In this talk, I will discuss the (geometric) intersection number between closed geodesics on finite volume hyperbolic surfaces. Specifically, I talk about the optimum upper bound on the intersection number in terms of the product of hyperbolic lengths. I also talk about the equidistribution of the intersection points between closed geodesics.

Spectra are among the most fundamental objects in algebraic topology and appear naturally in the study of generalized cohomology theories, algebraic K-groups and cobordism invariants.  I will first explain that spectra define a homotopical enlargement of algebra known as “higher algebra,” where one has topological analogues of algebraic structures like rings, modules, and tensor products.

A striking feature of higher algebra is that there are additional “chromatic characteristics” interpolating between characteristic 0 and characteristic p.  These intermediate characteristics have shed light on mod p phenomena in geometry, number theory, and representation theory.  On the other hand, the extension of algebraic ideas to higher algebra has been fruitful within algebraic topology: I will discuss joint work with Robert Burklund and Tomer Schlank which proves a higher analogue of Hilbert’s Nullstellensatz, thus identifying the ‘’algebraically closed fields’’ of intermediate characteristic.  In addition to initiating the study of “chromatic algebraic geometry,” this work resolves a form of Rognes’ chromatic redshift conjecture in algebraic K-theory.

Property testing is a fundamental subject in theoretical computer science and combinatorics, which studies when and how global properties of large objects (such as a massive data set or a huge graph) can be robustly inferred when given only local views of the object. Famous examples of property testing include testing whether a given graph is triangle-free or whether a given boolean function is linear.

In this talk, I'll present a generalization of the property testing model where the "local views" of an object are not given by deterministic evaluations, but instead by the probabilistic outcomes of measurements on a quantum state. This gives rise to a noncommutative model of property testing, and raises many interesting questions at the interface of complexity theory, quantum information, operator algebras, and more. Finally, I'll describe how the recent quantum complexity result MIP* = RE can be viewed through the lens of noncommutative property testing.

The Bruhat order encodes algebraic and topological information of Schubert varieties in the flag manifold and possesses rich combinatorial properties. In this talk, we discuss three interrelated stories regarding the Bruhat order: self-dual Bruhat intervals, Billey-Postnikov decompositions, and automorphisms of the Bruhat graph. This is joint work with Christian Gaetz. 

For convex complete intersections, the Gromov-Witten invariants are often computed using the Quantum Lefshetz Hyperplane theorem, which relates the invariants to those of the ambient space. However,  the convexity condition often fails when the target is an orbifold, even for genus 0, hence Quantum Lefshetz is no longer guaranteed. In this talk, I will showcase a method to compute these invariants, despite the failure of Quantum Lefshetz, for orbifold complete intersections in stack quotients of the form [V // G]. This talk will be based on joint works with Felix Janda (Notre Dame) and Yang Zhou (Fudan), and with Rachel Webb (Berkeley).

We consider generic translation surfaces of genus g>0 with n>1 marked points and take covers branched over the marked points such that the monodromy of every element in the fundamental group lies in a cyclic group of order d. Given a translation surface, the number of cylinders with waist curve of length at most L grows like L^2. By work of Veech and Eskin-Masur, when normalizing the number of cylinders by L^2, the limit as L goes to infinity exists and the resulting number is called a Siegel-Veech constant. The same holds true if we weight the cylinders by their area. Remarkably, the Siegel-Veech constant resulting from counting cylinders weighted by area is independent of the number of branch points n. All necessary background will be given and a connection to combinatorics will be presented. This is joint work with Aaron Calderon, Carlos Matheus, Nick Salter, and Martin Schmoll.

The study of affine Deligne-Lusztig varieties (ADLVs) $X_w(b)$ and their certain union $X(\mu,b)$ has been crucial in understanding mod-$p$ reduction of Shimura varieties; for instance, precise information about the connected components of ADLVs (in the hyperspecial level) has proved to be useful in Kisin's proof of the Langlands-Rapoport conjecture. On the other hand, first introduced in the context of enumerative geometry to describe the quantum cohomology ring of complex flag varieties, quantum Bruhat graphs have found recent applications in solving certain problems on the ADLVs. I will survey such developments and report on my work surrounding a dimension formula for $X(\mu,b)$ in the quasi-split case, as well as some partial description of the dimension and top-dimensional irreducible components in the non quasi-split case.

[pre-talk at 1:20PM]

I’ll talk about ongoing work in collaboration with the Ding lab of Biomedical Engineering at UCI. There are unresolved mysteries about the dynamics of RNA splicing, an important molecular process in the genetic machinery. These mysteries remain because the obtainable data for this process are not time series, but rather static spatial images of cells with stochastic particles.  From a modeling perspective, this creates a challenge of finding the right mathematical description that respects the stochasticity of individual particles but remains computationally tractable. I’ll share our approach of constructing a spatial Cox process with intensity governed by a reaction-diffusion PDE. We can do inference on this process with experimental images by employing variational Bayesian inference. Several outstanding issues remain about how to combine classical and modern statistical/data-science approaches with more exotic mechanistic models in biology.

Complete Calabi-Yau metrics are fundamental objects in Kahler geometry arising as singularity models or "bubbles" ​in degenerations of compact Calabi-Yau manifolds.  The existence of these metrics and their relationship with algebraic geometry are the subjects of several long standing conjectures due to Yau and Tian-Yau.  I will describe some recent progress towards the question of existence, and explain some future directions, highlighting connections with notions of algebro-geometric stability.​

Moving boundary (or often called “free boundary”) problems are ubiquitous in nature and technology. A computational perspective of moving boundary problems can provide insight into the “invisible” properties of complex dynamics systems, advance the design of novel technologies, and improve the understanding of biological and chemical phenomena. However, challenges lie in the numerical study of moving boundary problems. Examples include difficulties in solving PDEs in irregular domains, handling moving boundaries efficiently and accurately, as well as computing efficiency difficulties. In this talk, I will discuss three specific topics of moving boundary problems, with applications to ecology (population dynamics), plasma physics (ITER tokamak machine design), and cell biology (cell movement). In addition, some techniques of scientific computing will be discussed.

Positroids are certain representable matroids originally studied by Postnikov in connection with the totally nonnegative Grassmannian and now used widely in algebraic combinatorics.  The positroids give rise to determinantal equations defining positroid varieties as subvarieties of the Grassmannian variety. Rietsch, Knutson-Lam-Speyer, and Pawlowski studied geometric and cohomological properties of these varieties.  In this talk, we continue the study of the geometric properties of positroid varieties by establishing several equivalent conditions characterizing smooth positroid varieties using a variation of pattern avoidance defined on decorated permutations, which are in bijection with positroids.  This allows us to give several formulas for counting the number of smooth positroids according to natural statistics on decorated permutations.  Furthermore, we give a combinatorial method for determining the dimension of the tangent space of a positroid variety at the torus fixed points using an induced subgraph of the Johnson graph.  We will conclude with some open problems in this area.

This talk is based on joint work with Jordan Weaver and Christian Krattenthaler.  

We investigate the analogs of optimal transport theory in the setting of multivariable asymptotic random matrix theory.  Asymptotic random matrix theory concerns the behavior of randomly chosen $N \times N$ matrices in the limit as $N \to \infty$.  For several random matrices $X_1^{(N)}$, $\dots$, $X_d^{(N)}$, one can study the asymptotic behavior of expressions like $(1/N) \Tr(X_{i_1} \dots X_{i_k})$, and the appropriate limiting object is a non-commutative probability space, that is, a von Neumann algebra $A$ of "random variables" together with an expectation map $E: A \to \mathbb{C}$, analogous to the expected trace of a random matrix.  Meanwhile, optimal transport theory asks for the most efficient way to rearrange one distribution of mass $\mu$ on $\mathbb{R}^d$ into another such distribution $\nu$.  Such a scheme is often given by transporting the mass at point $x$ to point $f(y)$, for a smooth function $f: \mathbb{R}^d \to \mathbb{R}^d$.

Optimal transport is more challenging to make sense of in the non-commutative setting because, unlike classical probability theory, there are many non-isomorphic atomless non-commutative probability spaces, and in fact, space of non-commutative probability distributions fails basic separability and finite-dimensional approximation properties that one is used to in classical probability.  So there is often no possibility of transporting given non-commutative probability distribution $\mu$ to $\nu$ by some map $f$; nonetheless, for the relaxed problem of optimal couplings, we can recover a non-commutative analog of the Monge-Kantorovich duality characterizing optimal couplings. Furthermore, in the regime of convex free Gibbs laws (an analog of smooth log-concave probability measures on $\mathbb{R}^d$), non-commutative transport can be achieved by non-commutative smooth functions obtained as solutions to differential equations much like the classical case.  Moreover, the non-commutative analog of triangular transformations of measures led to new insight into the structure of the underlying von Neumann algebras.

We address a system of equations modeling a compressible fluid interacting with an elastic body in dimension three. We prove the local existence and uniqueness of a strong solution when the initial velocity belongs to the space $H^{2+\epsilon}$ and the initial structure velocity is in $H^{1.5+\epsilon}$, where $\epsilon\in(0,1/2)$. 

The Learn to Optimize paradigm leverages machine learning to accelerate the discovery of new optimization methods. This method's core idea is to use a neural network to simulate the optimization process or provide critical decisions during the process to solve the optimization problem. This talk will introduce two recent research works in learning to optimization.

The first is a theoretical work discussing the application of graph neural networks (GNNs) to linear and mixed integer programming. We prove that well-trained GNNs can solve linear programs but not mixed integer programs without proper fixes. The other is constructing a fixed-point iterative neural network to solve inverse problems and game problems.

 A {\em permutation statistic} is a complex-valued function on the symmetric group $\mathfrak{S}_n$.  We describe a notion of `locality' which measures the complexity of permutation statistics. Applications are given to the asymptotic behavior of families of statistics as the parameter $n$ grows. The key technical tool is an irreducible character evaluation on the symmetric group algebra which involves a novel combinatorics of `monotonic ribbon tableaux'. Joint with Zach Hamaker.

Bypassing the formalities of sigma-algebra and measures, I will show how one can see the Lebesgue integration as an initial object of some category. The talk is based on a paper by Tom Leinster.

