also find all NEs if there are finitely many ones. The method can also detect nonexistence if there is no NE.

This is a joint work with Dr. Xindong Tang.
 

In this talk we will discuss a new approach to the problem of emergence in hydrodynamic systems of collective behavior. The problem seeks to establish convergence to a flocking state in a system with self-organization governed by strictly local laws of communication. The typical results in this direction insist on propagation of flock connectivity which translates into a quantitative non-vacuum condition on macroscopic level. With the introduction of small noise one can relax such a condition considerably, and even allow for vacuum, in the context of the corresponding Fokker-Planck-Alignment equations. The flocking behavior becomes the problem of establishing hypocoercivity and relaxation of solutions to the global Maxwellian. We will describe a model which does precisely that in the non-perturbative settings.

The moduli stack of oriented formal groups embodies, in the world of spectral algebraic geometry, the fundamental chromatic connection between the stable homotopy category and formal groups. As such, it validates the folklore picture of Morava, Hopkins, et al. Somewhat surprisingly, it is also closely related to a more recent development: the "cofiber of $\tau$ philosophy" of Gheorghe-Isaksen-Wang-Xu.

In this talk, we will introduce the moduli stack of oriented formal groups, and explain how the algebro-geometric structure of its connective cover reflects and gives rise to the $\tau$-deformation structure of cellular motivic spectra over $\mathbb{C}$.

The optimal transport problem is known to suffer the curse of dimensionality. A recently proposed approach to mitigate the curse of dimensionality is to project the sampled data from the high dimensional probability distribution onto a lower-dimensional subspace, and then compute the optimal transport between the projected data. However, this approach requires to solve a max-min problem over the Stiefel manifold, which is very challenging in practice. In this talk, we propose a Riemannian block coordinate descent (RBCD) method to solve this problem. We analyze the complexity of arithmetic operations for RBCD to obtain an $\epsilon$-stationary point, and show that it significantly improves the corresponding complexity of existing methods. Numerical results on both synthetic and real datasets demonstrate that our method is more efficient than existing methods, especially when the number of sampled data is very large. We will also discuss how the same idea can be used to solve the projection robust Wasserstein barycenter problem.

Motivated by questions about a p-adic Fourier transform, we study invariant norms on the p-adic Schrödinger representations of Heisenberg groups. These Heisenberg groups are p-adic, and the Schrodinger representations are explicit irreducible smooth representations that play an important role in their representation theory.

Classically, the field of coefficients is taken to be the complex numbers and, among other things, one studies the unitary completions of the representations (which are well understood). By taking the field of coefficients to be an extension of the p-adic numbers, we can consider completions that better capture the p-adic topology, but at the cost of losing the Haar measure and the $L^2$-norm. Nevertheless, we establish a rigidity property for a family of norms (parametrized by a Grassmannian) that are invariant under the action of the Heisenberg group.

The irreducibility of some Banach representations follows as a result. The proof uses "q-arithmetics".

Let $\mu$ be a measure on a finite group $G$. We define the spectral gap of $\mu$ to be the operator norm of the map that sends $\phi \in L^2(G)^{\circ}$ to $\mu * \phi$. We say that $\mu$ is symmetric if $\mu(x) = \mu(x^{-1})$. Now fix $G = SL_2(\mathbb{Z}/n\mathbb{Z}) \times SL_2(\mathbb{Z}/n\mathbb{Z})$, with $n \in \mathbb{N}$ being arbitrary. Suppose that $\mu$ is a measure on $G$ such that it's pushforwards to the left and right component have spectral gaps lesser than $\lambda_0 < 1$ and $\mu$ takes a minimum of $\alpha_0$ on it's support. Further suppose that the support of $\mu$ generates $G$. Then we show that there are constants $L, \beta > 0$ depending only on $\lambda_0,\alpha_0$ such that $\mu^{(*)L\log |G|}(\Gamma) \leq \frac{1}{|G|^{\beta}}$, where $\Gamma$ is the graph of any automorphism of $SL_2(\mathbb{Z}/n\mathbb{Z})$. We will discuss this result and its implications. This talk is based on joint work with Professor Alireza Salehi-Golsefidy.

We consider Ricci flows admitting closed and smooth tangent flows in the sense of Bamler 20. The tangent flow in question can be either a tangent flow at infinity for an ancient Ricci flow, or a tangent flow at a singular point for a Ricci flow developing a finite-time singularity. In these cases, we show that the tangent flow is unique and the singularity is of Type I. This talk is based on a joint work with Zilu Ma and Yongjia Zhang.

We will discuss some of the known power savings for the number of $G$-extensions of a number field with discriminant bounded above by $X$. We will put a focus on the existence of secondary terms in the asymptotic growth rate, and in particular will discuss a proof of the existence of some secondary terms when $G$ is abelian.

Let M be a smooth submanifold of $\mathbb{R}^n$ equipped with the Euclidean(chordal) metric. This talk will consider the smallest dimension, m, for which there exists a bi-Lipschitz function f : M → $\mathbb{R}^m$ with biLipschitz constants close to one. We will begin by presenting a bound for the embedding dimension m from below in terms of the bi-Lipschitz constants of f and the reach, volume, diameter, and dimension of M. We will then discuss how this lower bound can be applied to show that prior upper bounds by Eftekhari and Wakin on the minimal low-distortion embedding dimension of such manifolds using random matrices achieve near-optimal dependence on dimension, reach, and volume (even when compared against nonlinear competitors). Next, we will discuss a new class of linear maps for embedding arbitrary (infinite) subsets of $\mathbb{R}^n$ with sufficiently small Gaussian width which can both (i) achieve near-optimal embedding dimensions of submanifolds, and (ii) be multiplied by vectors in faster than FFT-time. When applied to d-dimensional submanifolds of $\mathbb{R}^n$ we will see that these new constructions improve on prior fast embedding matrices in terms of both runtime and embedding dimension when d is sufficiently small.

This is joint work with Benjamin Schmidt (MSU) and Arman Tavakoli (MSU).

In classical ergodic theory, compact and weakly mixing actions/extensions have been well-studied. The main structural result from Furstenberg and Zimmer states that every action can be written as "a weakly mixing extension of a tower of compact extensions". We will discuss some of these classical results and their motivation, and consider similar notions for actions on von Neumann algebras which have been defined throughout the years. We will then complete (part of) the picture by establishing equivalence of several such notions, followed by some consequences and open questions. This is partially based on joint work with Asgar Jamneshan. 

Given a desired target distribution and an initial guess of that distribution, composed of finitely many samples, what is the best way to evolve the locations of the samples so that they accurately represent the desired distribution? A classical solution to this problem is to allow the samples to evolve according to Langevin dynamics, a stochastic particle method for the Fokker-Planck equation. In today’s talk, I will contrast this classical approach with a deterministic particle method corresponding to the porous medium equation. This method corresponds exactly to the mean-field dynamics of training a two layer neural network for a radial basis function activation function. We prove that, as the number of samples increases and the variance of the radial basis function goes to zero, the particle method converges to a bounded entropy solution of the porous medium equation. As a consequence, we obtain both a novel method for sampling probability distributions as well as insight into the training dynamics of two layer neural networks in the mean field regime.
 

In this talk, we will discuss an unstable approach to studying stable homotopy groups as pioneered by Curtis and Wellington. Using the Barratt-Priddy-Quillen theorem, we can identify the (co)homology of $BS_\infty$ with the (co)homology of the base point component of the loop space which represents stable homotopy. Using cohomology instead of homology to exploit the nice Hopf ring presentation of Giusti, Salvatore, and Sinha for the cohomology of symmetric groups, we find a width filtration, whose subquotients are simple quotients of Dickson algebras, which thus give a new filtration of stable homotopy. We make initial calculations and determine towers in the resulting width spectral sequence. We also make  calculations related to the image of J and conjecture that it is captured exactly by the lowest filtration in the width spectral sequence.

Deep neural networks have been widely used in massive engineering domains, but the training and deployment of neural networks are subject to many fundamental challenges. In the training phase, the large-scale optimization often consumes a huge amount of computing and energy resources. In practical deployment, we often need the capability of uncertainty quantification to ensure the safe operations in an uncertain environment. To address the first challenge, we need compressed training, but it is hard to determine the compression ratio automatically in the training phase. To address the second challenge, we often use Bayesian learning models, but the resulting uncertainty-aware model often leads to massive model copies which cause huge memory and computing overhead.

In this talk, we show that the interplay of compressed training and Bayesian learning can provide more sustainable neural network models. Firstly, we investigate end-to-end tensor compressed training. This approach can offer orders-of-magnitude parameter reduction in the training phase, but it is hard to determine the tensor rank and model complexity automatically. We show that efficient Bayesian formulation and solver can be developed to address this major challenge, enabling high-accuracy end-to-end compressed training as well as energy-efficient on-device training. Secondly, we investigate MCMC-type Bayesian training. Here the main challenge is how to use a small number of model copies to accurately represent model uncertainties. We provide an online and provable online sample thinning method based on kernelized Stein discrepancy. This method can reduce the model copies on the fly, and offers orders-of-magnitude memory and latency savings in the inference.

 

Speaker’s Bio:

Dr. Zheng Zhang is an Assistant Professor of Electrical and Computer Engineering at University of California, Santa Barbara. He received his PhD degree in Electrical Engineering and Computer Science from MIT in 2015. His research is focused on uncertainty quantification and tensor computation, with applications to multi-domain design automation, sustainable and trustworthy AI systems. He received the ACM SIGDA Outstanding New Faculty Award, IEEE CEDA Early CAREER Award, NSF Early Career Award, and three best journal paper awards from IEEE Transactions in the EDA research field. He is the receipt of ACM SIGDA Outstanding Dissertation Award in 2016, and MIT Microsystems Technology Lab PhD Dissertation Award in 2015.

In this talk, we will first introduce the complex continued fraction maps associated with some imaginary quadratic fields ($d=1, 2, 3, 7, 11$) and their dynamical properties. Baladi-Vallee analyzed (real) Euclidean algorithms and proved the central limit theorem for rational trajectories and a wide class of cost functions measuring algorithmic complexity. They used spectral properties of an appropriate bivariate transfer operator and a generating function for certain Dirichlet series whose coefficients are essentially the moment generating function of the cost on the set of rationals. We extend the work of Baladi-Vallee for complex continued fraction maps mentioned above. (This is joint work with Dohyeong Kim and Jungwon Lee.)

Let $N\geq3$ and $r\geq1$ be integers and $p\geq2$ be a prime such that $p\nmid N$. One can consider two different integral structures on the space of modular forms over $\mathbb{Q}$, one coming from arithmetic via $q$-expansions, the other coming from geometry via integral models of modular curves. Both structures are stable under the Hecke operators; furthermore, their quotient is finite torsion. Our goal is to investigate the exponent of the annihilator of the quotient. We will apply results due to Brian Conrad to the situation of modular forms of even weight and level $\Gamma(Np^{r})$ over $\mathbb{Q}_{p}(\zeta_{Np^{r}})$ to obtain an upper bound for the exponent. We also use Klein forms to construct explicit modular forms of level $p^{r}$ whenever $p^{r}>3$, allowing us to compute a lower bound which agrees with the upper bound. Hence we are able to compute the exponent precisely.

In this talk, I will give an introduction to optimal transport, which has evolved as one of the major frameworks to meaningfully compare distributional data. The focus will mostly be on machine learning, and how optimal transport can be used efficiently for clustering and supervised learning tasks. Applications of interest include image classification as well as medical data such as gene expression profiles.

The Gray Scott model is a set of reaction diffusion equations known to generate a wide variety of patterns. In this talk we consider a version of this model where diffusion is assumed to be nonlocal and can be described by convolution kernels that decay exponentially at infinity and have finite second moment. We prove the local well-posedness of the model on bounded one-dimensional domains with nonlocal Dirichlet and Neumann boundary constraints. We also present a numerical scheme that uses a quadrature-based finite difference to discretize the convolution operator. We show how the scheme allows us to approximate solutions to the nonlocal Gray Scott model both on bounded and unbounded domains.

We study unstable topological complex vector bundles over complex projective spaces. It is a classical problem in algebraic topology to count the number of rank $r$ bundles over $\mathbb{C}P^n$ (with $1 < r < n$) having fixed Chern class data. A particular case is when the Chern data is trivial, which we call the vanishing Chern enumeration. We apply a modern tool, Weiss calculus, to produce the vanishing Chern enumeration in the first two unstable cases (which belong to what we call the "metastable" range, following Mark Mahowald), namely rank $(n - 1)$ bundles over $\mathbb{C}P^n$ for $n > 2$, and rank $(n - 2)$ bundles over $\mathbb{C}P^n$ for $n > 3$.

The talk will describe some problems which arise in finding optimal quantum strategies for games.  In such problems one has a (noncommutative) algebra A which encodes quantum mechanical laws and a noncommutative polynomial b which corresponds to a particular game and tells its score.  The goal of the talk will be to give an idea of some  of the structure, methods and our results which arise in maximizing b.  To get more warning of what is in the talk see the last few years of my postings on arXiv with collaborators: Adam Bene Watts, Igor Klep and Vern Paulsen etal.

I will report on work in progress showing that the geodesic flow on any geometrically finite, rank one, locally symmetric space is exponentially mixing with respect to the Bowen-Margulis-Sullivan measure of maximal entropy. The method is coding-free and is instead based on a spectral study of transfer operators on suitably constructed anisotropic Banach spaces, ala Gouezel-Liverani, to take advantage of the smoothness of the flow. As a consequence, we obtain more precise information on the size of the essential spectral gap as well as the meromorphic continuation properties of Laplace transforms of correlation functions.