Teichmüller curves are algebraic curves in the moduli space of genus $g$ Riemann surfaces that are isometrically immersed for the Teichmüller metric. These curves arise from $\mathrm{SL}(2,\mathbb{R})$-orbits of highly symmetric translation surfaces, and the underlying surfaces have remarkable dynamical and algebro-geometric properties. A Teichmüller curve is algebraically primitive if the trace field of its affine symmetry group has degree $g$. In genus $2$, Calta and McMullen independently discovered an infinite family of algebraically primitive Teichmüller curves. However, in higher genus, such curves seem to be much rarer. We will discuss a result that shows that the regular $14$-gon yields the unique algebraically primitive Teichmüller curve in genus $3$ of a particular combinatorial type. All relevant notions will be explained during the talk.

Expected shortfall, measuring the average outcome (e.g., portfolio loss) above a given quantile
of its probability distribution, is a common financial risk measure. The same measure can be
used to characterize treatment effects in the tail of an outcome distribution, with applications
ranging from policy evaluation in economics and public health to biomedical investigations.
Expected shortfall regression is a natural approach of modeling covariate-adjusted expected
shortfalls. Because the expected shortfall cannot be written as a solution of an expected loss
function at the population level, computational as well as statistical challenges around
expected shortfall regression have led to stimulating research. We discuss some recent
developments in this area, with a focus on a new optimization-based semiparametric approach
to estimation of conditional expected shortfall that adapts well to data heterogeneity with
minimal model assumptions.

asymptotically cylindrical flows are ancient solutions to the mean curvature flow whose tangent flow at $-\infty$ are shrinking cylinders. In this talk, we study quantized behavior of asymptotically cylindrical flows. We show that the cylindrical profile function u of these flows have the asymptotics $u(y,\omega, \tau) = \frac{y^{T}Qy - 2tr Q}{|\tau|} + o(|\tau|^{-1})$ as $\tau\rightarrow -\infty$, where $Q$ is a constant symmetric $k\times k$ matrix whose eigenvalues are quantized to be either 0 or $-\frac{\sqrt{2(n-k)}}{4}$. Assuming non-collapsing, we can further draw two applications. In the zero rank case, we obtain the full classification. In the full rank case, we obtain the $SO(n-k+1)$ symmetry of the solution. This is joint work with Wenkui Du.

Prismatic cohomology is a new p-adic cohomology theory introduced by Bhatt and Scholze that specializes to various well-known cohomology theories such as étale, de Rham and crystalline. I will roughly recall the properties of this cohomology and explain how to prove its Poincaré duality.

[pre-talk at 1:20PM]

The matrix completion problems are about completing a partially filled matrix to achieve the lowest possible rank. As they can be interpreted as an understanding of a certain algebraic variety, we consider a corresponding algebraic matroid and desire to characterize its bases. Polyhedralizing via tropical algebra may help us to figure out this characterization. We begin the talk with brief introductions on matrix completion problems, algebraic matroids, and tropical algebra. No pre-knowledge is assumed. This is based on ongoing work with May Cai, Cvetelina Hill, and Josephine Yu. 

For the past 90 years, a fundamental question in classical homotopy theory is to understand the stable homotopy groups of spheres.  The most modern method to study these groups is to compare them with the ``motivic stable homotopy groups of spheres".  Motivic homotopy theory has its roots in algebraic geometry.  As a result of the recent advances, there is a reintegration of algebraic topology and algebraic geometry, with close connections to equivariant homotopy theory and number theory.

In this talk, I will introduce the classical and motivic stable homotopy categories and the connections between the two.  I will then talk about the rich properties and extra structures that are present in the motivic stable homotopy category.  The presence of these extra structures gives new computational tools that dramatically improve our understanding of the classical stable homotopy groups.  Moreover, the flow of information can be reversed as well, producing new results in motivic stable homotopy theory for general fields.

The Yang-Mills model is a theoretical framework for fundamental forces and elementary particles. It has made deep impacts in various branches of mathematics. A key challenge in mathematical physics is to construct the quantum Yang-Mills theory on four dimensional space and prove the existence of a "mass gap". In this talk, we will discuss stochastic quantization i.e. Langevin dynamics of the Yang-Mills model on two and three dimensional tori. This is a stochastic process on the space of "gauge orbits", induced by the solution to a nonlinear Lie algebra-valued stochastic PDE driven by space-time white noise. The presence of very singular random forcing as well as nonlinearities render it challenging to interpret what one even means by “solutions”, “state space”, “orbit space” and “gauge invariant observables”. We rigorously construct these objects by combining techniques from analysis, PDE, Stochastic PDE,  and especially the theory of regularity structures. The talk is based on joint work with Ajay Chandra, Ilya Chevyrev, Martin Hairer, among many other collaborators.

The focus of the talk is interior-point methods for inequality-constrained quadratic programs, and particularly the system of nonlinear equations to be solved for each value of the barrier parameter. Newton iterations give high quality solutions, but we are interested in modified Newton systems that are computationally less expensive at the expense of lower quality solutions.  We propose a structured modified Newton approach where each modified Jacobian is composed of a previous Jacobian, plus one low-rank update matrix per succeeding iteration. Each update matrix is, for a given rank, chosen such that the distance to the Jacobian at the current iterate is minimized, in both 2-norm and Frobenius norm. The approach is structured in the sense that it preserves the nonzero pattern of the Jacobian. The choice of update matrix is supported by results in an ideal theoretical setting. We also produce numerical results with a basic interior-point implementation to investigate the practical performance within and beyond the theoretical framework. In order to improve performance beyond the theoretical framework, we also motivate and construct two heuristics to be added to the method.

We extend the traditional framework of noncommutative geometry in order to deal with two types of approximation of metric spaces. On the one hand, we consider spectral truncations of geometric spaces, while on the other hand, we consider metric spaces up to finite resolution. In our approach, the traditional role played by C*-algebras is taken over by so-called operator systems. Essentially, this is the minimal structure required on a space of operators to be able to speak of positive elements, states, pure states, etc. We illustrate our methods in concrete examples obtained by spectral truncations of the circle and of metric spaces up to finite resolution. The former yield operator systems of finite-dimensional Toeplitz matrices, and the latter give suitable subspaces of the compact operators. We also analyze the cones of positive elements and the pure-state spaces for these operator systems, which turn out to possess a very rich structure.

The Maxwell-Vlasov equations model the evolution of a plasma and can be derived from a suitably chosen Lagrangian.  That Lagrangian decomposes as the sum of terms associated with the motion of the particles, the energy stored in the electromagnetic field, and the interaction of the particles with the field.  Previous structure-preserving approaches to modeling fluid dynamics, using the group structure of the configuration space, and electromagnetic fields in vacuum, using the de Rham complex, have proved effective, raising the question whether these results could be leveraged to obtain well-behaved numerical solutions of the Maxwell-Vlasov system, thought of as a composite.  With this goal in mind, we write the Maxwell-Vlasov Euler-Poincaré equations in a weak, variational form, then use approximation spaces suggested by the referenced works to obtain a semidiscrete version of the problem.  We then present work done towards solving the fully discrete problem and indicate future directions.

The Schubert polynomials and key polynomials form two important bases for the polynomial ring. Schubert and key polynomials are the​``bottom layers” of Grothendieck and Lascoux polynomials, two inhomogeneous polynomials. In this talk, we look at their​``top layers”. We develop a diagrammatic way to compute the degrees and the leading monomials of these top layers. Finally, we describe the Hilbert series of the space spanned by these top layers, involving a classical $q$-analogue of the Bell numbers.

The aim of this talk to show that several quantities: as the spectral radius of weakly irreducible tensors, maximum of d-homogeneous polynomial with nonnegative coefficients in the unit ball of the d-H¨older norm, are polynomially computable. This computability result is proven for a larger class of minimum of certain convex functions in R n, which was considered by several authors. This is a joint work with Stephane Gaubert, INRIA and Centre de Math´ematiques Appliqu´ees (CMAP), Ecole polytechnique, IP Paris, France.

I’ll give the audience three topics that I think are interesting, and they’ll vote on which one they want to hear about. So it’s a surprise!

 A translation surface is given by polygons in the plane, with sides identified by translations to create a closed Riemann surface with a flat structure away from finitely many singular points. Understanding geodesic flow on a surface involves understanding saddle connections. Saddle connections are the geodesics starting and ending at these singular points and are associated to a discrete subset of the plane. To measure the behavior of saddle connections of length at most $R$, we obtain precise decay rates as $R$ goes to infinity for the difference in angle between two almost horizontal saddle connections. This is based on joint work with Jon Chaika.

The Whittaker Plancherel theorem appeared as Chapter 15 in my two volume book, Real Reductive Groups. It was meant to be an application of Harish-Chandra’s Plancherel Theorem.  As it turns out, there are serious gaps in the proof given in the books. At the same time as I was doing my research on the subject, Harish-Chandra was also working on it. His approach was very different from mine and appears as part of Volume 5 of his collected works; which consists of three pieces of research by Harish-Chandra that were incomplete at his death and organized and edited by Gangolli and Varadarajan. Unfortunately,  it also does not contain a proof of the theorem. There was a complication in the proof of this result that caused substantial difficulties which had to do with the image of the analog of Harish-Chandra’s method of descent. In this lecture I will explain how one can complete the proof using a recent result of Raphael Beuzzart-Plessis. I will also give an application of the result to the Fourier transforms of automorphic functions at cusps.

(This seminar will be given remotely, but there will still be a live audience in the lecture room.)

[pre-talk at 1:20PM]

The moduli spaces of one-dimensional sheaves on P^2, first studied by Simpson and Le Potier, admit a Hilbert-Chow morphism to a projective base that behaves like a completely integrable system. Following a proposal of Maulik-Toda, one expects to obtain certain BPS invariants from the perverse filtration on cohomology induced by this morphism. This motivates us to study the cohomology ring structure of these moduli spaces. In this talk, we present a minimal set of tautological generators for the cohomology ring, and propose a “Perverse = Chern” conjecture concerning these generators, which specializes to an asymptotic product formula for refined BPS invariants of local P^2. This can be viewed as an analogue of the recently proved P=W conjecture for Hitchin systems. Based on joint work with Junliang Shen, and with Yakov Kononov and Junliang Shen.