The lecture will discuss recent joint work with Costante Bellettini at UCL. A main outcome of the work is a proof that for any closed Riemannian manifold $N$ of dimension $n \geq 3$ and any non-negative (or non-positive) Lipschitz function $g$ on $N$, there is a boundaryless $C^{2}$ hypersurface $M \subset N$ whose scalar mean curvature is prescribed by $g.$ More precisely, the hypersurface $M$ is the image of a quasi-embedding $\iota$ (of class $C^{2}$) admitting a global unit normal $\nu$ such that the mean curvature of $\iota$ at every point $x$ is $g(\iota(x))\nu(x)$. Here a 'quasi-embedding' is an immersion such that any point of its image near which the image is not embedded has an ambient neighborhood in which the image is the union of two $C^{2}$ embedded disks with each disk lying on one side of the other (so that any self-intersection is tangential). If $n \geq 7$, the singular set $\overline{M} \setminus M$ may be non-empty, but has Hausdorff dimension no greater than $n-7$. An important special case is the existence of a CMC hypersurface for any prescribed value of mean curvature. The method of proof is PDE theoretic. It utilises the elliptic and parabolic Allen-Cahn equations on $N$, and brings to bear on the question elementary, and yet very effective, variational and gradient flow principles in semi-linear elliptic and parabolic PDE theory--principles that serve as a conceptually and technically simpler alternative to the Geometric Measure Theory machinery pioneered by Almgren and Pitts to prove existence of a minimal hypersurface. For regularity conclusions the method relies on a new varifold regularity theory, a ''black-box'' tool of independent interest (also joint work with Bellettini). This theory provides multi-sheeted $C^{1, \alpha}$ regularity for mean-curvature-controlled codimension 1 integral varifolds $V$ near points where one tangent cone is a hyperplane of multiplicity $q \geq 2;$ this regularity holds whenever: (i) $V$ has no classical-singularities, i.e. no portion of $V$ is the union of three or more embedded hypersurfaces-with-boundary coming smoothly together along their common boundary, and (ii) the region where the mass density of $V$ is $< q$ is 'well-behaved' in a certain topological sense. A very important feature of this theory, crucial for its application to the Allen--Cahn method, is that $V$ is not assumed to be a critical point of a functional.

Learning operators between infinitely dimensional spaces is an important learning task arising in wide applications in machine learning, data science, mathematical modeling and simulations, etc. This talk introduces a new discretization-invariant operator learning approach based on data-driven kernels for sparsity via deep learning. Compared to existing methods, our approach achieves attractive accuracy in solving forward and inverse problems, prediction problems, and signal processing problems with zero-shot generalization, i.e., networks trained with a fixed data structure can be applied to heterogeneous data structures without expensive re-training. Under mild conditions, quantitative generalization error will be provided to understand discretization-invariant operator learning in the sense of non-parametric estimation.

Supersingular isogeny-based cryptosystems are strong contenders for post-quantum cryptography standardization. Such cryptosystems rely on the hardness of path-finding on supersingular isogeny graphs. The path-finding problem is known to reduce to the endomorphism ring problem. Can path-finding be reduced to knowing just one endomorphism? In this talk, we give explicit classical and quantum algorithms for path-finding to an initial curve using the knowledge of one endomorphism. An endomorphism gives an orientation of a supersingular elliptic curve. We use the theory of oriented supersingular isogeny graphs and algorithms for taking ascending/descending/horizontal steps on such graphs.

Interesting moduli spaces don't have many integral points. More precisely, if X is a variety over a number field, admitting a variation of Hodge structure whose associate period map is injective, then the number of S-integral points on X of height at most H grows more slowly than $H^ε$, for any positive ε.  This is a sort of weak generalization of the Shafarevich conjecture; it is a consequence of a point-counting theorem of Broberg, and the largeness of the fundamental group of X.  Joint with Ellenberg and Venkatesh.

Consider the following variant of Rock, Paper, Scissors (RPS) played by two players Rei and Norman.  The game consists of $3n$ rounds of RPS, with the twist being that Rei (the restricted player) must use each of Rock, Paper, and Scissors exactly $n$ times during the $3n$ rounds, while Norman is allowed to play normally without any restrictions. We show that a certain greedy strategy is the unique optimal strategy for Rei in this game, and that Norman's expected score is $\Theta(\sqrt{n})$.  We also prove several general theorems about semi-restricted games arising from digraphs.  This is joint work with Erlang Surya, Yuanfan Wang, Ji Zeng.

The 3D incompressible Euler equations in a bounded domain are most often supplemented with impermeable boundary conditions, which constrain the fluid to neither enter nor leave the domain. In this talk, I will explain how we obtain well-posedness of solutions in which the full value of the velocity is specified on inflow and the normal component is specified on outflow. We do this for multiply connected domains, and establish compatibility conditions to obtain arbitrarily high Holder regularity.

This is joint work with Gung-Min Gie and Anna Mazzucato.

Properly proximal groups were introduced recently by Boutonnet, Ioana, and Peterson, where they generalized several rigidity results to the setting of higher-rank groups. In this talk, I will describe how the notion of proper proximality fits naturally in the realm of von Neumann algebras. I will also describe several applications, including that the group von Neumann algebra of a non-amenable inner-amenable group cannot embed into a free group factor, which solves a problem of Popa. This is joint work with Srivatsav Kunnawalkam Elayavalli and Jesse Peterson.

Optimal Transport, a theory for optimal allocation of resources, is widely used in various fields such as astrophysics, machine learning, and imaging science. However, many applications impose elementwise constraints on the transport plan which traditional optimal transport cannot enforce. Here we introduce Supervised Optimal Transport (sOT) that formulates a constrained optimal transport problem where couplings between certain elements are prohibited according to specific applications. sOT is proved to be equivalent to an $l^1$ penalized optimization problem, from which efficient algorithms are designed to solve its entropy regularized formulation. We demonstrate the capability of sOT by comparing it to other variants and extensions of traditional OT in color transfer problem. We also study the barycenter problem in sOT formulation, where we discover and prove a unique reverse and portion selection (control) mechanism. Supervised optimal transport is broadly applicable to applications in which constrained transport plan is involved and the original unit should be preserved by avoiding normalization.

In recent years, our understanding of stable homotopy groups of spheres at $p=2$ increased drastically due to work of Isaksen, Wang, and Xu. A primary method they used is the "cofiber-of-tau philosophy" in the stable infinity category of 2-complete $\mathbb{C}$-motivic spectra. To a sufficiently nice spectrum $E$, Pstragowski produced an infinity-categorical deformation of spectra called "$E$-synthetic spectra," which exhibits and generalizes the cofiber-of-tau phenomena seen in $\mathbb{C}$-motivic spectra. $E$-synthetic spectra are closely related to the $E$-Adams spectral sequence and this relation has had many applications in recent years for Adams spectral sequence calculations. 

In this talk, we discuss some of the basic calculational features of synthetic spectra in the case of $E=H\mathbb{F}_2$, including how to compute bigraded synthetic homotopy groups and their applications to classical Adams spectral sequence calculations for $p=2$. In particular, we discuss our computation of the bigraded synthetic homotopy groups of 2-complete tmf, the connective topological modular forms spectrum.
 

In joint work with Jens Marklof, we prove new upper bounds for theta sums -- finite exponential sums with a quadratic form in the oscillatory phase -- in the case of smooth and box truncations. This generalizes results of Fiedler, Jurkat and Körner (1977) and Fedotov and Klopp (2012) for one-variable theta sums and, in the multi-variable case, improves previous estimates obtained by Cosentino and Flaminio (2015). Key inputs in our approach include the geometry of $\mathrm{Sp}(n, \mathbb{Z}) \backslash \mathrm{Sp}(n, \mathbb{R})$, the automorphic representation of theta functions and their growth in the cusp, and the action of the diagonal subgroup of $\mathrm{Sp}(n, \mathbb{R})$.

We will present some recent results on the construction of blow-up solutions for critical parabolic problems of geometric flavor. Initiated in the recent years, the inner/outer parabolic gluing is a very versatile parabolic version of the well-known Lyapunov-Schmidt reduction in elliptic PDE theory. The method allows to prove rigorously some formal matching asymptotics (if any available) for several PDEs arising in porous media, geometric flows, fluids, etc….I will give an overview of the strategy and will present several applications to (variations of) the harmonic map flow and Yamabe flow. I will also present some open questions.

A variety satisfies potential density if it contains a dense subset of rational points after extending its ground field by a finite degree. A collection of varieties satisfies uniform potential density if that degree can be uniformly bounded. I will discuss this property for connected algebraic groups of a fixed dimension and elliptic K3 surfaces. This is joint work with Kuan-Wen Lai.

In this talk, I will give an introduction to subdivision schemes, which are iterative refinement processes for interpolating or approximating discrete data points. Most result on subdivision schemes concern data in vector spaces and rules which are linear. I will present an adaptation of subdivision schemes to operate on manifold-valued data using the intrinsic geometry of the underlying manifold (such as the exponential map). Analysis of convergence and smoothness properties will be presented as well. Subdivision schemes find applications in computer graphics and 3D animated movies.

Data-driven machine learning methods such as those based on deep learning are playing a growing role in many areas of science and engineering for modeling time series, including fluid flows, and climate data. However, deep neural networks are known to be sensitive to various adversarial environments, and thus out of the box models and methods are often not suitable for mission critical applications. Hence, robustness and trustworthiness are increasingly important aspects in the process of engineering new neural network architectures and models. In this talk, I am going to view neural networks for time series prediction through the lens of dynamical systems. First, I will discuss deep dynamic autoencoders and argue that integrating physics-informed energy terms into the learning process can help to improve the generalization performance as well as robustness with respect to input perturbations. Second, I will discuss novel continuous-time recurrent neural networks that are more robust and accurate than other traditional recurrent units. I will show that leveraging classical numerical methods, such as the higher-order explicit midpoint time integrator, improves the predictive accuracy of continuous-time recurrent units as compared to using the simpler one-step forward Euler scheme. Finally, I will discuss extensions such as multiscale ordinary differential equations for learning long-term sequential dependencies and a connection between recurrent neural networks and stochastic differential equations.

Speaker’s Bio:

Michael W. Mahoney is at the University of California at Berkeley in the Department of Statistics and at the International Computer Science Institute (ICSI).  He is also an Amazon Scholar as well as head of the Machine Learning and Analytics Group at the Lawrence Berkeley National Laboratory.  He works on algorithmic and statistical aspects of modern large-scale data analysis.  Much of his recent research has focused on large-scale machine learning, including randomized matrix algorithms and randomized numerical linear algebra, scalable stochastic optimization, geometric network analysis tools for structure extraction in large informatics graphs, scalable implicit regularization methods, computational methods for neural network analysis, physics informed machine learning, and applications in genetics, astronomy, medical imaging, social network analysis, and internet data analysis.  He received his PhD from Yale University with a dissertation in computational statistical mechanics, and he has worked and taught at Yale University in the mathematics department, at Yahoo Research, and at Stanford University in the mathematics department.  Among other things, he was on the national advisory committee of the Statistical and Applied Mathematical Sciences Institute (SAMSI), he was on the National Research Council's Committee on the Analysis of Massive Data, he co-organized the Simons Institute's fall 2013 and 2018 programs on the foundations of data science, he ran the Park City Mathematics Institute's 2016 PCMI Summer Session on The Mathematics of Data, he ran the biennial MMDS Workshops on Algorithms for Modern Massive Data Sets, and he was the Director of the NSF/TRIPODS-funded FODA (Foundations of Data Analysis) Institute at UC Berkeley.  More information is available at https://www.stat.berkeley.edu/~mmahoney/.

The classical Segal--Bargmann transform (SBT) is an isomorphism between a real Gaussian Hilbert space and a reproducing kernel Hilbert space of holomorphic functions.  It arises in quantum field theory, as a concrete witness of wave-particle duality.  Introduced originally in the 1960s, it has been generalized and extended to many contexts: Lie Groups (Hall, Driver, late 1980s and early 1990s), free probability (Biane, early 2000s), and more recently $q$-Gaussian factors (Cébron, Ho, 2018).

In this talk, I will discuss current work with Charlesworth and Ho on a version of the SBT in bifree probability, a "two faced" version of free probability introduced by Voiculescu in 2014.  Our work leads to some interesting new combinatorial structures ("stargazing partitions"), as well as a detailed analysis of the resultant family of reproducing kernels.  In the end, the bifree SBT has a surprising connection with the $q$-Gaussian version for some $q\ne 0$.

Stochastic iterative methods lie at the core of large-scale optimization and its modern applications to data science. Though such algorithms are routinely and successfully used in practice on highly irregular problems (e.g., deep learning), few performance guarantees are available outside of smooth or convex settings. In this talk, I will describe a framework for designing and analyzing stochastic gradient-type methods on a large class of nonsmooth and nonconvex problems. The problem class subsumes such important tasks as matrix completion, robust PCA, and minimization of risk measures, while the methods include stochastic subgradient, Gauss Newton, and proximal point iterations. I will describe a number of results, including finite time efficiency estimates, avoidance of extraneous saddle points, and asymptotic normality of averaged iterates.

In this talk, we consider the maximization of a probability $\mathbb{P}\{ \zeta \mid \zeta \in K(x)\}$ over a closed and convex set $\mathcal X$, a special case of the chance-constrained optimization problem. We define $K(x)$ as $K(x) \triangleq \{ {\zeta} \in к \mid c(x,\zeta) \geq 0 \}$ where $\zeta$ is uniformly distributed on a convex and compact set $к$ and $c(x,\zeta)$ is defined as either {$c(x,\zeta) \triangleq  1-|\zeta^Tx|m$, $m\geq 0$} (Setting A) or $c(x,\zeta) \triangleq Tx - \zeta$ (Setting B). We show that in either setting, by leveraging recent findings in the context of non-Gaussian integrals of positively homogenous functions,  $\mathbb{P}\{ \zeta \mid \zeta \in K(x)\}$ can be expressed as the expectation of a suitably defined ${continuous}$ function $F({\bullet},\xi)$ with respect to an appropriately defined Gaussian density (or its variant), i.e. $\mathbb{E}_{\tilde p} [F(x,\xi)]$. Aided by a recent observation in convex analysis, we then develop a convex representation of the original problem requiring the minimization of ${g(\mathbb{E} [F(x,\xi)])}$ over $\mathcal X$  where ${g}$ is an appropriately defined smooth convex function. Traditional stochastic approximation schemes cannot contend with the minimization of ${g(\mathbb{E} [F(\bullet,\xi)])}$ over $\mathcal X$, since conditionally unbiased sampled gradients are unavailable. We then develop a regularized variance-reduced stochastic approximation (${\bf r-VRSA}$) scheme that obviates the need for such unbiasedness by combining iterative ${regularization}$ with variance-reduction. Notably,  (${\bf r-VRSA}$) is characterized by both almost-sure convergence guarantees, a convergence rate of $\mathcal{O}(1/k^{1/2-a})$ in expected sub-optimality where $a > 0$, and a sample complexity of $\mathcal{O}(1/\epsilon^{6+\delta})$ where $\delta > 0$.

To the best of our knowledge, this may be the first such scheme for probability maximization problems with convergence and rate guarantees. Preliminary numerics on a portfolio selection problem (Setting A) and a vehicle routing problem (Setting B) suggest that the scheme competes well with naivemini-batch SA schemes as well as integer programming approximation methods. This is joint work with Ibrahim Bardakci, Afrooz Jalilzadeh, and Constantino Lagoa. Time permitting, a brief summary of ongoing work will be provided on ongoing research in hierarchical optimization and games under uncertainty.

We discuss the notion of "tight inclusions" of dynamical systems which is meant to capture a certain tension between topological and measurable rigidity of boundary actions, and its relevance to Zimmer-amenable actions. Joint work with Mehrdad Kalantar

We interpret a harmonic Maass form as a variant of a local cohomology class of the modular curve. This is not only amenable to algebraic interpretation, but also nicely generalized to other Shimura varieties, avoiding the barrier of Koecher's principle, which could be useful for developing a generalization of Borcherds lifts. In this talk, we will exhibit how the theory looks like in the case of Hilbert modular varities.

We study two McKean-Vlasov equations involving hitting times. Let $(B(t); t \geq 0)$ be standard Brownian motion, and $\tau:= \inf\{t \geq 0: X(t) \leq 0\}$ be the hitting time to zero of a given process $X$. The first equation is $X(t) = X(0) + B(t) - \alpha \mathbb{P}(\tau \leq t)$.

We provide a simple condition on $\alpha$ and the distribution of $X(0)$ such that the corresponding Fokker-Planck equation has no blow-up, and thus the McKean-Vlasov dynamics is well-defined for all time $t \geq 0$. We take the PDE approach and develop a new comparison principle.

The second equation is $X(t) = X(0) + \beta t + B(t) + \alpha \log \mathbb{P}(\tau \leq t)$, $t \geq 0$, whose Fokker-Planck equation is non-local. We prove that if $\beta,1/\alpha > 0$ are sufficiently large, the McKean-Vlasov dynamics is well-defined for all time $t \geq 0$. The argument is based on a relative entropy analysis. This is joint work with Erhan Bayraktar, Gaoyue Guo and Wenpin Tang.

The cavity and TAP equations are high-dimensional systems of nonlinear equations of the local magnetization in the Sherrington-Kirkpatrick model. In the seminal work, Bolthausen introduced an iterative scheme that produces an asymptotic solution to the TAP equations if the model lies inside the Almeida-Thouless transition line. However, it was unclear if this asymptotic solution coincides with the true local magnetization. In this work, motivated by the cavity equations, we introduce a new iterative scheme and establish a weak law of large numbers. We show that our new scheme is asymptotically the same as the so-called Approximate Message Passing (AMP) algorithm that has been popularly adapted in compressed sensing, Bayesian inferences, etc. Based on this, we confirm that our cavity iteration and Bolthausen’s scheme both converge to the local magnetization as long as the overlap is locally uniformly concentrated. This is a joint work with Wei-Kuo Chen (University of Minnesota).

A fundamental problem in geometry is to classify geometric structures. In one complex variable, for example, the Riemann Mapping Theorem asserts that any simply connected region of the plane, other than the plane itself, is biholomorphically equivalent to the unit disk. This is far from true in higher dimensions, where the local CR geometry of the boundary obstructs the existence of biholomorphisms. In this talk, we shall survey some results and open problems characterizing the unit ball and ball quotients, up to biholomorphism, by properties of the Bergman kernel (e.g., the Ramadanov Conjecture and one concerning algebraicity of the kernel) and the Bergman metric (Cheng’s Conjecture).  Particular focus will be on generalizing some of the results to algebraic surfaces, weakly pseudoconvex domains and solving Cheng's conjecture for Stein spaces in dimension 2.

Most mathematicians spend the minimal amount of time on understanding the ins and outs of LaTeX. However, LaTeX can be…. finicky, and sometimes this can come back to bite us. In this chill, shorter-than-normal (one might even say lazy) talk, we give a few ideas on how to minimize later pain without having to read hundreds of pages on the innards of LaTeX.

In this talk I will present examples of property (T) type II1 factors with trivial fundamental group, thus, providing progress towards the well-known open questions of Connes'94 and Popa'06. We will show that the semidirect product feature is an algebraic feature that survive passage to group von Neumann algebras for a class of inductive limit of property (T) groups arising from geometric group theory. Using Popa's deformation/rigidity in conjunction with group theoretic methods we proved that the acting group can be completely recoverable from the von Neumann algebra as well as the limit action of the acting group. In addition, the fundamental group of the group von Neumann algebras associated to these limit groups are trivial, which contrasts the McDuff case.  This is based on a joint work with S. Das. 

We study a generalization of the classical Multidimensional Scaling procedure (cMDS) to the setting of general metric measure spaces which can be seen as natural 'continuous limits' of finite data sets. We identify certain crucial spectral properties of the generalized cMDS operator thus providing a natural and rigorous formulation of cMDS in this setting. Furthermore, we characterize the cMDS output of several continuous exemplar metric measures spaces such as high dimensional spheres and tori (both with their geodesic distance). In particular, the case of spheres (with geodesic distance) requires that we establish that its cMDS operator is trace class, a condition which is natural in the context when the cMDS operator has infinite rank. Finally, we establish the stability of the generalized cMDS method with respect to the Gromov-Wasserstein distance.

Quasi-Newton methods form the basis of many effective methods for unconstrained and constrained optimization.  Quasi-Newton methods require only the first-derivatives of the problem to be provided and update an estimate of the Hessian matrix of second derivatives to reflect new approximate curvature information found during each iteration.  In the years following the publication of the Davidon-Fletcher-Powell (DFP) method in 1963 the Broyden-Fletcher-Goldfarb-Shanno (BFGS) update emerged as the best update formula for use in unconstrained minimization.  More recently, a number of quasi-Newton methods have been proposed that are intended to improve on the efficiency and reliability of the BFGS method.  Unfortunately, there is no known analytical means of determining the relative performance of these methods on a general nonlinear function, and there is no accepted standard set of test problems that may be used to verify that results reported in the literature are comparable. In this talk we will discuss ongoing work to provide a thorough derivation, implementation, and numerical comparison of these methods in a systematic and consistent way. We will look in detail at several modifications, discuss their relative benefits, and review relevant numerical results.

Consider now the family of solutions $U_\nu$ to the associated Navier-Stokes equation with the no-slip condition on the flat boundaries, for small viscosities $\nu=1/ Re$, and initial values close in $L^2$ to $Ae_1$. Under a conditional assumption on the energy dissipation close to the boundary, Kato showed in 1984 that $U_\nu$ converges to $Ae_1$ when the viscosity converges to 0 and the initial value converges to $A e_1$. It is still unknown whether this inviscid is unconditionally valid. Actually, the convex integration method predicts the possibility of layer separation. It produces solutions to the Euler equation with initial values $Ae_1 $, but with layer separation energy at time T up to:

 $$\|U(T)-Ae_1\|^2_{L^2}\equiv A^3T.$$

In this work, we prove that at the double limit for the inviscid asymptotic $\bar{U}$, where both the Reynolds number $Re$ converges to infinity and the initial value $U_{\nu}$ converges to $Ae_1$ in $L^2$, the energy of layer separation cannot be more than:

$$\| \bar{U}(T)-Ae_1\|^2_{L^2}\lesssim A^3T.$$

Especially, it shows that, even if the limit is not unique, the shear flow pattern is observable up to time $1/A$. This provides a notion of stability despite the possible non-uniqueness of the limit predicted by the convex integration theory.

The result relies on a new boundary vorticity estimate for the Navier-Stokes equation. This new estimate, inspired by previous work on higher regularity estimates for Navier-Stokes, provides a non-linear control scalable through the inviscid limit.

This dissertation lies at the intersection of extremal combinatorics and probabilistic combinatorics.  Roughly speaking, extremal combinatorics studies how large a combinatorial object can be.  For example, a classical result of Mantel's says that every $n$-vertex triangle-free graph has at most $\frac{1}{4} n^2$ edges.  The area of probabilistic combinatorics encompasses both the application of probability to combinatorial problems, as well as the study of random combinatorial objects such as random graphs and random permutations.  In this dissertation we study problems related to extremal properties of random objects.  In particular we study a certain card guessing game, $F$-free subgraphs of random hypergraphs, and thresholds of random hypergraphs.  Minimal prerequisites will be assumed.

We give a comprehensive structural analysis of amenable subrelations of a treed quasi-measure preserving equivalence relation. The main philosophy is to understand the behavior of the Radon-Nikodym cocycle in terms of the geometry of the amenable subrelation within the tree. This allows us to extend structural results that were previously only known in the measure-preserving setting, e.g., we show that every nowhere smooth amenable subrelation is contained in a unique maximal amenable subrelation. The two main ingredients are an extension of Carrière and Ghys's criterion for nonamenability, along with a new Ping-Pong-style argument we call the "Paddle-ball lemma" that we use to apply this criterion in our setting. This is joint work with Anush Tserunyan.

We establish sufficient conditions for a locally-warped product structure to propagate backward in time under the Ricci flow. As an application, we show that if a gradient shrinking soliton is asymptotic to a cone whose cross-section is a locally warped product of Einstein manifolds, the soliton must itself be a warped product over the same manifolds.

The p-widths of a closed Riemannian manifold are a nonlinear analog of the spectrum of its Laplace--Beltrami operator, which was defined by Gromov in the 1980s and correspond to areas of a certain min-max sequence of hypersurfaces. By a recent theorem of Liokumovich--Marques--Neves, the p-widths obey a Weyl law, just like eigenvalues do. However, even though eigenvalues are explicitly computable for many manifolds, there had previously not been any ≥ 2-dimensional manifold for which all the p-widths are known. In recent joint work with Otis Chodosh, we found all p-widths on the round 2-sphere and thus the previously unknown Liokumovich--Marques--Neves Weyl law constant in dimension 2.

In classical real and complex dynamics, one studies topological and analytic properties of orbits of points under iteration of self-maps of $\mathbb R$ or $\mathbb C$ (or more generally self-maps of a real or complex manifold). In arithmetic dynamics, a more recent subject, one likewise studies properties of orbits of self-maps, but with a number theoretic flavor. Many of the motivating problems in arithmetic dynamics come via analogy with classical problems in arithmetic geometry: rational and integral points on varieties correspond to rational and integral points in orbits; torsion points on abelian varieties correspond to periodic and preperiodic points of rational maps; and abelian varieties with complex multiplication correspond to post-critically finite rational maps.

This analogy focuses on forward iteration, but sometimes surprising and interesting results can be found by thinking instead about pre-images of rational points under iteration. In this talk, we will give some background and motivation for the field of arithmetic dynamics in order to describe some of these "backwards iteration" results, including uniform boundedness for rational pre-images and open image results for Galois representations associated to dynamical systems.

We develop an efficient and accurate level set method to study numerically a crawling eukaryotic cell using a minimal model. This model describes the cell polarity and movement using a reaction-diffusion system coupled with a sharp-interface model. 

We employ an efficient finite difference method for the reaction-diffusion equations with no-flux boundary conditions. This results in a symmetric positive definite system, which can be solved by the conjugate gradient method accelerated by preconditioners. To track the long-time dynamics, we employ techniques of the moving computational window to keep the efficiency. Our level-set simulations capture well the cell crawling, the straight line trajectory, the circular trajectory, and other features. 

Our efficient and accurate computational techniques can be extended to a broad class of biochemical descriptions of cell motility, for which problems are posed on moving domains with complex geometry and fast simulations are very important. This is a joint work with Li-Tien Cheng and Bo Li.

We give a method for sampling points from an affine algebraic variety over a local field with a prescribed probability distribution. In the spirit of the previous work by Breiding and Marigliano on real algebraic manifolds, our method is based on slicing the given variety with random linear spaces of complementary dimension. We also provide an implementation of our sampling method and discuss a few applications, in particular we sample from algebraic p-adic matrix groups and modular curves.

Preservation of linear and quadratic invariants by numerical integrators has been well studied. However, many systems have linear or quadratic observables that are not invariant, but which satisfy evolution equations expressing important properties of the system. For example, a time-evolution PDE may have an observable that satisfies a local conservation law, such as the multisymplectic conservation law for Hamiltonian PDEs.

We introduce the concept of functional equivariance, a natural sense in which a numerical integrator may preserve the dynamics satisfied by certain classes of observables, whether or not they are invariant. After developing the general framework, we use it to obtain results on methods preserving local conservation laws in PDEs. In particular, integrators preserving quadratic invariants also preserve local conservation laws for quadratic observables, and symplectic integrators are multisymplectic.

Tissue growth, as it occurs during solid tumors, can be described at a number of different scales from the cell to the organ. For a large number of cells, 'fluid mechanical' approaches have been advocated in mathematics, mechanics or biophysics. We will give an overview of the modeling aspects and focuss on the links between those mathematical models. Then, we will focus on the `compressible' description describing the cell population density based on systems of porous medium type equations with reaction terms. A more macroscopic 'incompressible' description is based on a free boundary problem close to the classical Hele-Shaw equation. In the stiff pressure limit, one can derive a weak formulation of the corresponding Hele-Shaw free boundary problem and one can make the connection with its geometric form. The mathematical tools related to these questions include multi-scale analysis, Aronson-Benilan estimate, compensated compactness, uniform $L^4$ estimate on the pressure gradient and emergence of instabilities.

Tissue  growth, as it occurs during solid tumors, can be described at a number of different scales from the cell to the organ. For a large number of cells, 'fluid mechanical' approaches have been advocated in mathematics, mechanics or biophysics.

We will give an overview of the modeling aspects and focus on the links between those mathematical models. Then, we will focus on the `compressible' description  describing the cell population density based on systems of porous medium type equations with reaction terms.  A  more macroscopic 'incompressible' description is based on a free boundary problem close to the classical Hele-Shaw equation. In the stiff pressure limit, one can derive a weak formulation of the corresponding Hele-Shaw free boundary problem and one can make the connection with its geometric form.

The mathematical tools related to these questions include multi-scale analysis, Aronson-Benilan estimate, compensated compactness, uniform $L^4$ estimate on the  pressure gradient and emergence of instabilities.

The original version of the Quillen-Lichtenbaum Conjecture, proved by Voevodsky and Rost,  connects special values of Dedekind zeta functions and algebraic K-groups of number fields. In this talk, I will discuss a generalization of this conjecture to Dirichlet L-functions. The key idea is to twist algebraic K-theory spectra with the equivariant Moore spectra introduced in my thesis. This is joint work in progress with Elden Elmanto.