The growth of an infinite-dimensional algebra is a fundamental tool to measure its infinitude. Growth of noncommutative algebras plays an important role in noncommutative geometry, representation theory, differential algebraic geometry, symbolic dynamics and various recent homological stability results in number theory and arithmetic geometry.

We analyze the space of growth functions of algebras, answering a question of Zelmanov (2017) on the existence of certain 'holes' in this space, and prove a strong quantitative version of the Kurosh Problem on algebraic algebras. We use minimal subshifts with highly correlated oscillating complexities to resolve a question posed by Krempa-Okninski (1987) and Krause-Lenagan (2000) on the GK-dimension of tensor products.

An important property implied by subexponential growth (for both groups and algebras) is amenability. We show that minimal subshifts of positive entropy give rise to amenable graded algebras of exponential growth, answering a conjecture of Bartholdi (2007; naturally extending a wide open conjecture of Vershik on amenable group rings).

This talk is partially based on joint works with J. Bell and with E. Zelmanov.

Integrable systems, both classical and quantum, kept reemerging in mathematics and theoretical physics during the past several decades. In this talk, after briefly reviewing classical and quantum integrable systems, I will focus on two recent geometric incarnations of integrable systems based on quantum groups, solved by the algebraic Bethe ansatz method. One is motivated by studying 2- and 3-dimensional supersymmetric gauge theories and mathematically explained through enumerative geometry of quiver varieties. Another comes from an instance of geometric Langlands correspondence. Finally, I will explain the relationship between these two geometrizations and discuss their applications.

 Geometric Deep Learning is an emerging field of research that aims to extend the success of machine learning and, in particular, convolutional neural networks, to data with non-Euclidean geometric structure such as graphs and manifolds. Despite being in its relative infancy, this field has already found great success and is utilized by, e.g., Google Maps and Amazon’s recommender systems.


In order to improve our understanding of the networks used in this new field, several works have proposed novel versions of the scattering transform, a wavelet-based model of neural networks for graphs, manifolds, and more general measure spaces. In a similar spirit to the original scattering transform, which was designed for Euclidean data such as images, these geometric scattering transforms provide a mathematically rigorous framework for understanding the stability and invariance of the networks used in geometric deep learning. Additionally, they also have many interesting applications such as discovering new drug-like molecules, solving combinatorial optimization problems, and using single-cell data to predict whether or not a cancer patient will respond to treatment.
 

In 1938, Erdos, Ko and Rado proved a foundational result on the size of intersecting families of sets. Ever since, there has been a rich body of results proving similar theorems in different contexts. Notably, in geometries over finite fields like projective and polar spaces, such results were obtained by several groups of researchers. I will explain a very successful technique that can be used to prove these results, and indicate its shortcomings. We will show how a generalization of this problem recovers all known results and how algebraic combinatorics such as Iwahori-Hecke algebras and their representations come into play. The latter is based on joint work with Jan De Beule and Klaus Metsch.

We study Markov decision processes (MDPs) in the space of state-action distributions using algebraic and differential geometric methods. We provide an explicit description of the set of feasible state-action distributions of a partially observable problems with memoryless stochastic policies through polynomial constraints. In particular, this yields a formulation of the reward optimization problem as a polynomially constrained linear objective program. This allows us to study the combinatorial and algebraic complexity of the problem and we obtain explicit upper bounds on the number of critical points over every boundary component of the feasible set for a large class of problems. We demonstrate that the polynomial programming formulation of reward optimization can be solved using tools from constrained optimization and applied algebraic geometry, which exhibit stability improvements and provide globally optimal solutions. Further, we study the convergence of several natural policy gradient (NPG) methods with regular policy parametrization. For a variety of NPGs we show that the trajectories in state-action space are solutions of gradient flows with respect to Hessian geometries, based on which we obtain global convergence guarantees and convergence rates. In particular, we show linear convergence for unregularized and regularized NPG flows proposed by Kakade and Morimura and co-authors by observing that these arise from the Hessian geometries of conditional entropy and entropy respectively.

A modular form is a certain holomorphic function on the complex upper half plane which has an infinite group of discrete symmetries. I will discuss modular forms and some of their generalizations.  In particular, I will try to answer the following questions: What is a modular form? What are some examples of modular forms? What is a simple open question about modular forms?  The examples of modular forms I will give go under the name of "harmonic theta functions".  Time permitting, I will describe some exotic variants of these harmonic theta functions that are tied up with the octonions and the compact Lie group G_2.

In recent years we have experienced a remarkable growth on the number and size of available datasets. Such growth has led to the intense and challenging pursuit of estimators which are provably both computationally efficient and statistically accurate. Notably, the analysis of polynomial-time estimators has revealed intriguing phenomena in several high dimensional estimation tasks, such as their apparent failure of such estimators to reach the optimal statistical guarantees achieved among all estimators (that is the presence of a non-trivial “computational-statistical trade-off”).

In this talk, I will present new such algorithmic results for the well-studied planted clique model and for the fundamental sparse regression model. For planted clique, we reveal the surprising severe failure of the Metropolis process to work in polynomial-time, even when simple degree heuristics succeed. In particular, our result resolved a well-known 30-years old open problem on the performance of the Metropolis process for the model, posed by Jerrum in 1992. For sparse regression, we show the failure of large families of polynomial-time estimators, such as MCMC and low-degree polynomial methods, to improve upon the best-known polynomial-time regression methods. As an outcome, our work offers rigorous evidence that popular regression methods such as LASSO are optimally balancing their computational and statistical recourses.

The talk will give an overview of categorical semantics - a magical tool that puts our wanted loose ways of reasoning on solid ground. We’ll discuss how Grothendieck generic freeness lemma is “internally” a simple one-liner, in what secret sense Spec A is free, why intuitionistic mathematics is so natural, and maybe even how to approach algebraic geometry synthetically (i.e. with no set theory references).

 I will explain the relevance of the Lichnerowicz Laplacian in several situations and how one can easily understand the zeroth order term in the expression. This leads to simple proofs of some classical results and also to new results with more general curvature conditions.

Continuous-time Markov chains are frequently used as stochastic models for chemical reaction networks, especially in the growing field of systems biology. A fundamental problem for these Stochastic Chemical Reaction Networks (SCRNs) is to understand the dependence of the stochastic behavior of these systems on the chemical reaction rate parameters. Towards solving this problem, in this talk we describe theoretical tools called comparison theorems that provide stochastic ordering results for SCRNs. These theorems give sufficient conditions for monotonic dependence on parameters in these network models, which allow us to obtain, under suitable conditions, information about transient and steady state behavior. These theorems exploit structural properties of SCRNs, beyond those of general continuous-time Markov chains. Furthermore, we derive two theorems to compare stationary distributions and mean first passage times for SCRNs with different parameter values, or with the same parameters and different initial conditions. These tools are developed for SCRNs taking values in a generic (finite or countably infinite) state space and can also be applied for non-mass-action kinetics models. We illustrate our results with applications to models of chromatin regulation and enzymatic kinetics.

This talk is based on joint work with Simone Bruno, Felipe Campos, Domitilla Del Vecchio and Yi Fu.

Over a commutative ring of finite cardinality, how many $n
\times n$ matrices satisfy a polynomial equation? In this talk, I will explain how to solve this question when the ring is given by integers modulo a prime power and the polynomial is square-free modulo the prime.
Then I will discuss how this question is related to the distribution of the cokernel of a random matrix and the Cohen--Lenstra heuristics. This is joint work with Yunqi Liang and Michael Strand, as a result of a
summer undergraduate research I mentored.

[pre-talk at 1:20PM]

I’ll give a short intro to hyperbolic groups and small cancellation theory, and demonstrate how this theory can be used to construct groups with desirable properties. 

The Gaussian Elimination with Partial Pivoting (GEPP) is a classical algorithm for solving systems of linear equations. Empirical evidence strongly suggests that for a typical square coefficient matrix, GEPP is numerically stable. We obtain a (partial) theoretical justification of this phenomenon by showing that, given the random n×n standard Gaussian coefficient matrix A, the growth factor of the Gaussian Elimination with Partial Pivoting is at most polynomially large in n with probability close to one. This implies that with probability close to one the number of bits of precision sufficient to solve Ax=b to m bits of accuracy using GEPP is m+O(log(n)), which we conjecture to be optimal by the order of magnitude. We further provide tail estimates of the growth factor which can be used to support the empirical observation that GEPP is more stable than the Gaussian Elimination with no pivoting. Based on joint work with Han Huang.

How a Turing's reaction-diffusion process in a biological context leads to a rather surprising appearance of Ising-like colorings of the skin of Mediterranean lizards.

 The Balmer spectrum of a tensor triangulated category is a topological tool analogous to the usual spectrum of a commutative ring. It provides a universal theory of support, giving a categorical framework to (among others) the support varieties that have been used to great effect in modular representation theory. In this talk I will present an approach to the Balmer spectrum of the stable module category of a Hopf algebra using twisted tensor products and emphasizing examples. This will include an unpretentious introduction to twisted tensor products, the Balmer spectrum, and the relevance of both in representation theory.

Recall that two finitely generated groups G and H are quasi-isometric, if they admit a topological coupling, i.e. an action of G times H on a locally compact topological space such that each factor acts properly and cocompactly. This topological definition of quasi-isometry was given by Gromov, and at the same time he proposed a measure theoretic analogue of this definition, called the measure equivalence, which is closely related to the notion of orbit equivalence in ergodic theory. Despite the similarity in the definition of measure equivalence and quasi-isometry, their relationship is rather mysterious and not well-understood. We study the relation between these two notions in the class of right-angled Artin groups. In this talk, we show if H is a countable group with bounded torsion which is integrable measure equivalence to a right-angled Artin group G with finite outer automorphism group, then H is finitely generated, and H and G are quasi-isometric. This allows us to deduce integrable measure equivalence rigidity results from the relevant quasi-isometric rigidity results for a large class of right-angled Artin groups. Interestly, this class of groups are rigid for a reason which is quite different from other cases of measure equivalence rigidity. We will also do a quick survey of relevant measure equivalence rigidity and quasi-isometric rigidity results of other classes of groups, motivating our choice of right-angled Artin groups as a playground. This is joint work with Camille Horbez.

I will present a recent work on variational problems involving nonlocal energy functionals that appear in nonlocal mechanics. The well-posedness of variational problems is established via a careful study of the associated energy spaces, which are nonstandard. I will discuss some difficulties in proving fundamental structural properties of the function spaces such as compactness. For a sequence of parametrized nonlocal functionals in suitable form, we compute their variational limit and establish a rigorous connection with classical models.