Donaldson-Thomas (DT) theory is an enumerative theory which produces a virtual count of stable coherent sheaves on a Calabi-Yau 3-fold. Motivic Donaldson-Thomas theory, originally introduced by Kontsevich-Soibelman, is a categorification of the DT theory. This categorification contains more refined information of the moduli space. In this talk, I will explain the role of d-critical locus structure in the definition of motivic DT invariant, following the definition by Bussi-Joyce-Meinhardt. I will also discuss results on this structure on the Hilbert schemes of zero dimensional subschemes on local toric Calabi-Yau threefolds. This is based on joint works with Sheldon Katz. The results have substantial overlap with recent work by Ricolfi-Savvas, but techniques used here are different. 

Branching Brownian motion (BBM) is a random particle system which incorporates both the tree-like structure and the diffusion process. In this talk, we will consider two variant models of BBM, BBM with absorption and BBM with an inhomogeneous branching rate. In the first model, we will study the transition from the slightly subcritical regime to the critical regime and obtain a Yaglom type asymptotic result of the expected number of particles conditioned on survival as the process gets closer to being critical. In the second model, we will see how it can be used to study the evolution of populations undergoing selection. We will provide a mathematically rigorous justification for the biological observation that the distribution of the fitness levels of individuals in a population evolves over time like a traveling wave with a profile defined by the Airy function. The second part of the talk is based on joint work with Jason Schweinsberg.

In this talk, we will discuss the dynamics on Teichmüller space and moduli space of square-tiled surfaces. For square-tiled surfaces, one can explicitly write down the $SL(2,\mathbb{R})$-orbit on the moduli space. To study the dynamics of Teichmüller flow of the $SL(2,\mathbb{R})$-action, we study its derivative, namely the Kontsevich--Zorich cocycle (KZ cocycle). In this talk, we will define what a KZ cocycle is, and by following explicit examples, we will show how one can compute the KZ monodromy. This is part of an ongoing work with Anthony Sanchez.

Modern data analysis involves large sets of structured data, where the structure carries critical information about the nature of the data. These relationships between entities, such as features or data samples, are usually described by a graph structure. While many real-world data are intrinsically graph-structured, e.g. social and traffic networks, there is still a large number of applications, where the graph topology is not readily available. For instance, gene regulations in biological applications or neuronal connections in the brain are not known. In these applications, the graphs need to be learned since they reveal the relational structure and may assist in a variety of learning tasks. Graph learning (GL) deals with the inference of a topological structure among entities from a set of observations on these entities, i.e., graph signals. Most of the existing work on graph learning focuses on learning a single graph structure, assuming that the relations between the observed data samples are homogeneous. However, in many real-world applications, there are different forms of interactions between data samples, such as single-cell RNA sequencing (scRNA-seq) across multiple cell types. This talk will present a new framework for multiview graph learning in two settings: i) multiple views of the same data and ii) heterogeneous data with unknown cluster information. In the first case, a joint learning approach where both individual graphs and a consensus graph are learned will be developed. In the second case, a unified framework that merges classical spectral clustering with graph signal smoothness will be developed for joint clustering and multiview graph learning. 
This is joint work with Abdullah Karaaslanli, Satabdi Saha and Taps Maiti.

This talk is mainly about a new Minkowski formula for  hypersurfaces with free boundary or capillary boundary supported on the unit sphere. With it we have classified all stable free boundary CMC hypersurfaces.  Using it we have introduced a inverse curvature flow, which is used to prove Alexandrov-Fenchel type inequalities for newly introduced quermassintegrals  for free boundary  hypersurfaces. At the end we will talk about various generalizations. The talk is based on the joint work with J. Scheuer and C. Xia and other collaborators.

In this talk, I will give an introduction to optimal transport, which has evolved as one of the major frameworks to meaningfully compare distributional data. The focus will mostly be on machine learning, and how optimal transport can be used efficiently for clustering and supervised learning tasks. Applications of interest include image classification as well as medical data such as gene expression profiles.

We recently described the construction and theoretical analysis of a framework (competitive-refractory dynamics model) derived from the canonical neurophysiological principles of spatial and temporal summation. The framework models the competing interactions of signals incident on a target downstream node (e.g. a neuron) along directed edges coming from other upstream nodes that connect into it. The model takes into account how temporal latencies produce offsets in the timing of the summation of incoming discrete events due to the geometry (physical structure) of the network, and how this results in the activation of the target node. It provides a computable representation of how local computations result in global network dynamics. Grounded in this neurophysiological model, we are beginning to explore the use some aspects of category theory and related ideas in order to abstract up and understand how the brain might produce generative (emergent) non-trivial computational properties. In particular, we are interested in understanding the emergence of creativity and imagination.

https://mathweb.ucsd.edu/~bli/research/mathbiosci/MBBseminar/

We will discuss a $p$-adic cohomology theory for rigid analytic varieties with overconvergent structure (dagger spaces) over a local field of characteristic $p$. After explaining the motivation, we will define a site (Robba site) and discuss its basic properties.

The development of vaccines for COVID-19 has enabled us to nearly return to pre-pandemic life. However, while vaccines are becoming globally widespread, high alert levels prevail. Even with vaccines, monitoring for the evolution of mutations or detecting new outbreaks calls for continued vigilance. Therefore, testing is likely to prevail to be a vital mechanism to inform decision making in the near future. In order to conserve scarce testing resources, many nations have endorsed so-called group/pooling test methods. Such methods can be expressed using linear algebra. The basic principle underlying pooling tests is the observation that to efficiently detect positive cases among a population with a very low occurrence prevalence, it can be advantageous to test groups of samples instead of testing all individual samples. We develop matrix designs, which encode all relevant information for doing pooling tests and that enable high compression rates when exactly identifying up to a certain number of positive cases.

Analyzing a linear model is a fundamental topic in statistical inference and has been well-studied. However, the complex nature of modern data brings new challenges to statisticians, i.e., the existing theories and methods may fail to provide consistent results. Focusing on a high dimensional linear model with i.i.d. errors or heteroskedastic and dependent errors, this talk introduces a new ridge regression method called `the debiased and thresholded ridge regression' that fits the linear model. After that, it introduces new bootstrap algorithms that generate consistent simultaneous confidence intervals/performs hypothesis testing for the linear model. This talk also applies bootstrap algorithm to construct the simultaneous prediction intervals for future observations. 

Another topic of this talk is about properties of a residual-based bootstrap prediction interval. It derives the asymptotic distribution of the difference between the conditional coverage probability of a nominal prediction interval and the conditional coverage probability of a prediction interval obtained via a residual-based bootstrap. This result shows that the residual-based bootstrap prediction interval has about $50\%$ possibility of yielding conditional under-coverage. Moreover, it introduces a new bootstrap prediction interval that has the desired asymptotic conditional coverage probability and the possibility of conditional under-coverage.

Relative Toeplitz algebras of directed graphs were introduced by Spielberg in 2002 to describe certain subalgebras corresponding to subgraphs. They can also be used to describe quotients of graph algebras corresponding to subgraphs. We use the latter relationship to answer a question posed in a recent paper regarding pushout diagrams of graphs that give rise to pullback diagrams of the respective graph C*-algebras. We introduce a new category of relative graphs to this end, and we prove our results using graph groupoids and their C*-algebras. This is joint work with Jack Spielberg.

This talk will be devoted to the existence of pulsating travelling front solutions for spatially periodic heterogeneous reaction-diffusion equations in arbitrary dimension. In the bistable case, such a pulsating front indeed exists and it also describes the large time dynamics of solutions of the Cauchy problem. However, unlike in the homogeneous case the periodic problem is no longer invariant by rotation, so that the front speed may be different depending on its direction. This in turn raises some difficulties in the spreading shape of solutions of the evolution problem, which may exhibit strongly asymmetrical features. In the general multistable case, that is when there is a finite but arbitrary number of stable steady states, the notion of a single front is no longer sufficient and we instead observe the appearance of a so-called propagating terrace. This roughly refers to a finite family of stacked fronts connecting intermediate stable steady states and whose speeds are ordered. The presented results come from a series of work with W. Ding, A. Ducrot, H. Matano and L. Rossi.

To each integral vector v in $\mathbb{Z}^n$, we attach several natural objects of geometric/arithmetic nature. For example:

1. The direction of v (i.e., its radial projection to the unit sphere),
2. The orthogonal lattice to v (i.e., the proper rescaling of the lattice of integral points in the orthogonal hyperplane to v),
3. The residue class of v modulo a fixed integer k.

Each of these objects resides in a natural “homogeneous space” which supports a “uniform probability measure”. This allows one to ask statistical questions regarding these objects as v varies in some meaningful set of integral vectors. I will survey some classical and more recent results along these lines where there are limit laws governing the statistics. In some cases one obtains the uniform measure as the limit and in some cases a non-uniform limit. Interesting examples include the integral points on quadratic surfaces and the sequence of “best approximations” of an irrational line. In the talk I will try to explain how homogeneous dynamics can be used to tackle such questions.

The random d-process corresponds to a natural algorithmic model for generating d-regular graphs: starting with an empty graph on n vertices, it evolves by sequentially adding new random edges so that the maximum degree remains at most d.

In 1999 Wormald conjectured that the final graph of the random d-process is "similar" to a uniform random d-regular graph.

We show that this conjecture does not extend to a natural generalization of this process with mixed degree restrictions, i.e., where each vertex has its own degree restriction (under some mild technical assumptions). 
Our proof uses the method of switchings, which is usually only applied to uniform random graph models -- rather than to stochastic processes.

Based on joint work in progress with Mike Molloy and Erlang Surya.

The mapping class group, $\mathrm{Map}(S)$, of a surface $S$, is the group of all isotopy classes of homeomorphisms of $S$ to itself. A mapping class group is a topological group with the quotient topology inherited from the quotient map of $\mathrm{Homeo}(S)$ with the compact-open topology.

For surfaces of finite type, $\mathrm{Map}(S)$ is countable and discrete. Surprisingly, the topology of $\mathrm{Map}(S)$ is more interesting if $S$ is an infinite-type surface; it is uncountable, topologically perfect, totally disconnected, and more importantly, has the structure of a Polish group. In recent literature, this last class of groups is called "big mapping class groups.''

In this talk, I will give a brief introduction to big mapping class groups and explain our results on the topological structure of conjugacy classes. This was a joint work with Jesús Hernández Hernández, Michael Hrušák, Manuel Sedano, and Ferrán Valdez.

For any 3D steady gradient Ricci soliton, if it is asymptotic to a ray we prove that it must be isometric to the Bryant soliton. Otherwise, it is asymptotic to a sector and called a flying wing. We show that all flying wings are O(2)-symmetric. Hence, all 3D steady gradient Ricci solitons are O(2)-symmetric.

I will talk about recent work towards a conjecture of Gonek regarding negative shifted moments of the Riemann zeta function. I will explain how to obtain asymptotic formulas when the shift in the Riemann zeta function is big enough, and how we can obtain non-trivial upper bounds for smaller shifts. Joint work with H. Bui.

K-theoretic enumerative invariants are defined by holomorphic Euler characteristics of coherent sheaves on moduli spaces. In this talk, I will give an introduction to quantum K-theory whose definition involves moduli spaces of stable maps to given target spaces. I will mention its connections to birational geometry, combinatorics, and number theory. In joint work with Yang Zhou, we proved wall-crossing formulas of quantum K-invariants for any orbifold GIT quotient. These formulas can be used to compute quantum K-invariants. When the target space is an orbifold, quantum K-theory turns out to be quite different from its cohomological counterpart--quantum cohomology. I will present some of these new phenomena.

Classical comparison theorem established the isomorphism between singular cohomology and de Rham cohomology for smooth manifolds. The p-adic analogy aims to compare p-adic etale cohomology to de Rham cohomology for projective smooth schemes. In this talk, I will review and explain p-torsion version of such comparison and discuss how prismatic cohomology, recently invented by Bhatt-Scholze,  can help to promote such p-torsion comparison.  This is joint work with Shizhang Li.

We study the $L^2$-Betti numbers of fiber bundles $F\rightarrow E\rightarrow B$ of manifolds. Under certain conditions (e.g., when $F$ is simply connected), $b^{(2)}_{\ast}(E)$ can be computed using the twisted $L^2$-Betti numbers of $B$. We relate the twisted and untwisted $L^2$-Betti numbers of $B$ when $\pi_1(B)$ is locally indicable. As an application, we compute $b^{(2)}_{\ast}(E)$ when $B$ is either a surface or a non-positively curved $3$-manifold. This is a joint work with Dawid Kielak.

Sampling-based inference and learning techniques, especially Bayesian inference, provide an essential approach to handling uncertainty in machine learning (ML).   As these techniques are increasingly used in daily life, it becomes essential to safeguard the ML systems with various trustworthy-related constraints, such as fairness, safety, interpretability. We propose a family of constrained sampling algorithms which generalize Langevin Dynamics (LD) and Stein Variational Gradient Descent (SVGD) to incorporate a moment constraint or a level set  specified by a general nonlinear function. By exploiting the gradient flow structure of LD and SVGD, we derive algorithms for handling constraints, including a  primal-dual gradient approach and the constraint controlled gradient descent approach.  We investigate the continuous-time mean-field limit of these algorithms and show that they have $O(1/t)$ convergence under mild conditions.

Speaker Bio:
Dr. Xin Tong is an associate professor at the National University of Singapore, department of mathematics. He received his Ph.D. degree from Princeton University in 2013. Prior to his position at the National University of Singapore, he was a postdoc at the Courant Institute of New York University. His recent research focuses on the analysis and derivation of stochastic algorithms.

We prove the conjecture of Marian-Oprea-Pandharipande on the Segre series associated to a rank zero class.  Hence the rank zero Segre integrals on the Hilbert schemes of points for all surfaces are determined.

In enumerative geometry, Virasoro constraints first appeared in the context of moduli of stable curves and maps. These constraints provide a rich set of conjectural relations among Gromov-Witten descendent invariants. Recently, the analogous constraints were formulated in several sheaf theoretic contexts; stable pairs on 3-folds, Hilbert scheme of points on surfaces, and higher rank sheaves on surfaces with only (p,p)-cohomology. In joint work with A. Bojko, M. Moreira, we extend and reinterpret Virasoro constraints in sheaf theory using Joyce's vertex algebra. This new interpretation yields a proof of Virasoro constraints for curves and surfaces with only (p,p) cohomology by means of wall-crossing formulas.

Pre-talk for graduate students 12:30 - 1:00pm. 

I will show you how to construct a full factor $M$ such that $M$ and $L(F_2)$ do not have any isomorphic ultrapowers.  The construction uses a combination of techniques from deformation/rigidity and free entropy theory.  We also provide the first example of a $\mathrm{II}_1$ factor that is full such that its ultrapower is strongly $1$-bounded.  This is joint work with Adrian Ioana and Ionut Chifan.

We develop hybrid projection methods for computing solutions to large-scale inverse problems, where the solution represents a sum of different stochastic components. Such scenarios arise in many imaging applications (e.g., anomaly detection in atmospheric emissions tomography) where the reconstructed solution can be represented as a combination of two or more components and each component contains different smoothness or stochastic properties. In a deterministic inversion or inverse modeling framework, these assumptions correspond to different regularization terms for each solution in the sum. Although various prior assumptions can be included in our framework, we focus on the scenario where the solution is a sum of a sparse solution and a smooth solution. For computing solution estimates, we develop hybrid projection methods for solution decomposition that are based on a combined flexible and generalized Golub-Kahan processes. This approach integrates techniques from the generalized Golub-Kahan bidiagonalization and the flexible Krylov methods. The benefits of the proposed methods are that the decomposition of the solution can be done iteratively, and the regularization terms and regularization parameters are adaptively chosen at each iteration. Numerical results from photoacoustic tomography and atmospheric inverse modeling demonstrate the potential for these methods to be used for anomaly detection.

We discuss Bregman distance extensions of the primal-dual three-operator (PD3O) and Condat-Vu proximal algorithms. When used with standard proximal operators these algorithms include several important methods as special cases. Extensions to generalized Bregman distances are attractive if the complexity per iteration can be reduced by matching the Bregman distance to the structure in the problem. As an example, we apply the proposed method to the centering problem in sparse semidefinite programming. The logarithmic barrier function for the cone of positive semidefinite completable sparse matrices is used as a distance-generating kernel. For this distance, the complexity of evaluating the Bregman proximal operator is shown to be roughly proportional to the cost of a sparse Cholesky factorization. This is much cheaper than the standard proximal operator with Euclidean distances, which requires an eigenvalue decomposition.

LIFE HACK: Attend Food For Thought (FFT) on Wednesday at 4:00 PM. Studies show that attending FFT improves mood by 43%, attending FFT boosts cognition by 15%, attending FFT decreases stress by 28%, and that 120% of statistics that people quote are 150% true! If you attend FFT this week, we'll talk about a few other graduate student life hacks that hopefully can improve your life by just a little bit. See you there!

An invariant random order is a shift-invariant measure on the space of all orders on a group. It is easy to show that on an amenable group, any invariant random order could be invariantly extended to an invariant random total order. Recently, Glaner, Lin and Meyerovitch showed that this is no longer true for $\mathrm{SL}_3(\mathbb{Z})$. I will explain, how starting from their construction, one can show that this order extension property does not hold for non-amenable groups, and discuss an analog of this result for measure preserving equivalence relations.

We introduce a "mesoscale flatness criterion" for hypersurfaces with bounded mean curvature, discussing its relation to and differences with classical blow-up and blow-down theorems, and then we exploit this tool for a complete resolution of relative isoperimetric sets with large volume in the exterior of a compact obstacle. This is joint work with Francesco Maggi (UT Austin).

(Joint with Stefan Patrikis.) In this talk, we discuss a strong form of independence of $\ell$ for canonical $\ell$-adic local systems on Shimura varieties, and sketch a proof of this for Shimura varieties arising from adjoint groups whose simple factors have real rank $\geq 2$. Notably, this includes all adjoint Shimura varieties which are not of abelian type. The key tools used are the existence of companions for $\ell$-adic local systems and the superrigidity theorem of Margulis for lattices in Lie groups of real rank $\geq 2$.

The independence of $\ell$ is motivated by a conjectural description of Shimura varieties as moduli spaces of motives. For certain Shimura varieties that arise as a moduli space of abelian varieties, the strong independence of $\ell$ is proven (at the level of Galois representations) by recent work of Kisin and Zhou, refining the independence of $\ell$ on the Tate module given by Deligne's work on the Weil conjectures.

We will discuss about resolutions of coherent sheaves by line bundles from strong full exceptional sequences over rational surfaces. We call them Gaeta resolutions. We then apply the results towards the study of the moduli space of sheaves, in particular Le Potier's strange duality conjecture. We will show that the strange morphism is injective in some new cases. One of the key steps is to show that certain Quot schemes are finite and reduced. The next key step is to enumerate the length of the finite Quot scheme, by identifying the Quot scheme as the moduli space of limit stable pairs, where we are able to calculate the (virtual) fundamental class. This is based on joint work with Thomas Goller.

Pre-talk for graduate students: 3:30pm - 4:00pm

Adjoint systems are widely used to inform control, optimization, and design in systems described by ordinary differential equations and differential-algebraic equations. In this talk, we begin by exploring the geometric properties of adjoint systems associated to ordinary differential equations by investigating their symplectic and Hamiltonian structures. We then extend this to adjoint systems associated to differential-algebraic equations and develop geometric methods for such systems by utilizing presymplectic geometry to characterize the fundamental properties of such systems, such as the adjoint variational quadratic conservation laws admitted by these systems, which are key to adjoint sensitivity analysis. We develop structure-preserving numerical methods for such systems by extending the Galerkin Hamiltonian variational integrator construction of Leok and Zhang to the presymplectic setting. Such methods are natural, in the sense that reduction, forming the adjoint system, and discretization commute for suitable choices of these processes. We conclude with a numerical example. This is joint work with Prof. Melvin Leok.

Waldhausen's algebraic K-theory of manifolds satisfies a homotopical lift of the classical h-cobordism theorem and provides a critical link in the chain of homotopy theoretic constructions that show up in the classification of manifolds and their diffeomorphisms. I will give an overview of joint work with Goodwillie, Igusa and Malkiewich about the equivariant homotopical lift of the h-cobordism theorem.

Abstract: In this talk, I shall introduce the notion of polynomials with Lorentzian signature. This class is a generalization to the remarkable class of Lorentzian polynomials. The hyperbolic polynomials and conic polynomials are shown to be polynomials with Lorentzian signature. Using the notion of polynomials with Lorentzian signature I shall describe how to compute the permanents of a special class of nonsingular matrices via hyperbolic programming. The nonsingular $k$ locally singular matrices are contained in the  special class of nonsingular matrices for which computing the permanents can be done via hyperbolic programming.

Geometric invariant theory (GIT) is the main tool for taking quotients by group actions in algebraic geometry. In this talk I will try to show how GIT actually works by showing lots of examples.

We prove various antirigidity and rigidity results around the orbit equivalence of wreath product actions. Let $F$ be a nonabelian free group. In particular, we show that the wreath products $A \wr F$ and $B \wr F$ are orbit equivalent for any pair of nontrivial amenable groups $A$, $B$. This is joint work with Robin Tucker-Drob.

I will first give an introduction to the notion of the curvature operator of the second kind and review some known results, including the proof of Nishikawa's conjecture stating that a closed Riemannian manifold with positive (resp. nonnegative) curvature operator of the second is diffeomorphic to a spherical space form (resp. a Riemannian locally symmetric space). Then I will talk about my recent works on the curvature operator of the second kind on Kahler manifolds and product manifolds. Along the way, I will mention some interesting questions and conjectures.

Stein's method for distributional approximation has become a valuable tool in probability and statistics by providing finite sample distributional bounds for a wide class of target distributions in a number of metrics. A key step in popular versions of the method involves making couplings constructions, and a family of couplings of Chen and Roellin vastly expanded the range of applications for which Stein's method for normal approximation could be applied. This family subsumes both Stein's classical exchangeable pair, and the size bias coupling. A further simple generalization includes zero bias couplings, and also allows for situations where the coupling is not exact. The zero bias versions result in bounds for which often tedious computations of a variance of a conditional expectation is not required. An example to the Lightbulb process shows that even though the method may be simple to apply, it may yield improvements over previous results that had achieved bounds with optimal rates and small, explicit constants.

Let $G$ be a $p$-adic reductive group with $\mathfrak{g} = \mathrm{Lie}(G)$. I will summarize work with Matthias Strauch in which we construct an exact functor from category $\mathcal{O}^\infty$, the extension closure of the Bernstein-Gelfand-Gelfand category $\mathcal{O}$ inside the category of $U(\mathfrak{g})$-modules, into the category of admissible locally analytic representations of $G$. This expands on an earlier construction by Sascha Orlik and Matthias Strauch. A key role in our new construction is played by $p$-adic logarithms on tori, and representations in the image of this functor are related to some that are known to arise in the context of the $p$-adic Langlands program.

[pre-talk at 1:20PM]

Geometers are interested in the problem of finding a "best"
metric on a manifold. In dimension 2, the best metric is usually one
which possesses the most symmetries, such as the round metric on a
sphere, or a flat metric on a torus. In higher dimensions, there are
many more classes of geometrically interesting metrics. I will give a
general overview of a certain class of Einstein metrics in dimension 4
which have special holonomy, and which are known as "gravitational
instantons." I will then discuss certain aspects of their classification
and connections with algebraic surfaces.

A variety $X$ over an algebraically closed field $k$ of characteristic $p>0$ is Witt-liftable if it is the closed fiber of a flat morphism $\mathcal{X}\to\mathrm{Spec}W(k)$, where $W(k)$ denotes the ring of Witt vectors of $k$. The existence of such a lift allows us to study $X$ using techniques from complex geometry. Although it is well-known that such a lift does not always exist, it is conjectured that every globally F-split variety is Witt-liftable. We show a stronger result in dimension two, and apply this to the study of singularities of globally F-split del Pezzo and Calabi-Yau surfaces. This is a joint work with F. Bernasconi, T. Kawakami, and J. Witaszek.

Pre-talk: 3:30-4:00pm

Adjoint systems are widely used to inform control, optimization, and design in systems described by ordinary differential equations or differential-algebraic equations. In this session, we explore the geometric properties and develop methods for such adjoint systems. In particular, we utilize symplectic and presymplectic geometry to investigate the properties of adjoint systems associated with ordinary differential equations and differential-algebraic equations, respectively. We show that the adjoint variational quadratic conservation laws, which are key to adjoint sensitivity analysis, arise from (pre)symplecticity of such adjoint systems. We discuss various additional geometric properties of adjoint systems, such as symmetries and variational characterizations. For adjoint systems associated with a differential-algebraic equation, we relate the index of the differential-algebraic equation to the presymplectic constraint algorithm of Gotay and Nester. As an application of this geometric framework, we discuss how the adjoint variational quadratic conservation laws can be used to compute sensitivities of terminal or running cost functions. Furthermore, we develop structure-preserving numerical methods for such systems using Galerkin Hamiltonian variational integrators which admit discrete analogues of these quadratic conservation laws. We additionally show that such methods are natural, in the sense that reduction, forming the adjoint system, and discretization all commute, for suitable choices of these processes. We utilize this naturality to derive a variational error analysis result for the presymplectic variational integrator that we use to discretize the adjoint DAE system. This is joint work with Prof. Melvin Leok.

In the post-talk discussion session, we plan to discuss future directions; in particular, exploring the geometry of adjoint systems for infinite-dimensional spaces with the application of PDE-constrained optimization in mind.

We will introduce the notion of nonlocal symmetry of a graph G, defined as winning quantum correlation for the G-automorphism game that cannot be produced classically. We investigate the differences and similarities between this and the quantum symmetry of the graph G, defined as non-commutativity of the algebra of functions on the quantum automorphism group of G. We show that quantum symmetry is a necessary but not sufficient condition for nonlocal symmetry. In particular, we show that the complete graph on four points does not exhibit nonlocal symmetry. We will also see that the complete graph on five or more points does have nonlocal symmetry. This talk is based on joint work with David Roberson.

The Zakharov system is a quadratically coupled system of a Schrödinger and a wave equation, which is related to the focussing cubic Schrödinger equation. We consider the associated Cauchy problem in the energy-critical dimension d=4 and prove that it is globally well-posed in the full (non-radial) energy space for any initial data with energy and wave mass below the ground state threshold. The result is based on a uniform Strichartz estimate for the Schrödinger equation with potentials solving the wave equation. A key ingredient in the non-radial setting is a bilinear Fourier extension estimate. This is joint work with Timothy Candy and Kenji Nakanishi.

Thanks to Voiculescu’s freeness, one knows that the normalized eigenvalue counting measure of a selfadjoint non-commutative polynomial in iid GUE’s converges in the  limit of large dimension, and there exist many tools to compute its limiting distribution. On the other hand, on the limiting space (a free product algebra), lots of progress has been made in understanding non-commutative rational fractions. A question by Speicher is whether these rational fractions admit matrix models too. I will explain why the natural candidate is actually a matrix model. In other words, bearing in mind that we already understand the asymptotics of the eigenvalue counting measure of a matrix model obtained as sums, scalings products of iid random matrices, we will show that we can do the same if we allow in addition multiple uses of the matrix inverse when creating our matrix model. 

This is based on arXiv/2103.05962, written in collaboration with Tobias May, Akihiro Miyagawa, Felix Parraud and Sheng Yin.

Étale homotopy theory was invented by Artin and Mazur in the 1960s as a way to associate to a scheme X, a homotopy type with fundamental group the étale fundamental group of X and whose cohomology captures the étale cohomology of X with locally constant constructible coefficients. In this talk we'll explain how to construct a stratified refinement of the étale homotopy type that classifies constructible étale sheaves and gives rise to a new definition of the étale homotopy type. The stratified étale homotopy type also plays a role in the reconstruction of schemes: in nice cases, schemes can be completely reconstructed from their stratified étale homotopy types. This is joint work with Clark Barwick and Saul Glasman.