The development of accurate and efficient numerical solutions to the wave equation is a fundamental area of scientific research with applications in several fields, including music, computer graphics, architecture, and telecommunications. A key challenge in wave simulation research concerns the proper handling of wave propagation on an unbounded domain. This challenge is known as the infinite domain problem. In this talk, I present a novel geometric framework for solving this problem based on the classical Kelvin transformation. I express the wave equation as a Laplace problem in Minkowski spacetime and show that the problem is conformally invariant under Kelvin transformations using the Minkowski metric while the boundedness of the spacetime is not. These two properties of the Kelvin transformation in Minkowski spacetime ensure that harmonic functions which span infinite spacetime can be simulated using finite computational resources with no loss of accuracy.

 In this talk, I will discuss connections between configuration spaces, an important class of topological space, and combinatorial algebras arising from the theory of reflection groups. In particular, I will present work relating the cohomology rings of some classical configuration spaces - such as the space of n ordered points in Euclidean space - with Solomon descent algebra and the peak algebra. The talk will be centered around two questions. First, how are these objects related? Second, how can studying one inform the other? This is joint, on-going work with Marcelo Aguiar and Vic Reiner.

Recent radical evolution in distributed sensing, computation, communication, and actuation has fostered the emergence of cyber-physical network systems. Regardless of the specific application, one central goal is to shape the network's collective behavior through the design of admissible local decision-making algorithms. This is nontrivial due to various challenges such as local connectivity, system complexity and uncertainty, limited information structure, and the complex intertwined physics and human interactions.

In this talk, I will present our recent progress in formally advancing the systematic design of distributed coordination in network systems via harnessing special properties of the underlying problems and systems. In particular, we will present three examples and discuss three types of properties, i) how to use network structure to ensure the performance of the local controllers; ii) how to use the information and communication structure to develop distributed learning rules; iii) how to use domain-specific properties to further improve the efficiency of the distributed control and learning algorithms.

Bio: Na Li is a Gordon McKay professor in Electrical Engineering and Applied Mathematics at Harvard University.  She received her Bachelor degree in Mathematics from Zhejiang University in 2007 and Ph.D. degree in Control and Dynamical systems from California Institute of Technology in 2013. She was a postdoctoral associate at Massachusetts Institute of Technology 2013-2014.  She has hold a variety of shorter term visiting appointments including  the Simons Institute for the Theory of Computing, MIT, and Google Brain. Her research lies in control, learning, and optimization of networked systems, including theory development, algorithm design, and applications to real-world cyber-physical societal system.  She has been an associate editor for IEEE Transactions on Automatic Control, Systems & Control letters, IEEE Control Systems Letters, and served on the organizing committee for numerous conferences.  She received NSF career award (2016), AFSOR Young Investigator Award (2017), ONR Young Investigator Award(2019),  Donald P. Eckman Award (2019), McDonald Mentoring Award (2020), the IFAC Manfred Thoma Medal (2023), along with some other awards.

In this talk, I will introduce a few games for you and your partner (if any exists) to play. They can help you get to know each other and descriptive set theory better. It’s all fun and games.

This talk will provide a basic introduction to the world of nonlinear PDEs.

 I will discuss compact uniformly recurrent subgroups and compact invariant random subgroups in locally compact groups, and present results from ongoing projects with Pierre-Emanuel Caprace and Waltraud Lederle, and with Tal Cohen.

We prove the existence of nontrivial closed surfaces with constant anisotropic mean curvature with respect to elliptic integrands in closed smooth 3-dimensional Riemannian manifolds. The constructed min-max surfaces are smooth with at most one singular point. The constant anisotropic mean curvature can be fixed to be any real number. In particular, we partially solve a conjecture of Allard [Invent. Math.,1983] in dimension 3.

 In this talk we will discuss several stochastic epidemic models recently developed to account for general infectious durations, infection-age dependent infectivity and/or progress loss of immunity/varying susceptibility, extending the standard epidemic models. We construct individual based stochastic models, and prove scaling limits for the associated epidemic dynamics in large populations. Each individual is associated with a random function/process that represents the infection-age dependent infectivity force to exert on other individuals. We extend this formulation to associate each individual with a random function that represents the loss of immunity/varying susceptibility. A typical infectivity function first increases and then decreases from the epoch of becoming infected to the time of recovery, while a typical immunity/susceptibility function gradually increases from the time of recovery to the time of losing immunity and becoming susceptible. The scaling limits are deterministic and stochastic Volterra integral equations. We also discuss some new PDEs models arising from the scaling limits. (This talk is based on joint work with Etienne Pardoux, Raphael Forien, and Arsene Brice Zosta Ngoufack.)

It is a folklore conjecture that the sup norm of a Laplace eigenfunction on a compact hyperbolic surface grows more slowly than any positive power of the eigenvalue.  In dimensions three and higher, this was shown to be false by Iwaniec-Sarnak and Donnelly.  I will present joint work with Farrell Brumley that strengthens these results, and extends them to locally symmetric spaces associated to $\mathrm{SO}(p,q)$.

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 I will explain how various results in arithmetic statistics by Bhargava, Gross, Shankar, and others on 2-Selmer groups of Jacobians of (hyper)elliptic curves can be organized and reproved using the theory of graded Lie algebras, following the earlier work of Thorne. This gives uniform proof of these results and yields new theorems for certain families of non-hyperelliptic curves. I will also mention some applications to rational points on certain families of curves.

The talk will involve a mixture of representation theory, number theory
and algebraic geometry and I will assume no familiarity with arithmetic
statistics.

 In this talk, I will discuss injectivity and injective envelope of objects in different categories. I will present our recent work, which attempts to answer the question “What happens to injective objects under particular functors?” This is based on joint work with David Blecher and Mehrdad Kalantar. 

 We design a monotone meshfree finite difference method for linear elliptic PDEs in non-divergence form on point clouds via a nonlocal relaxation method. The key idea is a combination of a nonlocal integral relaxation of the PDE problem with a robust meshfree discretization on point clouds. Minimal positive stencils are obtained through a linear optimization procedure that automatically guarantees the stability and, therefore, the convergence of the meshfree discretization.  A major theoretical contribution is the existence of consistent and positive stencils for a given point cloud geometry. We provide sufficient conditions for the existence of positive stencils by finding neighbors within an ellipse (2d) or ellipsoid (3d) surrounding each interior point, generalizing the study for Poisson’s equation by Seibold in 2008. It is well-known that wide stencils are in general needed for constructing consistent and monotone finite difference schemes for linear elliptic equat ions. Our result represents a significant improvement in the stencil width estimate for positive-type finite difference methods for linear elliptic equations in the near-degenerate regime (when the ellipticity constant becomes small), compared to previously known works in this area. Numerical algorithms and practical guidance are provided with an eye on the case of small ellipticity constant. Numerical results will be presented in both 2d and 3d, examining a range of ellipticity constants including the near-degenerate regime.

We generalize a result of Mazorchuk and Tenner, showing that the “run” statistic influences the shape of the RSK tableaux of an arbitrary permutation. We define and construct the “canonical reduced word” of a boolean permutation, and show that the RSK tableaux for that permutation can be read off directly from this reduced word. We also describe those tableaux that can correspond to boolean permutations in terms of “uncrowded sets.” We then extend this work to fully commutative permutations, showing that each fully commutative permutation has a well-defined “boolean core,” related to the right weak order. The contents of the second row of the insertion tableaux of fully commutative permutations are partially ordered as subsets, with respect to the right weak order. We explore the partial order of these subsets, with particular interest in when they change from uncrowded to crowded. This is joint work with Gunawan, Russell and Tenner, based on recent work in arXiv:2207.05119 and arXiv:2212.05002.

The purpose of this talk is to give a linear algebra algorithm to find out if a rank of a given tensor over a field $F$ is at most $k$ over the algebraic closure of $F$, where $k$ is a given positive integer. We estimate the arithmetic complexity of our algorithm. 

 Horospherical group actions on homogeneous spaces are famously known to be extremely rigid. In finite volume homogeneous spaces, it is a special case of Ratner's theorems that all horospherical orbit closures are homogeneous. Rigidity further extends in rank-one to infinite volume but geometrically finite spaces. The geometrically infinite setting is far less understood. We consider $\mathbb{Z}$-covers of compact hyperbolic surfaces and show that they support quite exotic horocycle orbit closures. Surprisingly, the topology of such orbit closures delicately depends on the choice of a hyperbolic metric on the covered compact surface. In particular, our constructions provide the first examples of geometrically infinite spaces where a complete description of non-trivial horocycle orbit closures is known. Based on joint work with James Farre and Yair Minsky.

We first describe the mathematical foundation of DSGRN (Dynamic Signatures Generated by Regulatory Networks), an approach that provides a queryable description of global dynamics of a network over its entire parameter space. We also describe a connection to Boolean network models that allows us to view DSGRN as a platform for bifurcation theory of Boolean maps. Finally, we compare DSGRN to RACIPE, an approach based on sampling parameters for ODE models.  We discuss several applications to systems biology as well as design of robust networks in synthetic biology.

Analogies between number theory and graph theory have been studied for quite some time now.  During the past few years, it has been observed in particular that there is an analogy between classical Iwasawa theory and some phenomena in graph theory.  In this talk, we will explain this analogy and present some of the results that have been obtained so far in this area.

[pre-talk at 1:20PM]

Problems in extremal graph theory typically aim to maximize some graph parameter under local restrictions. In order to prove lower bounds for these kinds of problems, several techniques have been developed. The most popular one, initiated by Paul Erdős, being the probabilistic method. While this technique has enjoyed tremendous success, it does not always provide sharp lower bounds. Frequently, algebraically and geometrically defined graphs outperform random graphs. We will show how historically, geometry over finite fields has been a rich source of such graphs. I will show a broad class of graphs defined from geometry of finite fields, which has found several recent applications in extremal graph theory. Often, certain interesting families of graph had in fact already been discovered and studied, years before their value in extremal graph theory was realized. I will demonstrate some instances of this phenomenon as well, which indicates that there might still be uncharted territory to explore.