Counting objects is a fundamental but challenging problem. In this paper, we propose diffusion-based, geometry-free, and learning-free methodologies to count the number of objects in images. The main idea is to represent each object by a unique index value regardless of its intensity or size, and to simply count the number of index values. First, we place different vectors, referred to as seed vectors, uniformly throughout the mask image. The mask image has boundary information of the objects to be counted. Secondly, the seeds are diffused using an edge-weighted harmonic variational optimization model within each object. We propose an efficient algorithm based on an operator splitting approach and alternating direction minimization method, and theoretical analysis of this algorithm is given. An optimal solution of the model is obtained when the distributed seeds are completely diffused such that there is a unique intensity within each object, which we refer to as an index. For computational efficiency, we stop the diffusion process before a full convergence, and propose to cluster these diffused index values. We refer to this approach as Counting Objects by Diffused Index (CODI). We explore scalar and multi-dimensional seed vectors. For Scalar seeds, we use Gaussian fitting in histogram to count, while for vector seeds, we exploit a high-dimensional clustering method for the final step of counting via clustering. The proposed method is flexible even if the boundary of the object is not clear nor fully enclosed. We present counting results in various applications such as biological cells, agriculture, concert crowd, and transportation. Some comparisons with existing methods are presented.

We prove insane theorems by counting to 15.

The classification of horocycle invariant measures on finite volume hyperbolic surfaces with negative curvature is known since the work of Furstenberg and Dani in the seventies: they are either the Haar measure or are supported on periodic orbits. This problem fits inside the more general problem of the classification of horospherical measures in finite volume homogenous spaces.

In this talk, I will explain how similar questions arise in the moduli space of abelian differentials (and more generally in any affine invariant manifolds) and will discuss a notion of horospherical measures in that context. I will then report on progress toward a classification of those horospherical measures and related topological results. This is a joint work with J. Smillie, P. Smillie and B. Weiss.

In 1978, Katz gave a construction of the $p$-adic $L$-function of a CM field by using a $p$-adic analog of the Maass--Shimura operators acting on $p$-adic Hilbert modular forms. However, this $p$-adic Maass--Shimura operator is only defined over the ordinary locus, which restricted Katz's choice of $p$ to one that splits in the CM field. In 2021, Andreatta and Iovita extended Katz's construction to all $p$ for quadratic imaginary fields using overconvergent differential operators constructed by Harron--Xiao and Urban, which act on elliptic modular forms. Here we give a construction of such overconvergent differential operators which act on Hilbert modular forms.

[Pre-talk at 1:20PM]

The Goss zeta function is a characteristic-p analogue of the Riemann zeta function for function fields. In the spirit of the Riemann hypothesis, Goss has made several conjectures concerning the distribution of its zeros. We discuss the history of these questions and some recent progress we have made in collaboration with Joe Kramer-Miller. Our main result is a comparison of the distribution of zeros between the higher-genus and genus-zero cases. As a consequence, we are able to prove Goss' conjectures in a large number of previously unknown cases.

I will talk about a recent work jointly with Tian on Kaehler-Ricci flow on Fano G-manifolds. We prove that on a Fano G-manifold, the Gromov-Hausdorff limit of Kaehler-Ricci flow with initial metric in $2\pi c_1(M)$ must be a Q-Fano horosymmetric variety which admits a singular Keahler-Ricci soliton. Moreover, we show that the complex structure of limit variety can be induced by $C^*$-degeneration via the soliton vector field. A similar result can be also proved for Kaehler-Ricci flows on any Fano horosymmetric manifolds.

What are the noncommutative rings that are most analogous to polynomial rings? One class of such rings are the regular algebras first defined by Artin and Schelter in 1987. Since then such algebras have been extensively studied. We give a survey of these interesting examples and their associated projective geometry.

Efficient optimization has become one of the major concerns in machine learning, and there has been a lot of focus on first-order optimization algorithms because of their low cost per iteration. In 1983, Nesterov's Accelerated Gradient method (NAG) was shown to converge in $\mathcal{O}(1/k^2)$ to the minimum of the convex objective function $f$, improving on the $\mathcal{O}(1/k)$ convergence rate exhibited by the standard gradient descent methods, which is the phenomenon referred to as acceleration. It was shown that NAG limits to a second order ODE, as the time-step goes to 0, and that the objective function $f(x(t))$ converges to its optimal value at a rate of $\mathcal{O}(1/t^2)$ along the trajectories of this ODE. In this talk, we will discuss how the convergence of $f(x(t))$ can be accelerated in continuous time to an arbitrary convergence rate $\mathcal{O}(1/t^p)$ in normed spaces, by considering flow maps generated by a family of time-dependent Bregman Lagrangian and Hamiltonian systems which is closed under time rescaling. We will then discuss how this variational framework can be exploited together with the time-invariance property of the family of Bregman dynamics using adaptive geometric integrators to design efficient explicit algorithms for accelerated optimization. We will then discuss how these results and computational methods can be generalized from normed spaces to Riemannian manifolds. Finally, we will discuss some practical considerations which can be used to improve the performance of the algorithms. 

I will recall a construction of certain operator algebras arising naturally as multiparameter deformations of  operator algebras of Coxeter groups, initially motivated by the study of cohomology of groups acting on buildings. We will explain that for right-angled Coxeter groups, at a certain range of multiparameters, the resulting von Neumann algebra is a factor, thus completing earlier results of Garncarek, and of Caspers, Klisse and Larsen. This result, of interest in itself, has several consequences and interpretations for the representation theory of both right-angled Coxeter groups and of certain groups acting on buildings. I will also outline further questions/results related to the classification of the related C*-algebras.

Based on joint work with Sven Raum.

If a man is on a rocket with a certain finite speed, he will travel through the entire universe (no matter how large the universe is) in the blink of an eye. Notoriously weird and unrealistic claim from relativity, but takes just about 20 minutes to really understand this.

In many important actions of groups there are no invariant measures. For example: the action of a free group on its boundary and the action of any discrete infinite group on itself. The problem we will discuss in this talk is 'On a given action, how invariant measure can be? '. Our measuring of non-invariance will be based on entropy (f-divergence).

In the talk I will describe the solution of this problem for the Free group acting on its boundary and on itself. For doing so we will introduce the notion of ultra-limit of $G$-spaces, and give a new description of the Poisson-Furstenberg boundary of $(G,k)$ as an ultra-limit of $G$ action on itself, with 'Abel sum' measures. Another application will be that amenable groups possess KL-almost-invariant measures (KL stands for the Kullback-Leibler divergence). All relevant notions, including the notion of Poisson-Furstenberg boundary and the notion of Ultra-filters will be explained during the talk. This is a master thesis work under the supervision of Yehuda Shalom.

The Harish-Chandra-Itzykson-Zuber integral, also called spherical integral is defined as the expectation of exp(Tr(AUBU*)) for A and B two self adjoint matrices and U Haar-distributed on the unitary/orthogonal/symplectic group. It was initially introduced by Harish-Chandra to study Lie groups and it also has an interpretation in terms of Schur functions. Since then, it has had many kinds of applications, from physics to statistical learning. In this talk we will look at the asymptotics of these integrals when one of the matrices remains of small rank. We will also see how to use these asymptotics to prove large deviation principles for the largest eigenvalues for random matrix models that satisfy a sub-Gaussian bound. This talk is mainly based on a collaboration with Justin Ko. 

In geometric measure theory, the monotonicity formula for the area functional is a basic tool upon which many other basic fundamental facts depend. Some of them also follow from a weaker analytic tool, which is the Michael-Simon inequality. For anisotropic integrands (which generalize the area), monotonicity does not hold, while the latter inequality is conjectured to be true (under appropriate assumptions); actually, the latter is more essential to geometric measure theory, in that it turns out to be equivalent to the compactness of the classes of rectifiable and integral varifolds. In this talk we present some partial results, one of which is a slight improvement of a posthumous result of Almgren, namely the validity of this inequality for convex integrands close enough to the area, for surfaces in $R^3$. Our technique relies on a nonlinear inequality bounding the $L^1$-norm of the determinant of a function, from the plane to $2x2$ matrices, with the $L^1$-norms of the divergence of the rows, provided the matrix obeys some pointwise nonlinear constraints. This is joint work with Guido De Philippis (NYU).

In 2015 Greither and Popescu constructed a new class of Iwasawa modules, which are the number field analogues of $p-$adic realizations of Picard 1- motives constructed by Deligne. They proved an equivariant main conjecture by computing the Fitting ideal of these new modules over the appropriate profinite group ring. This is an integral, equivariant refinement of Wiles' classical main conjecture. As a consequence they proved a refinement of the Brumer-Stark conjecture away from 2. All of the above was proved under the assumption that the relevant prime $p$ is odd and that the appropriate classical Iwasawa $\mu$–invariants vanish. Recently, Dasgupta and Kakde proved the Brumer-Stark conjecture, away from 2, unconditionally, using a generalization of Ribet's method. We use the Dasgupta-Kakde results to prove an unconditional equivariant main conjecture, which is a generalization of that of Greither and Popescu. As applications of our main theorem we prove a generalization of a certain case of the main result of Dasgupta-Kakde and we compute the Fitting ideal of a certain naturally arising Iwasawa module. This is joint work with Cristian Popescu.

[Pre-talk at 1:20PM]

Logarithmic Sobolev inequalities, first introduced by Gross in 70s, have rich connections to probability, geometry, as well as information theory. In recent years, logarithmic Sobolev inequalities for quantum Markov semigroups attracted a lot of attentions for its applications in quantum information theory and quantum many-body systems. In this talk, I'll present a simple, information-theoretic approach to modified logarithmic Sobolev inequalities for both quantum Markov semigroup on matrices, and classical Markov semigroup on matrix-valued functions. In the classical setting, our results implies every sub-Laplacian of a Hörmander system admits a uniform  modified logarithmic Sobolev constant for all its matrix valued functions. For quantum Markov semigroups, we improve a previous result of Gao and Rouzé by replacing the dimension constant by its logarithm. This talk is based on a joint work with Marius Junge, Nicholas, LaRacunte, and Haojian Li.

A fundamental question in classical stable homotopy theory is to understand the stable homotopy groups of the spheres. A relatively new method is via the motivic approach. Motivic stable homotopy theory has an algebro-geometric root and closely connects to questions in number theory. Besides, it relates to the classical and the equivariant theories. The motivic category has good properties and allows different computational tools. I will talk about some of these properties and computations, and will show how it relates to the classical and equivariant categories.

We apply Quantum Linear System Algorithms (QLSAs) to Newton systems within IPMs to gain quantum speedup in solving LO problems. Due to their inexact nature, QLSAs can be applied only to inexact variants of IPMs, which are inexact infeasible methods due to the inexact nature f their computations. We propose Inexact-Feasible IPMs (IF-IPM) for LO and SDO problems, using novel Newton systems to generate inexact but feasible steps. We show that this method enjoys the to-date best iteration complexity. Further, we explore how QLSAs can be used efficiently in iterative refinement schemes to find an exact optimal solution without excessive calls to QLSAs. Finally, we experiment with the proposed IF-IPM’s efficiency using IBMs QISKIT environment.

Bio of the Speaker:

Dr. Terlaky is a George N. and Soteria Kledaras ’87 Endowed Chair Professor Department of Industrial and Systems Engineering, Lehigh University, and Director of the Quantum Computing Optimization Laboratory.

Dr. Terlaky has published four books, edited over ten books and journal special issues and published over 200 research papers. Topics include theoretical and algorithmic foundations of operations research, computational optimization, nuclear reactor core reloading optimization, oil refinery and VLSI design optimization, robust radiation therapy treatment optimization, inmate assignment optimization, quantum computing.

His research interest includes high performance optimization methods, optimization modeling, optimization problems in engineering sciences and service systems, and quantum computing optimization.

Dr. Terlaky is Editor-in-Chief of the Journal of Optimization Theory and Applications. He has served as associate editor of ten journals and has served as conference chair, conference organizer, and distinguished invited speaker at conferences all over the world. He was general Chair of the INFORMS 2015 Annual Meeting, a former Chair of INFORMS’ Optimization Society, Chair of the ICCOPT Steering Committee of the Mathematical Optimization Society, Chair of the SIAM AG Optimization. He received the MITACS Mentorship Award; Award of Merit of the Canadian Operational Society, Egerváry Award of the Hungarian Operations Research Society, H.G. Wagner Prize of INFOMRS, Outstanding Innovation in Service Science Engineering Award of IISE. He is Fellow of INFORMS, SIAM, IFORS, The Fields Institute, and elected Fellow of the Canadian Academy of Engineering. Currently he is serving as Vice President of INFORMS.

 In 2020, more than a hundred years after Einstein's publication of his theory of gravitation, half of the Nobel prize in Physics was awarded to Sir Roger Penrose "for the discovery that black hole formation is a robust prediction of the general theory of relativity". In this talk, I will present the basic mathematical formalism of general relativity, black holes, and their connections with modern analysis.

Stationary symmetric $\alpha$-stable ($S \alpha S$) processes is an important class of stochastic processes, including Gaussian processes, Cauchy processes and Lévy processes. In an analogy to that the ergodicity of a Gaussian process is determined by its spectral measure, it was shown by Rosinski and Samorodnitsky that the ergodicity of a stationary $S \alpha S$ process is characterized by its spectral representation. While this result is known when the process is indexed by $\mathbb{Z}$ or $\mathbb{R}$, the classical techniques fail when it comes to processes indexed by non-Abelian groups and it was an open question whether the ergodicity of stationary $S \alpha S$ processes indexed by amenable groups admits a similar characterization.

In this talk I will introduce the fundamentals of stable processes, the ergodic theory behind their spectral representation, and the key ideas of the characterization of the ergodicity for processes indexed by amenable groups. If time permits, I will explain how to use a recent construction of A. Danilenko in order to prove the existence of weakly-mixing but not strongly-mixing stable processes indexed by many groups (Abelian groups, Heisenberg group).

Let $E/\mathbb{Q}$ be an imaginary quadratic extension and
suppose $G$ is the unitary group attached to hermitian space over $E$ of
signature $(2,n)$. The symmetric domain $X$ attached to $G$ is a
quaternionic Kahler manifold in the sense of differential geometry. In
the late nineties N. Wallach studied harmonic analysis on $X$ in the
context of this quaternionic structure. He established a multiplicity
one theorem for spaces of generalized Whittaker periods appearing in the
cohomology of certain quaternionic $G$-bundles on $X$.

We prove new cases of Wallach's multiplicity one statement for some
degenerate generalized Whittaker periods and give explicit formulas for
these periods in terms of modified K-Bessel functions. Our results can
be interpreted as giving a refined Fourier expansion for automorphic
forms on $G$ which are quaternionic discrete series at infinity. As an
application we study the cusp forms on $G$ which arise as theta lifts of
holomorphic modular forms on quasi-split $\mathrm{U}(1,1)$. We show that
these cusp forms can be normalized so that all their Fourier
coefficients are algebraic numbers. (joint with Anton Hilado and Pan Yan)

We will discuss Einstein manifolds which are invariant under an isometric Lie group action. Our main goal is to explain the proof of the 1975 Alekseevskii Conjecture on non-compact homogeneous Einstein spaces, recently obtained in collaboration with Christoph Böhm (Münster). To that end, we will also present new structure results for Einstein metrics on principal bundles. The talk will conclude with open questions and future directions.