In this talk I will discuss two different topics: a) the topic of precise asymptotics for linear waves on black hole spacetimes, and b) the topic of construction of spacetimes containing curvature singularities. If time permits, I will try to make connections with more general problems for quasilinear wave equations (for both topics).

 Inductive limits are a central construction in C*-theory because they allow one to use well-understood building blocks to naturally construct intricate C*-algebras whose properties remain tractable. One task for the structural theory of operator algebras is to determine which classes of operator algebras arise as inductive limits of nice operator algebras.

The model result in this direction is Connes' 1970's theorem showing that many classes of von Neumann algebras, including semi-discrete von Neumann algebras, arise as inductive limits of finite dimensional von Neumann algebras. For C*-algebras, an analogous result fails outright: many nuclear C*-algebras are not inductive limits of finite dimensional C*-algebras. However, by generalizing our notion of an inductive system, we can in fact describe any nuclear C*-algebra as the limit of a system of finite dimensional C*-algebras. Though seemingly abstract, these generalized inductive systems arise naturally from completely positive approximations of nuclear C*-algebras.

This is based in part on joint work with Wilhelm Winter. 

The vorticity-streamfunction formulation for incompressible inviscid fluids is the basis for many fluid simulation methods in computer graphics, including vortex methods, streamfunction solvers, spectral methods, and Monte Carlo methods. We point out that current setups in the vorticity-streamfunction formulation are insufficient at simulating fluids on general non-simply-connected domains. This issue is critical in practice, as obstacles, periodic boundaries, and nonzero genus can all make the fluid domain multiply connected. These scenarios introduce non-trivial cohomology components to the flow in the form of harmonic fields. The dynamics of these harmonic fields have been previously overlooked. In this talk, we derive the missing equations of motion for the fluid cohomology components. We elucidate the physical laws associated with the new equations, and show their importance in reproducing physically correct behaviors of fluid flows on domains with general topology.

We discuss Milnor's link invariants through a geometric lens using intersections of Seifert surfaces. Our work is thus of a similar flavor as that of Cochran from 1990, who based his work on particular choices of Seifert surfaces. But like Mellor and Melvin in 2003, who considered only the first invariant (after linking number), we allow for more arbitrary choices. We conjecture that Milnor’s invariants can be recovered geometrically using the work of Monroe and Sinha on linking of letters and Sinha and Walters on Hopf invariants. We expect our approach to recover Cochran’s work and to extend work of Polyak, Kravchenko, Goussarov, and Viro on Gauss diagrams.

 Information theory provides a mathematical framework for quantifying information processing tasks, such as storage, computation, and communication. Connecting the abstract theory to concrete physical systems often gives insight into a system's physics; conversely, physics can often inspire new ideas in information theory itself. This perspective has been particularly fruitful in quantum gravity, for which the essential question is to understand how information is stored and processed by gravitating systems, such as black holes or even the Universe itself. In this talk we will see how quantum gravitational considerations lead to an extended Nyquist-Shannon sampling theorem for fields on Lorentzian manifolds. Applying the results to the physics of the early Universe leads to predictions for cosmological signatures of quantum gravity that can be tested with present-day observations of the cosmos.

In this work, we present a difference of convex algorithm for solving bilevel programs in which the upper level objective functions are difference of convex functions, and the lower level programs are fully convex. This nontrivial class of bilevel programs provides a powerful modelling framework for dealing with applications arising from hyperparameter selection in machine learning. Thanks to the full convexity of the lower level program, the value function of the lower level program turns out to be convex and hence the bilevel program can be reformulated as a difference of convex bilevel program. We propose an algorithm for solving the reformulated difference of convex program and show its convergence to stationary points under very mild assumptions.

In this talk, I will explore the rich interplay between differential equations and machine learning. I will highlight the use of collective dynamics and partial differential equations as powerful tools for improving machine learning algorithms and models. (i) In the first half of the talk, I will introduce a novel dynamical system that draws inspiration from collective intelligence observed in biology. This system offers a compelling alternative to gradient-based optimization. It enables gradient-free optimization to efficiently find global minimum in non-convex optimization problems. (ii) In the second half of the talk, I will build the connection between Hamilton-Jacobi-Bellman equations and the multi-armed bandit (MAB) problems. MAB is a widely used paradigm for studying the exploration-exploitation trade-off in sequential decision-making under uncertainty. This is the first work that establishes this connection in a general setting. I will present an efficient algorithm for solving MAB problems based on this connection and demonstrate its practical applications.

I'm gonna introduce p-adic Analysis and I'm gonna talk about the Skolem-Mahler-Lech theorem and I'm gonna blow your mind.

We discuss about smooth actions on manifold by higher rank lattices. We mainly focus on lattices in $\mathrm{SL}_n(\mathbb{R})$ ($n$ is at least $3$). Recently, Brown-Fisher-Hurtado and Brown-Rodriguez Hertz-Wang showed that if the manifold has dimension at most $(n-1)$, the action is either isometric or projective. Both cases, we don't have chaotic dynamics from the action (zero entropy). We focus on the case when one element of the action acts with positive topological entropy. These dynamical properties (positive entropy element) significantly constrains the action. Especially, we deduce that if there is a smooth action with positive entropy element on a closed $n$-manifold by a lattice in $\mathrm{SL}_n(\mathbb{R})$ ($n$ is at least $3$) then the lattice should be commensurable with $\mathrm{SL}_n(\mathbb{Z})$. This is the work in progress with Aaron Brown.

 The successive minima of an order in a degree n number field are n real numbers encoding information about the Euclidean structure of the order. How many orders in degree n number fields are there with almost prescribed successive minima, fixed Galois group, and bounded discriminant? In this talk, I will address this question for n = 3, 4, 5. The answers, appropriately interpreted, turn out to be piecewise linear functions on certain convex bodies. If time permits, I will also discuss function field analogues of this problem.

[pre-talk at 1:20PM]

Expander graphs are perhaps one of the most widely useful classes of graphs ever considered. In this talk, we will focus on a fairly weak notion of expanders called sublinear expanders, first introduced by Komlós and Szemerédi around 25 years ago. They have found many remarkable applications ever since. In particular, we will focus on certain robustness conditions one may impose on sublinear expanders and some applications of this very recent idea, which include: 

- recent progress on one of the most classical decomposition conjectures in combinatorics, the Erdős-Gallai Conjecture,

- Rainbow Turan problem for cycles, raised by Keevash, Mubayi, Sudakov and Verstraete, including an application of this result to additive number theory and 

- essentially tight answers to the classical Erdős unit distance and distinct distances problems in "almost all" real normed spaces of any fixed dimension.

Moduli of cubic surfaces can be compactified from the point of view of geometric invariant theory (GIT), and from the point of view of the ball quotient. The Kirwan desingularization resolves the GIT singularities to yield a smooth Kirwan compactification, while the toroidal compactification of the ball quotient is also smooth. We show that these two smooth compactifications are, however, not isomorphic. Based on joint work with S. Casalaina-Martin, K. Hulek, R. Laza

Inspired by the rise on the interest for nonlocal models, mainly Peridynamics, we decided to study a functional framework suitable for (variational) nonlocal models, such as that of nonlocal hyperelasticity. This has lead to a nonlocal framework based on truncated fractional gradients (i.e. nonlocal gradients with a fractional singularity defined over bounded domains), in which continuous and compact embeddings and, in particular, nonlocal Poincaré inqualities has been obtained thanks to a nonlocal version of the fundamental theorem of calculus. As a consequence, the existence of minimizers of nonlocal polyconvex vectorial functionals is obtained, and more recently quasiconvex functionals. Some of these last results have been obtained from a result that relates nonlocal gradients with classical ones and vice-versa. These results are accompanied by a study of the localization (recovering of the classical model) when s goes to 1 (actually, continuity on s with s being the fractional index of differentiability).

We will discuss quantum no-signaling correlations introduced by Duan and Winter with focus on its different subclasses (quantum commuting, quantum and local). They will appear as strategies of quantum-to-quantum non-local games. We will then discuss concurrent quantum non-local games, as quantum versions of synchronous/bisynchronous non-local games focusing on quantum graph homomorphism/isomorphism games, and provide tracial characterisations of their perfect strategies belonging to various correlation classes; e.g. the perfect strategies for quantum graph isomorphism game are given by tracial states on the universal C∗-algebra of the projective free unitary quantum group. This is a joint work with Michael Brannan, Sam Harris and Ivan Todorov.

The animation of delicate vortical structures of gas and liquids has been of great interest in computer graphics. However, common velocity-based fluid solvers can damp the vortical flow, while vorticity-based fluid solvers suffer from performance drawbacks. We propose a new velocity-based fluid solver derived from a reformulated Euler equation using covectors. Our method generates rich vortex dynamics by an advection process that respects the Kelvin circulation theorem. The numerical algorithm requires only a small local adjustment to existing advection-projection methods and can easily leverage recent advances therein. The resulting solver emulates a vortex method without the expensive conversion between vortical variables and velocities. We demonstrate that our method preserves vorticity in both vortex filament dynamics and turbulent flows significantly better than previous methods, while also improving preservation of energy.

Among all nD Euclidean shapes carrying a degree of shade function, only those black and white ones, defined by a single polynomial inequality are determined by finitely many moments. In the single variable situation, a landmark moment transform discovered by A. Markov untangles the above uniqueness feature. The 2D case is not less interesting, touching some spectral inverse problem for pairs of non-commuting self-adjoint transforms. A super-resolution phenomenon will be described in some detail with optimization and asymptotic expansion techniques.

I will begin by defining the notion of the differential power of an ideal. Focusing on polynomial rings in two or three variables over a field of characteristic zero, I will discuss when the differential power of an ideal becomes principal. I will use differential powers to define the differential closure of an ideal and discuss how it relates to the radical of an ideal.

 The possible asymptotic distributions of a random dynamical system are described by stationary measures, and in this talk, we will be interested in the properties of these measures — in particular, whether they are absolutely continuous. First, I will quickly describe the case of Bernoulli convolutions, which can be seen as generalizations of the Cantor middle third set, and then the case of random iterations of matrices in $\mathrm{SL}(2, \mathbb{R})$ acting on the real projective line, where the stationary measure is unique under certain conditions and is called the Furstenberg measure. It had been conjectured that the Furstenberg measure is always singular when the random walk has finite support. There have been several counter-examples, and the aim of the talk will be to describe that of Bourgain, where the measure even has a very regular density. I will explain why the construction works for any simple Lie group, using the work of Boutonnet, Ioana, and Salehi Golsefidy on local spectral gaps in simple Lie groups.

 Translators are known as candidates of Type II blow-up model for mean curvature flows. Various examples of mean curvature flow translators have been constructed in the convex case and semi-graphical case, most of which have either infinite entropy or higher multiplicity asymptotics near infinity. In this talk, we shall present the construction of a new family of translators with prescribed end. This is based on the joint work with Ao Sun.

The Central Limit Theorem (CLT) plays an indispensable role in classical statistical inference for finite-dimensional parameters, including calibration of confidence sets and statistical tests. However, the validity of the CLT in high dimensional problems where the dimension (p) of the observations diverges to infinity with the sample size (n) is no longer guaranteed. There is extensive recent work on the problem, following the seminal paper by Chernozhukov, Chetverikov, and Kato (2013; Annals of Statistics), that attempts to establish the CLT (or Gaussian Approximation) in high dimensions under various growth conditions on the dimension p. In this talk, we present some new results on Gaussian Approximation for different classes of sets, providing insights into specific distributional characteristics of the underlying high dimensional random vectors that determine the optimal growth rates.

I will talk about my recent joint work with Aubert where we prove the Local Langlands Conjecture for G_2 (explicitly). This  uses our earlier results on Hecke algebras attached to Bernstein components of reductive p-adic groups, as well as an expected property on cuspidal support, along with a list of characterizing properties. In particular, we obtain "mixed" L-packets containing F-singular supercuspidals and non-supercuspidals.

[pre-talk at 1:20PM]

The growth of an E. coli colony on hard agar exhibits robust expansion kinetics and morphology despite the complex interactions between millions of cells experiencing compact confinement under a spatiotemporally varying nutrient gradient. To probe the mechanistic origins of such robust colony characteristics, an agent-based model along with a set of reaction-diffusion equations to model the spatiotemporal dynamics of extracellular metabolites is employed. For colonies grown on glucose minimal medium plates, we find glucose depletion driven by anaerobiosis to be the primary factor contributing to the experimentally observed saturation in vertical expansion, while the excreted fermentation products play a crucial role for cell maintenance at the aerated colony surface. We also establish that the well-known linear expansion of the colony radius is not limited by nutrients. Overall, our study emphasizes that in addition to mechanical interactions, the spatiotemporally varying metabolite gradients and the mode of metabolism of the individual cells determine the emergent morphology and expansion kinetics of the macroscopic colony.

I will talk about Manolescu's work on the triangulation conjecture. Using equivariant homology theory, it is proved that there exist non-triangulable manifolds in high dimensions. I will introduce the equivariant stable homotopy theory and some of its applications.

The slopes of modular forms are the $p$-adic valuations of the eigenvalues of the Hecke operators $T_p$. The study of slopes plays an important role in understanding the geometry of the eigencurves, introduced by Coleman and Mazur.

The study of the slope began in the 1990s when Gouvea and Mazur computed many numerical data and made several interesting conjectures. Later, Buzzard, Calegari, and other people made more precise conjectures by studying the space of overconvergent modular forms. Recently, Bergdall and Pollack introduced the ghost conjecture that unifies the previous conjectures in most cases. The ghost conjecture states that the slope can be predicted by an explicitly defined power series. We prove the ghost conjecture under a certain mild technical condition. In the pre-talk, I will explain an example in the quaternionic setting which was used as a testing ground for the proof.

This is joint work with Ruochuan Liu, Liang Xiao, and Bin Zhao.

The q-Canonical Commutation Relation (q-CCR) is an interpolation between the CCR and the CAR with a parameter q. In the 1990s, M. Bożejko and R. Speicher found that the q-CCR is represented on the q-Fock space. The q-Gaussians are realized as the field operators with the vacuum state, which forms a non-commutative distribution. 

On the other hand, a conjugate system is a notion of free probability introduced by D. Voiculescu. This carries important information about a non-commutative distribution of given operators and has many implications for the generated von Neumann algebra.  

In this talk, I will present a concrete formula for the conjugate system for the q-Gaussians. This talk is based on the joint work with R. Speicher.

We will discuss various constructions of modular forms in the context of classical and exceptional theta correspondences. 

Geometric numerical integrators are numerical methods used to solve ordinary and partial differential equations that preserve geometric properties of the underlying dynamical systems. These methods are designed to accurately approximate the trajectories of the systems while conserving important physical or mathematical properties such as energy, momentum, symplecticity, or volume. In this talk, I will be talking about two classes of geometric numerical integrators both of which suffer from parasitic instabilities namely G-symplectic general linear methods for Hamiltonian systems and Variational Integrators for degenerate Lagrangian systems. I will also discuss the strategies to control the effect of parasitism in these methods.

The $C_2$-equivariant stable stems were first studied by Bredon and Landweber, in the 1960s. From the start, it was clear that these groups exhibited certain periodic behavior closely related to James periodicity for stunted projective spaces. This was made more explicit and extensively applied to computations by Araki and Iriye in the late 1970s / early 1980s. In the past decade, this phenomena has been lifted to $\mathbb{R}$-motivic homotopy theory under the guise of "$\tau$-periodicity", and plays a central role in Behrens and Shah's work relating $\mathbb{R}$-motivic and $C_2$-equivariant homotopy theory.

In this talk, I will review some of the above story, and then explain how similar periodic phenomena occurs in $G$-equivariant stable homotopy theory for an arbitrary finite group $G$.

How many essentially distinct ways are there to arrange $n$ convex sets in $\mathbb{R}^d$? Here, `essentially distinct' means with different intersection pattern'. We discuss this question both in the dimension $1$, where it amounts to counting the interval graphs, and in higher dimenions. Based on the joint works with Amzi Jeffs.  Plain text abstract: How many essentially distinct ways are there to arrange n convex sets in R^d? Here, `essentially distinct' means `with different intersection pattern'. We discuss this question both in the dimension 1, where it amounts to counting the interval graphs, and in higher dimenions. Based on the joint works with Amzi Jeffs.

The classical Kakeya needle problem asks: what is the smallest set in the plane inside which a unit-length needle can be translated and rotated through a full 360-degree turn? In this talk, we will show how to construct such sets of arbitrarily small measures (known as Kakeya sets). We will then describe some applications to several important problems in analysis where this seemingly innocuous rotation property can lead to very counterintuitive and pathological results.

The Quantum Unique Ergodicity conjecture of Rudnick and Sarnak says that eigenfunctions of the Laplacian on a compact manifold of negative curvature become equidistributed as the eigenvalue tends to infinity. In the talk, I will discuss recent work on this problem for arithmetic quotients of the three-dimensional hyperbolic space. I will discuss our key result that Hecke eigenfunctions cannot concentrate on certain proper submanifolds. Joint work with Lior Silberman.

We consider {0,1}-matrices. For matrices A and B, we say A contains B as a configuration if there is a submatrix of A that is a column and row permutation of B. For instance, if A and B are incidence matrices of graphs G and H respectively, then A contains B as a configuration if and only if G contains H as a subgraph. In this talk, we study some extremal problems for matrices and hypergraphs (set systems).

The birational classification of foliated surface is pretty much complete, thanks to the work of Brunella, Mendes, McQuillan. The next obvious step in this endeavour, in analogy with the classical case of projective varieties and log pairs, is to construct moduli spaces for foliated varieties (starting from the general type case). The first question to ask, on the road towards constructing such a moduli space, is how to show that foliated varieties of fixed Kodaira dimension are bounded, that is, they come in finitely many algebraic families ― provided, of course, that we fix certain appropriate numerical invariants. It turns out that, to best answer this question, rather than working with the canonical divisor of a foliation it is better to consider linear systems of the form $|nK_X + mK_F|, n,m >0, as those encode a lot of the positivity features that classically the canonical divisor of a projective variety displays.
In this talk, I will introduce this framework and explain how this approach leads to answering the question about boundedness for foliated surfaces. Time permitting, I will address also what happens when we try to construct a moduli functor, or rather, what we have been finding out, so far. This talk features joint work with C. Spicer, work with J. Pereira, and work in progress with M. McQuillan and C. Spicer.

 Modified Newton methods are designed to extend the desirable properties of classical Newton method to a wider class of optimization problems. If the Hessian of the objective function is singular at the solution, these methods tend to behave like gradient descent and rapid local convergence is lost. An adaptive regularization technique is described that yields a modified Newton method that retains superlinear local convergence on non-convex problems without the nonsingularity assumption at the solution. The minimum norm perturbation and symmetric indefinite factorization used to construct a sufficiently positive definite approximate Hessian are discussed, and numerical results comparing regularized and standard modified Newton methods will be presented. Lastly, a well-behaved pathological example will be used to illustrate an assumption required for superlinear convergence.

I will present a recent joint work with Andrew Lawrie from MIT. We consider the harmonic map heat flow for maps from the plane $R^2$ to the sphere $S^2$, under the so-called equivariant symmetry. It is known that solutions to the initial value problem exhibit bubbling along a sequence of times - the solution decouples into a superposition of harmonic maps concentrating at different scales and a body map that accounts for the rest of the energy. We prove that this bubble decomposition occurs continuously in time. The main new ingredient in the proof is the notion of a collision interval motivated by our recent work on the soliton resolution problem for equivariant wave maps.

Numerical semigroups are combinatorial objects that lead to deep and subtle questions.  With tools from complex, harmonic, and functional analysis, probability theory, algebraic combinatorics, and computer-aided design, we answer virtually all asymptotic questions about factorization lengths in numerical semigroups.  Our results yield uncannily accurate predictions, along with unexpected results about symmetric functions, trace polynomials, and the statistical properties of certain AF C$^*$-algebras. 

Partially supported by NSF Grants DMS-1800123 and DMS-2054002.  Joint work (in various combinations) with K.~Aguilar, A.~B\"ottcher, \'A. Ch\'avez, L.~Fukshansky, M.~Omar, C.~O'Neill, J.~Vol\v{c}i\v{c} and undergraduate students J.~Hurley, G.~Udell, T.~Wesley, S.~Yih.