Differential item functioning (DIF) refers to the situation where responses to a given question on an exam (or survey or similar) differ between several groups. For several decades now, social scientists and education researchers have employed a standard battery of statistical tools to detect DIF from sample data, but essentially all of these standard tools rely on theoretical asymptotic results and presuppose sample sizes that are rarely achieved by real data sets. In this talk, we'll discuss how ideas dating back to Diaconis and Sturmfels, in which techniques from computational algebra are brought to bear in statistics, provide an alternative method to detect DIF which avoids asymptotics and is more robust with smaller sample sizes. This is joint work with Luis David Garcia-Puente, Minho Kim, and Flavia Sancier-Barbosa.

In a seminal work, Leray demonstrated the existence of global weak solutions to the Navier-Stokes equations in three dimensions. Are Leray's solutions unique? This is a fundamental question in mathematical hydrodynamics, which we answer in the negative within the "forced" category, by exhibiting a one-parameter family of distinct Leray solutions with zero initial velocity and identical body force. This is joint work with Elia Brué and Maria Colombo.

Given a topological group G, one considers the space of its closed subgroups, called the Chabauty space. I will talk about the structure and features of this space, and show how various algebraic and topological properties of a group are expressed there. 

The Garsia-Procesi module $R_\lambda$ has a well known basis of Artin monomials indexed by λ-subYamanouchi words, which correspond to the inv-statistic of the Haglund-Haiman-Loehr combinatorial formula for the modified Macdonald polynomials $H_\lambda(X;q,t)$ at $t=0$. We introduce a new basis for $R_\lambda$ of Garsia-Stanton descent monomials, giving a major-index type formula of the modified Hall-Littlewood polynomial $H_\lambda(x;q,t)$, and discuss the subtle connection to $H_\lambda(x;q,t)$ at $q=0$ via Robinson-Schensted-Knuth insertion. Our formula was discovered while searching for a basis of the Garsia-Haiman module by extending a similar result of Carlsson and Oblomkov for the diagonal coinvariants $DH_n$. This is joint work with E. Carlsson.

 A hybrid system is a dynamical system that exhibits both continuous and discrete dynamic behavior. The state of a hybrid system changes either continuously by the flow described by a differential equation or discretely following some jump conditions. A canonical example of a hybrid system is the bouncing ball, imagined as a point-mass, under the influence of gravity. In this talk, we explore the solutions and algorithms to the extensions of this example in 3-dimension, where the body of interest is rigid and convex in general and the plane may be tilted. In particular, the solutions utilize the theory of nonsmooth Lagrangian mechanics to derive the differential equations and jump conditions, which heavily depend on the collision detection function. The proposed algorithm called Lie group variational collision integrator (LGVCI) is developed using the combination of techniques and knowledge from variational collision integrators and Lie group variational integrators. Furthermore, we also developed a sensible and practical regularization (by analysis and applying $\epsilon$-rounding on signed distance functions) for collision response for convex rigid bodies with corners, and this completely avoids the need for nonsmooth convex analysis, and computations of tangent and normal cones. We have extensive numerical experiments and animations from our algorithm demonstrating that LGCVI are symplectic-momentum preserving and have long-time, near energy conservation.

This is a joined work with Professor Melvin Leok, and we are looking to apply and extend this work in the fields of control & optimal control theory and robotics, especially in the realm of bipedal robots. There will be further discussions on these topics in the section of future directions of the talk.

A separable $C^*$-algebra $A$ is said to be $\ell$-open ( or $\ell$-closed) when the image of Hom(A, B) is open (or closed) in Hom(A, B/I), for all separable $C^*$-algebras B and ideals I. The concept of semiprojectivity has been used many times in the classification of C*-algebras. Bruce Blackadar introduced $\ell$-open and $\ell$-closed $C^*$-algebras as a superclass of semiprojective $C^*$-algebras.

In recent work with A. Tikuisis, we characterize $\ell$-open and $\ell$-closed $C^*$-algebras and deduce that $\ell$-open $C^*$-algebras are $\ell$-closed as conjectured by Blackadar. Moreover, we show that the notion of $\ell$-open $C^*$-algebras and semiprojective $C^*$-algebras coincide for commutative unital $C^*$-algebras.

Topological K-theory is one of the classical motivating examples of a commutative ring spectrum, and it has a natural equivariant generalization. The equivariant structure here has the strongest possible type of compatibility with the multiplication, making K-theory an example of a ``genuine-commutative" ring spectrum.  There's quite a lot of structure involved here, so in order to understand it, we employ a classic strategy and rationalize.   After rationalizing, we can use algebraic models due to Barnes--Greenlees--Kedziorek and to Wimmer to show that all of the additional ``norm" structure is determined by the equivariant homotopy groups and the underlying multiplication.  This is joint work with Christy Hazel, Jocelyne Ishak, Magdalena Kedziorek, and Clover May.

The Kardar-Parisi-Zhang (KPZ) equation is a fundamental stochastic PDE related to many important models like random growth processes, Burgers turbulence, interacting particles system, random polymers etc. In this talk, we focus on how the tall peaks and deep valleys of the KPZ height function grow as time increases. In particular, we will ask what is the appropriate scaling of the peaks and valleys of the (1+1)-d KPZ equation and whether they converge to any limit under those scaling. These questions will be answered via the law of iterated logarithms and fractal dimensions of the level sets. The talk will be based on joint works with Sayan Das and Jaeyun Yi. If time permits, I will also mention an interesting story about the (2+1)-d and (3+1)-d case (work in progress with Jaeyun Yi).

There are two types of algorithms in Reinforcement Learning (RL): value-based and policy-based. As nonlinear function approximations, such as Deep Neural Networks, become popular in RL, algorithmic instability is often observed in practice for both types of algorithms. One reason is that most algorithms are based on the contraction property of the Bellman operator, which may no longer hold in nonlinear approximation. In this talk, we will introduce two algorithms based on the Bellman residual whose performance is independent of the contraction property of the Bellman operator. In both algorithms, we formulate the RL into an unconstrained optimization problem. The first algorithm is value-based, where we assume the underlying dynamics is smooth. We proposed an algorithm called Borrowing From the Future (BFF), and we proved that it has an exponentially fast convergence rate in model-free control. The second algorithm is policy-based. We proposed an algorithm called variational actor-critic with flipping gradients. We prove that it is guaranteed to converge to the optimal policy when the state space is finite. 

This talk will cover some foundational frameworks, strategies, and empirical findings related to undergraduate mathematics education and mathematics education more broadly. Specifically, I will present some evidence-based research on how to promote student engagement (what is active learning?), instructional design theory, teacher talk moves, and equity frameworks. Lastly, I’ll give some resources related to teaching undergraduate mathematics. 

We present families of origamis of genus 3 that have arithmetic Kontsevich-Zorich monodromy in the sense of Sarnak. It is known this is true for origamis of genus 2, however the techniques for higher genera should be different. We present an outline of the proof for the existence of these families.

A lot of graph inference problems consist in finding a low-rank structure planted in the adjacency matrix of the graph. When sparse enough, the simple study of the adjacency matrix is not enough; the individual variance of each vertex influences too much the overall spectrum of $A$. In contrast, we show how the non-backtracking matrix $B$ recovers these low-rank structures more consistently. This generalizes the results of Bordenave et al. (2015) to a much wider range of settings, beyond the classical stochastic block model.

Consider a countable group G acting on the unit sphere S in the space of L^2 functions on G by the regular representation. Given a homology class h in the quotient space S/G, one defines the spherical Plateau solutions for h as the intrinsic flat limits of volume minimizing sequences of cycles representing h. In some special cases, for example when G is the fundamental group of a closed hyperbolic manifold of dimension at least 3, the spherical Plateau solutions are essentially unique and can be identified. However not much is known about the properties of general spherical Plateau solutions. I will discuss the questions of existence and structure of non-trivial spherical Plateau solutions.

The Emerton-Gee stack for $\mathrm{GL}_2$ is a stack of $(\varphi, \Gamma)$-modules whose reduced part $\mathcal{X}_{2, \mathrm{red}}$ can be viewed as a moduli stack of mod $p$ representations of a $p$-adic Galois group. We compute criteria for codimension one intersections of the irreducible components of $\mathcal{X}_{2, \mathrm{red}}$, and interpret them in sheaf-theoretic terms. We also give a cohomological criterion for the number of top-dimensional components in a codimension one intersection.

[pre-talk at 1:20PM]

Mirror symmetry was a phenomenon discovered by physicists
around 1989: they observed that certain kinds of six-dimensional
geometric objects known as Calabi-Yau manifolds seemed to come in
pairs, with a strange relationship between different kinds of
geometric objects on the pairs. Since then, the subject has blossomed
into a vast field, with many different approaches and philosophies. I
will give a brief introduction to the subject, and explain how one of
these approaches, developed with Bernd Siebert, has led to a general
construction of mirror pairs.

I am interested in developing numerical methods for solving
PDEs (e.g. Einstein's equation) on manifolds with topology $\mathbb{R} \times M$, where $M$ is a three-dimensional manifold with arbitrary topology.  This talk will describe the basic methods we have developed for constructing convenient representations of these manifolds suitable for this numerical work, and some simple examples will be shown.  There won't be time in this talk to describe everything we have done, so I will focus on just one issue: how to construct $C^0$ reference metrics on these manifolds.  We now have methods that can construct such metrics automatically for a fairly large collection of manifolds.  Unfortunately, these methods fail in general, so improved methods are needed.

I will discuss my ongoing joint work with Ionut Chifan and Daniel Drimbe on various rigidity aspects of von Neumann algebras arising from graph product groups whose underlying graph is a certain cycle of cliques and whose vertex groups are wreath-like product property (T) groups. In particular, I will describe all symmetries of these von Neumann algebras by establishing formulas in the spirit of Genevois and Martin’s results on automorphisms of graph product groups. In doing so, I will highlight the methods used from Popa’s deformation/rigidity theory as well as new techniques pertaining to graph product algebras.

The Kostka semigroup consists of pairs of partitions with at most r parts that have a positive Kostka coefficient. For this semigroup, Hilbert basis membership is an NP-complete problem. We introduce KGR graphs and conservative subtrees, through the Gale-Ryser theorem on contingency tables, as a criterion for membership. In our main application, we show that if a partition pair is in the Hilbert basis then the partitions are at most $r$ wide. We also classify the extremal rays of the associated polyhedral cone; these rays correspond to a (strict) subset of the Hilbert basis. In an appendix, the second and third authors show that a natural extension of our main result on the Kostka semigroup cannot be extended to the Littlewood-Richardson semigroup. This furthermore gives a counterexample to recent speculation of P. Belkale concerning the semigroup controlling nonvanishing conformal blocks.

A pervading question in the study of stochastic PDE is how small-scale random forcing in an equation combines to create nontrivial statistical behavior on large spatial and temporal scales. I will discuss recent progress on this topic for several related stochastic PDEs - stochastic heat, KPZ, and Burgers equations - and some of their generalizations. These equations are (conjecturally) universal models of physical processes such as a polymer in a random environment, the growth of a random interface, branching Brownian motion, and the voter model. The large-scale behavior of solutions on large scales is complex, and in particular depends qualitatively on the dimension of the space. I will describe the phenomenology, and then describe several results and challenging problems on invariant measures, growth exponents, and limiting distributions.

 Inspired by work of Rogers in the classical geometry of numbers, we'll describe how to obtain variance bounds for classical geometric counting problems in the settings of translation surfaces and hyperbolic surfaces, and give some applications to understanding correlations between special trajectories on these types of surfaces. Parts of this will be joint work with Y. Cheung and H. Masur; S. Fairchild and H. Masur; and F. Arana-Herrera, and all of this has been inspired by joint work with G. Margulis.

In this talk we define the fundamental geometric constructions behind the generalized Kahler geometry introduced by Hitchin and Gualtier and set up an appropriate generalization of the Calabi problem. Similarly to Cao's approach to the solution of the classical Calabi problem, we study the existence and convergence of the generalized Kahler-Ricci flow (GKRF) on relevant backgrounds. In particular, we prove that on a Kahler Calabi-Yau background, the GKRF converges to the unique classical Ricci-Flat structure. This result has non-trivial applications to understanding the space of generalized Kahler structures, and as a special case yields the topological structure of natural classes of Hamiltonian symplectomorphisms on hyperKahler manifolds. Based on a joint work with Apostolov, Fu and Streets.

We study the capacity of the range of a simple random walk in three and higher dimensions. It is known that the order of the capacity of the random walk range in n dimensions is similar to that of the volume of the random walk range in n-2 dimensions. We show that this correspondence breaks down for the law of the iterated logarithm for the capacity of the random walk range in three dimensions. We also prove the law of the iterated logarithm in higher dimensions.

This is joint work with Amir Dembo. 

We will discuss Galois cohomology groups of “intermediate” quotients of an induced module, which sit between Iwasawa cohomology up a tower and cohomology over the ground field. Special elements in Iwasawa cohomology that arise from Euler systems become divisible by a certain Euler factor upon norming down to the ground field. In certain instances, there are reasons to wonder whether this divisibility can also hold for the image in intermediate cohomology. Using “intermediate” Coleman maps, we shall see that the situation locally at $p$ is as nice as one could imagine.

We will discuss Galois cohomology groups of “intermediate” quotients of an induced module, which sit between Iwasawa cohomology up a tower and cohomology over the ground field. Special elements in Iwasawa cohomology that arise from Euler systems become divisible by a certain Euler factor upon norming down to the ground field. In certain instances, there are reasons to wonder whether this divisibility can also hold for the image in intermediate cohomology. Using “intermediate” Coleman maps, we shall see that the situation locally at $p$ is as nice as one could imagine.

[pre-talk at 1:20PM]

Understanding the structure of conjugacy classes is essential in the study of a group. We will see how conjugacy classes of a group can be understood using group actions, and analyze the conjugacy classes for a variety of examples, including the group of symmetries of a tree and the group of almost symmetries of a tree, following a joint work with Waltraud Lederle.