Schubert polynomials originated in the study of the cohomology ring for the complete flag manifold by Bernstein, Gelfand, and Gelfand and Demazure, with their combinatorics, developed extensively by Lascoux and Schutzenberger. For each permutation, there is a Schubert polynomial which, when evaluated at certain Chern classes, gives the cohomology class of a Schubert subvariety of the flag manifold. Thus the Schubert structure constants enumerate flags in a suitable triple intersection of Schubert varieties. As such, they are known from geometry to be nonnegative. A fundamental open problem in algebraic combinatorics is to give a positive combinatorial formula for these structure constants. 

Recently, I conjectured a formula for Schubert structure constants in the classical flag manifold that occurs in the product of an arbitrary Schubert class by one pulled back from a Grassmannian projection. In this talk, I’ll present joint work with Nantel Bergeron in which we define an insertion algorithm on Kohnert diagrams, proving this conjecture. 

This talk should be generally accessible, with no prior knowledge of Schubert varieties or Schubert polynomials required.

The Segal conjecture suggests that the 2-completed $C_2$-equivariant sphere is equivalent to its homotopy completion. However, the genuine $C_2$-equivariant Adams spectral sequence for the $C_2$-equivariant sphere is not isomorphic to the Borel one. In this talk, I will show that the Borel $C_2$-equivariant Adams spectral sequence can be obtained from the genuine one through a degree shifting of the negative cone with connecting differentials shortened. I will also show that the Borel $C_2$-equivariant Adams spectral sequence is related to some classical Adams spectral sequences, whose $E_2$-terms are computable through the Curtis algorithm.

In 1903, Max Dehn settled the following question: which rectangles can you tile with finitely many square tiles (possibly of different sizes)? In this talk, we'll see two (relatively) modern proofs of his result. The first involves redefining the area of a rectangle in a way that would make a measure theorist's skin crawl, and the second involves something even more sacrilegious: physics.

Equidistribution properties of translates of orbits for subgroup actions on homogeneous spaces are intimately linked to the mixing features of the global action of the ambient group. The connection appears already in Margulis' thesis (1969), displaying its full potential in the work of Eskin and McMullen during the nineties. On a quantitative level, the philosophy underlying this linkage allows transferring mixing rates to effective estimates for the rate of equidistribution, albeit at the cost of a sizeable loss in the exponent. In joint work with Ravotti, we instead resort to a spectral method, pioneered by Ratner in her study of quantitative mixing of geodesic and horocycle flows, in order to obtain the precise asymptotic behavior of averages of regular observables along expanding circles on compact hyperbolic surfaces. The primary goal of the talk is to outline the salient traits of this method, illustrating how it leads to the relevant asymptotic expansion. In addition, we shall also present applications of the main result to distributional limit theorems and to quantitative error estimates on the corresponding hyperbolic lattice point counting problem; predictably, the latter fails to improve upon the currently best-known bound, achieved via finer methods by Selberg more than half a century ago.

Consider a smooth manifold with a Riemannian metric satisfying some sort of curvature or other geometric constraints like, for example, positive scalar curvature, non-negative Ricci or negative sectional curvature, being Einstein, Kähler, Sasaki, etc. A natural question to ponder is then what the space of all such metrics does look like. Moreover, one can also pose this question for corresponding moduli spaces of metrics, i.e., quotients of the former by (suitable subgroups of) the diffeomorphism group of the manifold, acting by pulling back metrics. The study of spaces of metrics and their moduli has been a topic of interest for differential geometers, global and geometric analysts, and topologists alike, and I will introduce to and survey in detail the main results and open questions in the field with a focus on non-negative Ricci or sectional curvature as well as other lower curvature bounds on closed and open manifolds, and, in particular, also discuss broader non-traditional approaches from metric geometry and analysis to these objects and topics

$p$-adic representations are important objects in number theory, and stable lattices serve as a connection between the study of ordinary and modular representations. These stable lattices can be understood as stable vertices in Bruhat-Tits buildings. From this viewpoint, the study of fixed point sets in these buildings can aid research on $p$-adic representations. The simplicial distance holds an important role as it connects the combinatorics of lattices and the geometry of root systems. In particular, the fixed-point sets of Moy-Prasad subgroups are precisely the simplicial balls. In this talk, I'll explain those findings and compute their simplicial volume under certain conditions.

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We construct a family of operator systems and k-AOU spaces generated by a finite number of projections satisfying a set of linear relations. This family is universal in the sense that the map sending the generating projections to any other set of projections that satisfy the same relations is completely positive. These operator systems are constructed as inductive limits of explicitly defined operator systems. By choosing the linear relations to be the non-signaling relations from quantum correlation theory, we obtain a hierarchy of ordered vector spaces dual to the hierarchy of quantum correlation sets. By considering another set of relations, we also find a new necessary condition for the existence of a SIC-POVM.

I will survey recent results regarding the dynamics of positive definite functions and character rigidity of irreducible lattices in higher-rank semisimple algebraic groups. These results have several applications to ergodic theory, topological dynamics, unitary representation theory, and operator algebras. In the case of lattices in higher-rank simple algebraic groups, I will explain the key operator algebraic novelty, which is a noncommutative Nevo-Zimmer theorem for actions on von Neumann algebras. I will also present a noncommutative Margulis' factor theorem and discuss its relevance regarding Connes' rigidity conjecture for group von Neumann algebras of higher-rank lattices.

Semidefinite programming (SDP) forms a class of convex optimization problems with remarkable modeling power. Apart from its classical applications in combinatorics and control, it also enjoys a range of applications in data science. This talk first discusses various concrete SDPs in data science and their conditioning. In particular, we show that even though Slater’s constraint qualification condition may fail, these SDPs satisfy an important regularity, strict complementarity, which ensures the good conditioning of the problem. In the second part of the talk, based on the regularity and computational structure shared by these problems, we design time- and space-efficient algorithms to solve these SDPs.

We describe a principled approach for constructing non-vacuum initial data sets for the Cauchy problem in general relativity. The core idea has an interesting history of having been known in the '70s,
forgotten by the mathematical relativity community for decades, and now
independently rediscovered and rigorously demonstrated. We show how
it explains why certain techniques for generating initial data
worked well in the past, but also how it leads to new equations
with applealing physical properties when generating initial data containing
fluids. The talk will be targeted at a broad audience.

I’ll discuss recent progress on the existence theory for harmonic maps, in particular the existence of harmonic maps of optimal regularity from manifolds of dimension n>2 to every non- aspherical closed manifold containing no stable minimal two-spheres. As an application, we’ll see that every manifold carries a canonical family of sphere-valued harmonic maps, which (in dimension<6) stabilize at a solution of a spectral isoperimetric problem generalizing the conformal maximization of Laplace eigenvalues on surfaces. Based on joint work with Mikhail Karpukhin.

 I will discuss recent advances in our understanding of the stable homotopy groups of spheres using formal groups, modular forms, and  (time permitting) motives.

Syzygies of algebraic varieties have long been a topic of intense interest among algebraists and geometers alike. Starting with the pioneering work of Mark Green on curves, numerous attempts have been made to extend these results to higher dimensions. Ein and Lazarsfeld proved that if A is a very ample line bundle, then K_X + mA satisfies property N_p for any m>=n+1+p. It has ever since been an open question if the same holds true for A ample and basepoint free. In joint work with Purnaprajna Bangere we give a positive answer to this question.

In this talk we'll investigate the primeness of generalized wreath product II1 factors using techniques from deformation/rigidity theory. We give general conditions relating tensor decompositions of generalized wreath products to stabilizers of the associated group action, and use this to find new examples of prime II1 factors.

In this talk, I will give a survey of recent and upcoming results on various linear, semilinear, and quasilinear wave equations on a wide class of dynamical spacetimes in various even and odd spatial dimensions. These results include asymptotics for a wide range of nonlinearities. For many of these results, the spacetimes under consideration have only weak asymptotic flatness conditions and are allowed to be large perturbations of the Minkowski spacetime, provided that an integrated local energy decay estimate holds. We explain the dichotomy between even- and odd-dimensional wave behaviour. Part of this work is joint with Mihai Tohaneanu and Jared Wunsch.

Configuration spaces are not only fundamental objects in mathematics, but appear in numerous areas such as robot motion planning, DNA sequencing, sensor networks, and origami designs. Our story is motivated by objects in algebraic geometry, namely the configuration space of surface deformations. To understand this space, we consider a combinatorial viewpoint based on scribbling loops on the surface. This leads to a classification of such spaces that can be realized as convex polytopes, capturing elegant hidden algebraic
structures.  These spaces now appear across a broad spectrum of research, including amplituhedra, geometric group theory, and phylogenetics networks. The entire talk is heavily infused with visual imagery and concrete examples.

Topological Hochschild homology(THH), which is a lift of Hochschild homology to the stable category, has recently seen a large paradigm shift in the way it and related invariants are studied.
This is due to several groundbreaking results, among which are the results of Bhatt, Morrow, and Scholze which introduce the quasisyntomic topology and compute the quasisyntomic filtration of topological Hochschild homology, topological negative cyclic homology ($TC^{-}$), topological periodic homology(TP), and topological cyclic homology(TC). Building on these results, I will talk about the quasisyntomic filtration on the quasisyntomic sheaves $THH(-)^{hC_n}$ and $THH(-)^{tC_n}$ where $C_n$ is the finite cyclic group of order $n$. These give cohomology theories that interpolate between the quasisyntomic filtrations on $THH$ and those on $TC^{-}$ and $TP$, and as it turns out can be expressed in terms of sheaves already of central importance to the study of the cohomology theories introduced by Bhatt, Morrow, and Scholze.
As applications, I will explain how these filtrations can be used to extend the results of Angeltveit, Gerhardt, and Hesselholt on the K-theory of truncated polynomial algebras, and a result on topological restriction homology.

I'm gonna introduce p-adic Analysis and I'm gonna talk about the Skolem-Mahler-Lech theorem and I'm gonna blow your mind.

The fractal uncertainty principle (FUP) was introduced by Dyatlov and Zahl which states that a function cannot be localized near a fractal set in both position and frequency spaces. It has rich applications in spectral gaps and quantum chaos on hyperbolic manifolds and has recently been an active area of research in harmonic analysis. I will talk about the history of the fractal uncertainty principle and explain its applications to spectral gaps. Then I will talk about our recent work, joint with Aidan Backus and James Leng, which proves a general fractal uncertainty principle for small fractal sets, improving the volume bound in higher dimensions. This generalizes the work of Dyatlov--Jin using Dolgopyat's method. As an application, we give effective essential spectral gaps for convex cocompact hyperbolic manifolds in higher dimensions with Zariski dense fundamental groups.