For more than hundred years, various concepts were developed to understand the fields of geometric objects and invariant differential operators between them for conformal Riemannian, projective geometries, hypersurface type CR geometries, etc.. More recently, general tools were presented for the entire class of the so called parabolic geometries, i.e., the Cartan geometries modelled on homogeneous spaces G/P with P a parabolic subgroup in a semi-simple Lie group G. All these geometries determine a class of distinguished affine connections, which carry an affine structure modelled on differential 1-forms . They correspond to reductions of P to its reductive Levi factor, and we call them Weyl structures similarly to the conformal case. The standard definition of differential invariants in this setting is as affine invariants of these connections, which do not depend on the choice within the class. In the lecture, I shall describe a universal calculus which provides an important first step to determine such invariants. The lecture will follow the recent preprint https://arxiv.org/abs/2210.16652 with Andreas Cap, but I will try to stress the cases relevant for the CR structures.

This talk will feature an idiosyncratic take on an underlying theme in my research program, that topology and geometry in higher dimensions can be used in describing arithmetic phenomena in lower ones. I hope to explain why there might be such a phenomenon, while indicating how unexpectedly deep it appears to be. For instance, here’s an interesting question that doesn’t appear to have been much studied but ties in closely with joint work with Akshay Venkatesh: when do two integer polynomials in a single variable x that are products of powers of x and cyclotomic polynomials sum to a third? Curiously, the path towards an answer appears to intertwine with the homology of modular curves, as well as a chain complex computing the homology of a circle.

This talk will feature an idiosyncratic take on an underlying theme in my research program, that topology and geometry in higher dimensions can be used in describing arithmetic phenomena in lower ones. I hope to explain why there might be such a phenomenon, while indicating how unexpectedly deep it appears to be. For instance, here’s an interesting question that doesn’t appear to have been much studied but ties in closely with joint work with Akshay Venkatesh: when do two integer polynomials in a single variable x that are products of powers of x and cyclotomic polynomials sum to a third? Curiously, the path towards an answer appears to intertwine with the homology of modular curves, as well as a chain complex computing the homology of a circle.

Algebraic cycles on varieties are central objects in algebraic geometry and number theory. Problems around them are notoriously difficult. In the case of Shimura varieties, the study of modular forms whose coefficients are algebraic cycles and the closely related study of the automorphy of representations spanned by algebraic cycles are central to the advancement of knowledge in this area. I will discuss the background and history of these topics, as well as some recent progress and applications.

Algebraic cycles on varieties are central objects in algebraic geometry and number theory. Problems around them are notoriously difficult. In the case of Shimura varieties, the study of modular forms whose coefficients are algebraic cycles and the closely related study of the automorphy of representations spanned by algebraic cycles are central to the advancement of knowledge in this area. I will discuss the background and history of these topics, as well as some recent progress and applications

Linear regression on network-linked observations has been essential in modeling the relationships between responses and covariates with additional network structures. Many approaches either lack inference tools or rely on restrictive assumptions of social effects. More importantly, these methods usually assume that networks are error-free. I introduce a regression model with nonparametric network effects based on subspace assumptions. This model does not assume the network structure to be precisely observed and is provably robust to network observational errors. An inference framework is established under the general requirement of network observational errors, and corresponding robustness is studied in detail when observational errors arise from random network models. Results reveal a phase-transition phenomenon of inference validity in relation to network density when no prior knowledge of the network model is available. I also show that significant improvements can be achieved when the network model is known. I then briefly discuss an ensemble network estimation strategy, network mixing, which can improve the adaptivity of the proposed method. The regression model is applied to investigate social impacts on students' perceptions of school safety based on observed friendship relations. It enables reliable analysis thanks to the nonparametric network effects and the robustness to network observational errors.

The notion of Strong Approximate Transitivity (SAT) for group actions on probability measure spaces was introduced by Jaworski in the early 90's. A canonical example of an SAT group action is provided by a group acting on its Poisson boundary (with respect to some "nice" probability measure on the group).   
In this talk, I will discuss a noncommutative analogue of the SAT property, and its connection with noncommutative Poisson boundary inclusions. 

 Projected-search methods for bound-constrained optimization are based on performing a search along a piecewise-linear continuous path obtained by projecting a search direction onto the feasible region. A potential benefit of a projected-search method is that the direction of the search path may change multiple times at the cost of computing a single direction.
In this talk, we present a new interior method for general nonlinearly constrained optimization that combines a shifted primal-dual interior method with a projected-search method for bound-constrained optimization. The method is based on the formulation of a primal-dual penalty-barrier function that incorporates shifts on both primal and dual variables.  A modified Newton direction is used in conjunction with a new projected-search algorithm that employs a non-monotone flexible quasi-Armijo line search for the minimization of the penalty-barrier function. Computational results indicate that the proposed method requires fewer iterations than a conventional interior method, thereby reducing the number of times that the search direction need be computed.

The celebrated Brunn-Minkowski inequality states that for compact subsets $X$ and $Y$ of $\Bbb{R}^d$, $m(X+Y)^{1/d} \geq m(X)^{1/d}+m(Y)^{1/d}$ where $m(\cdot)$ is the Lebesgue measure. We will introduce a conjecture generalizing this inequality to every locally compact group where the exponent is believed to be sharp. In a joint work with Yifan Jing and Chieu-Minh Tran, we prove this conjecture for a large class of groups (including e.g. all real linear algebraic groups). We also prove that the general conjecture will follow from the simple Lie group case. For those groups where we do not know the conjecture yet (one typical example being the universal covering of $SL_2(\Bbb{R})$), we also obtain partial results. In this talk I will discuss this inequality and explain important ingredients, old and new, in our proof.

The algebraic K-theory of the category of finite type $n$ spectra is a fundamental object containing structural information about the stable homotopy category. However, until recently almost nothing was known about it for $n>1$, primarily because it is not the K-theory of a connective ring. In this talk, I will explain how, for $n=2$, it can be computed in terms of K-theory of discrete rings and topological cyclic homology. In particular, we can read off the K groups in low degrees and find that there is an infinite family of 2-torsion classes in $K_0$ at the prime 2. I will also explain how to construct type 2 spectra representing these $K_0$ classes.

In this talk, we introduce the dynamic stochastic variational inequality (DSVI). The DSVI is an ODE whose right hand side is defined by the natural mapping of a VI (referred to as the first-stage VI) and is coupled with another stochastic VI (referred to as the second-stage SVI). The DSVI provides a unified modeling framework for various applications involving equilibrium/optimality conditions (VI), dynamics (ODE), and uncertainties (stochasticity). We establish solution existence and uniqueness for two classes of DSVIs: the first class is defined by a strongly monotone SVI in the second stage, and the second class pertains to a box-constrained stochastic linear VI with the P-property in the second stage. Preliminary results on switching dynamics of the DSVI are presented. We develop sample average approximation (SAA) and time-stepping schemes to compute the DSVI. The uniform convergence and exponential convergence are established for the SAA under suitable conditions. A time-stepping EDIIS (energy direct inversion on the iterative subspace) method is proposed to solve the differential VI arising from the SAA of the DSVI. Our results are illustrated by an instantaneous dynamic user equilibrium  problem in transportation engineering. This is a joint work with Dr. Xiaojun Chen of the Hong Kong Polytechnic University.

Tensor categories have played an important role in areas as diverse as topology, mathematical physics, operator algebras, and representation theory.
This is an introductory talk. I will mostly talk about the classification of tensor categories with given tensor product rules and module categories for certain important examples.

Deep learning's success has revealed a number of phenomena that appear to conflict with classical intuitions in the fields of optimization and statistics.  First, the objective functions formulated in deep learning are highly nonconvex but are typically amenable to minimization with first-order optimization methods like gradient descent.  And second, neural networks trained by gradient descent are capable of 'benign overfitting': they can achieve zero training error on noisy training data and simultaneously generalize well to unseen data.  In this talk we go over our recent work towards understanding these phenomena. 

The Hilbert transform $H$ is a basic example of Fourier multipliers.  Its behaviour on Fourier series is the following:

$$
\sum_{n\in \mathbb{Z}}a_n e^{inx} \longmapsto \sum_{n\in \mathbb{Z}}m(n)a_n e^{inx},
$$
with $m(n)=-i\,{\rm sgn} (n)$.
Riesz proved that $H$ is a bounded operator on $L_p(\mathbb{T})$ for all $1<p<\infty$.
We study  Hilbert transform type Fourier multipliers on group algebras and their boundedness on corresponding non-commutative $L_p$ spaces.
The pioneering work in this direction is due to Mei and Ricard who proved $L_p$-boundedness of Hilbert transforms on free group von Neumann algebras using a Cotlar identity. In this talk, we introduce a generalised Cotlar identity and a new geometric form of Hilbert transform for groups acting on tree-like structures. This class of groups includes amalgamated free products, HNN extensions, left orderable groups and many others.

\bigskip

\noindent{\small
Joint work with Adri\'an Gonz\'alez and Javier Parcet.

Differential forms are basic objects of manifolds and encode invariants. This talk will motivate the usefulness of considering pairs of differential forms together with a map linking them. We will show how this can lead to novel functionals and geometric flows. As an application, it leads to new notions of flat connections and Morse theory on symplectic manifolds. This talk is based on joint works with Jiawei Zhou, David Clausen and Xiang Tang.

The purpose of this talk is to illustrate an application of the powerful machinery of geometric measure theory to a conjecture in Gromov–Witten theory arising from physics. Very roughly speaking, the Gromov–Witten invariants of a symplectic manifold (X,ω) equipped with a tamed almost complex structure J are obtained by counting pseudo-holomorphic maps from mildly singular Riemann surfaces into (X,J). It turns out that Gromov–Witten invariants are quite complicated (or “have a rich internal structure”). This is true especially for if (X,ω) is a symplectic Calabi–Yau 3–fold (that is: dim X = 6, c_1(X,ω) = 0). In 1998, using arguments from M–theory, Gopakumar and Vafa argued that there are integer BPS invariants of symplectic Calabi–Yau 3–folds. Unfortunately, they did not give a direct mathematical definition of their BPS invariants, but they predicted that they are related to the Gromov–Witten invariants by a transformation of the generating series. The Gopakumar–Vafa conjecture asserts that if one defines the BPS invariants indirectly through this procedure, then they satisfy an integrality and a (genus) finiteness condition. The integrality conjecture has been resolved by Ionel and Parker. A key innovation of their proof is the introduction of the cluster formalism: an ingenious device to side-step questions regarding multiple covers and super-rigidity. Their argument could not resolve the finiteness conjecture, however. The reason for this is that it relies on Gromov’s compactness theorem for pseudo-holomorphic maps which requires an a priori genus bound. It turns out, however, that Gromov’s compactness theorem can (and should!) be replaced with the work of Federer–Flemming, Allard, and De Lellis–Spadaro–Spolaor. This upgrade of Ionel and Parker’s cluster formalism proves both the integrality and finiteness conjecture. This talk is based on joint work with Eleny Ionel and Aleksander Doan.

If we choose at random an integral binary form $f(x, z)$ of fixed degree $d$, what is the probability that the superelliptic curve with equation  $C \colon: y^m = f(x, z)$ has a $p$-adic point, or better, points everywhere locally? In joint work with Lea Beneish, we show that the proportion of forms $f(x, z)$ for which $C$ is everywhere locally soluble is positive, given by a product of local densities. By studying these local densities, we produce bounds which are suitable enough to pass to the large $d$ limit. In the specific case of curves of the form $y^3 = f(x, z)$ for a binary form of degree 6, we determine the probability of everywhere local solubility to be 96.94%, with the exact value given by an explicit infinite product of rational function expressions.

[pre-talk at 1:20PM]

One of the major tasks that a cell faces during its lifecycle is how to spatially localize its components. Correct spatial organization of various proteins (cellular polarity) is fundamental not only for the correct cell shape but also to carry out essential cellular functions, such as the spatial coordination of cell division. We present a mathematical model of the core mechanism responsible for the regulation of polarized growth dynamics in the model organism, fission yeast. The model is based on the competition of growth zones of polarity protein Cdc42 localized at the cell tips for a common substrate (inactive Cdc42) that diffuses in the cytosol. To explore the underlying mechanism for oscillations and the effect of diffusion and noise, we consider three model frameworks including a 1D deterministic model, a 2D deterministic model, and a stochastic model. We simulate and analyze these models using numerical bifurcation tools, PDEs, and stochastic simulation algorithms.

Many fundamental biophysical processes, from cell division to cellular motility, involve dynamics of thin structures immersed in a very viscous fluid. Various popular models have been developed to describe this interaction mathematically, but much of our understanding of these models is only at the level of numerics and formal asymptotics. Here we seek to develop the PDE theory of filament hydrodynamics.

First, we propose a PDE framework for analyzing the error introduced by slender body theory (SBT), a common approximation used to facilitate computational simulations of immersed filaments in 3D. Given data prescribed only along a 1D curve, we develop a novel type of boundary value problem and obtain an error estimate for SBT in terms of the fiber radius. This places slender body theory on firm theoretical footing.

Second, we consider a classical elastohydrodynamic model for the motion of an immersed inextensible filament. We highlight how the analysis can help to better understand undulatory swimming at a low Reynolds numbers. This includes the development of a novel numerical method to simulate inextensible swimmers.

Poonen's Closed Point Sieve has proven to be a powerful technique for producing structural and combinatorial results for varieties over finite fields. In this talk we discuss three results which come, in part, as a consequence of this technique. First we will discuss semiample Bertini Theorems over finite fields and examine the probability with which a semiample complete intersection is smooth. Next we apply the Closed Point Sieve to compute the probability with which a high degree projective hypersurface over $\mathbb{F}_{2^k}$ is locally Frobenius split (a characteristic $p$ analog of log canonical singularities). In doing so we show that most such hypersurfaces are only mildly singular. The final part, which is based on joint work with Kiran Kedlaya and James Upton, discusses $p$-adic coefficient objects in rigid cohomology. Namely, we show (under a geometric tameness hypothesis) that the overconvergence of a Frobenius isocrystal can be detected by the restriction of that isocrystal to the collection of smooth curves on a variety.

We present recent results about finite jet determination of CR maps of positive codimension from real-analytic CR manifolds into real-algebraic subsets in complex space, or more generally Nash subsets. One instance of such results is the unique jet determination of germs of CR maps from minimal real-analytic CR submanifolds in $\C^N$ into Nash subsets in $\C^{N'}$ of D'Angelo finite type, for arbitrary $N,N'\geq 2$. This is joint work with B. Lamel.