This talk is an exposition on the partial progress of Yau's conjecture on the lower estimate of the first eigenvalue of embedded minimal hypersurfaces in a unit sphere.

In this talk we investigate stochastic chemical reaction networks with scaling methods. This approach is used to study the stability properties of the associated Markov processes, but also to investigate the transient behavior of these networks. It may give insight on the impact of complex features of these networks such as their polynomial rates, leading to the coexistence of multiple timescales. Several examples are discussed.

Please contact Professor Williams for zoom information.

 A guiding question in number theory, specifically in arithmetic statistics, is counting number fields of fixed degree and Galois group as their discriminants grow to infinity.  We will discuss the history of this question and take a closer look at the story in the case of quartic fields. In joint work with Florea, Serrano Lopez, and Varma, we extend and make explicit the counts of extensions of an arbitrary number field that was done over the rationals by Cohen, Diaz y Diaz, and Olivier.

Rochelle Gutiérrez defines four dimensions of equity in mathematics education: access, achievement, identity, and power. In this talk, we briefly explore these dimensions and specific examples of how they can motivate course design choices. In particular, I will discuss my redesign of UW-Madison's college algebra course from 2016 to 2018 and course design choices for UCSD's Math 2 in Winter 2022.

Recall that an algebraic variety is rational if it is birational to a projective space. In this talk, I will review classical results and I will report on progress in the last decade. This includes rationality of hypersurfaces,  deformation properties, and equivariant properties.

In K-12 school contexts, responsive math pedagogy typically refers to instruction where the teacher continuously elicits information about what students currently know and understand and responds in ways that move them forward in relation to developmental and grade-level mathematical goals. It is claimed that recognizing what students know and are able to do and leveraging that to move towards higher level reasoning and problem solving ensures equity and access to mathematics for all students. Inspired by this approach, I discuss my teaching, research, and leadership efforts aimed at fostering a form of responsive pedagogy in collegiate math contexts. I discuss how these efforts can be relevant to mathematics teaching and learning at UCSD, especially in relation to increasing the retention and persistence of students from historically underrepresented backgrounds. 

A Fleming-Viot process is a system of n particles driven by independent copies of a driving Markov process. When one of the particles hits the boundary of the domain, it is killed, and some other particle branches. There is only one infinite path (spine) in the branching structure. Under some assumptions, when n goes to infinity, the distribution of the spine converges to the distribution of the driving Markov process conditioned to avoid the boundary of the domain forever. 

Based on joint articles with M. Bieniek, J. Englander, and T. Tadic.

Ricci flow deforms the Riemannian structure of a manifold in the direction of its Ricci curvature and tends to regularize the metric. This provides useful information about the underlying space. Ricci solitons are special solutions to the Ricci flow and arise naturally in the singularity analysis of the flow. We shall discuss some curvature and entropy gap theorems of gradient Ricci solitons. This talk is based on joint works with Yongjia Zhang and Zilu Ma, Eric Chen and Man-Chun Lee.

Gene expression is the fundamental biological process by which RNA and protein molecules are produced in a cell based on the information encoded in the DNA. The regulation of gene expression dictates which genes are expressed and when, which is crucial for cells to perform specific functions and adapt to changes in their environment. Single-cell experiments reveal that RNA production occurs in bursts whose size and timing is random.  Over the last thirty years, a plethora of mathematical models of stochastic gene expression have been developed in order to understand the origin of this noise. However, solving these models analytically becomes progressively more difficult as their complexity is increased. In this talk, I will show how many of these models can be solved using the queueing theory, which in turn can help us to solve the inverse problem of inferring the kinetics of gene expression from single-cell measurements of RNA numbers.

We introduce a new lattice basis reduction algorithm with approximation guarantees analogous to the LLL algorithm and practical performance that far exceeds the current state of the art. We achieve these results by iteratively applying precision management techniques within a recursive algorithm structure and show the stability of this approach. We analyze the asymptotic behavior of our algorithm, and show that the heuristic running time is $O(n^{\omega}(C+n)^{1+\varepsilon})$ for lattices of dimension $n$, $\omega\in (2,3]$ bounding the cost of size reduction, matrix multiplication, and QR factorization, and $C$ bounding the log of the condition number of the input basis $B$. This yields a running time of $O\left(n^\omega (p + n)^{1 + \varepsilon}\right)$ for precision $p = O(\log \|B\|_{max})$ in common applications. Our algorithm is fully practical, and we have published our implementation. We experimentally validate our heuristic, give extensive benchmarks against numerous classes of cryptographic lattices, and show that our algorithm significantly outperforms existing implementations.

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In this talk, I will give an overview of some tools used in probabilistic combinatorics, and illustrate their use in two different projects. First, in joint work with Marcus Michelen and Will Perkins, we establish an asymptotic formula for the number of integer partitions of a general type: namely, partitions of an integer n where the sums of the kth powers of the parts are also fixed, for some collection of values k. Second, in joint work with Lina Li and Jinyoung Park, we give enumerative and structural results for colorings of the middle layers of the Hamming cube. This talk does not assume a lot of background, and it’s definitely okay if you don’t know much probability or combinatorics!

Let $f_1,\ldots, f_r \in k[x_1,\ldots, x_n]$ be homogeneous polynomial of degree $d$. Ananyan and
Hochster (2016) proved that there exists a bound $N = N(r, d)$ where if collective strength
of $f_1, \ldots, f_r \geq N$, then $f_1, \ldots, f_r$ are regular sequence. In this paper, we study the explicit
bound $N(r, d)$ when $r = 3$ and $d = 2, 3$ and show that $N(3, 2) = 2$ and $N(3; 3) > 2$.

The study of regularity in free probability boils down to the question of how much information about a *-algebra can be gleaned from probabilistic properties of its generators. Some of the first results in this theme come from the theory of Voiculescu's free entropy: generators satisfying certain entropic assumptions generate von Neumann algebras which are non-Gamma, or prime, or do not admit Cartan subalgebras. Free Stein dimension -- a quantity I introduced with Nelson -- is a more recent quantity in a similar vein, which is robust under polynomial transformations and not trivial for variables which do not embeddable in R^\omega.

In this talk, I will recall the motivation and definition of free Stein dimension, and spend some time focusing on how (approximate) algebraic relations between generators can be used to provide upper bounds on the Stein dimension; of particular interest are commutation, and good behaviour under conjugation, and I will mention how these results apply in some interesting examples. Time permitting I will discuss how free Stein dimension behaves under ``building block'' operations such as direct sums and tensor products with finite dimensional algebras. This is joint work with Brent Nelson.

Many seemingly ad hoc constructions in algebra become simpler and much more natural through the lens of bicategories. In this talk I'll describe a series of papers with Kate Ponto touching on Euler characteristics, 2 dimensional field theories, and topological Hochschild homology, which never could have been written without thinking bicategorically. Particular focus will be put on iterated traces (relating to 2d field theories) and the structure of topological Hochschild homology.
 

I will discuss recent joint work with Hayes wherein we show that any sofic group G that is initially sub-amenable (a limit of amenable groups in Grigorchuk's space of marked groups) admits two embeddings into the universal sofic group S that are not conjugate by any automorphism of S. Time permitting, I will also characterise precisely when two almost homomorphisms of an amenable group G are conjugate, in terms of certain IRS's associated to the two actions of G. One of the applications of this is to recover the result of Becker-Lubotzky-Thom around permutation stability for amenable groups. The main novelty of our work is the usage of von Neumann algebraic techniques in a crucial way to obtain group theoretic consequences.

The classical Diophantine problem of determining  which integers can be written as a sum of two rational cubes has a long history; it includes works of Sylvester,  Selmer, Satgé, Leiman  and the recent work of Alpöge-Bhargava-Shnidman-Burungale-Skinner.  In this talk, we will  use  Selmer groups of elliptic curves and integral binary cubic forms to study some cases of the rational cube sum problem.  This talk is based on  joint works with D. Majumdar, P. Shingavekar and B. Sury.

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In this talk we will discuss how to define Brownian motions on a curved space. We will briefly discuss some definitions on Riemannian manifolds and then focus on a construction on Lie groups. If time permits, we will discuss quasi-invariance with respect to left/right-multiplication and how this is related to the geometry of the group. (p.s. a background in probability is not required)

For example, in the case where \calM = BPGL_n, this gives a recipe to extend fibrations in Brauer-Severi varieties, whereas for \calM = [A^1/Gm] this gives a way to extend families of line bundles with a section. The talk is based on a joint work with Andrea Di Lorenzo.

 A natural class of dynamical systems is obtained by iterating polynomial maps, which can be viewed as maps from projective space to itself. One can ask which other projective varieties admit endomorphisms of degree greater than 1. This seems to be an extremely restrictive property, with all known examples coming from toric varieties (such as projective space) or abelian varieties. We describe what is known in this direction, with the new ingredient being the "Bott vanishing" property. Joint work with Tatsuro Kawakami.

We give an introduction to the theory of zeta functions for function fields of characteristic p > 0. I will discuss the history of these zeta functions, what is known about their special values, and the question of the distribution of their zeros. I will also present some recent work with Joe Kramer Miller which constitutes a Riemman hypothesis for the zeta functions of "ordinary" function fields.

In this talk, we show the extension (in spirit of Caffarelli-Silvestre) of fractional power of operators defined on Banach spaces. Starting with the Balakrishnan definition, we use semigroup method to prove the extension. This is a joint work with Pablo Raul Stinga.
 

I will talk about $N_\infty$ operads, a generalization of $E_\infty$ operads in the equivariant world, following Blumberg-Hill. We show that the admissible sets of an $N_\infty$ operad capture the data of the norm maps of its algebra. This ties the notion of norms in the (incomplete) Tambara functors, which are the commutative algebra objects in Mackey functors.
 

When considering topological spaces with algebraic structures, there are certain properties which are invariant under homotopy equivalence, such as homotopy-associativity, and others that are not, such as strict associativity. A natural question is: which properties, in general, are homotopy invariant? As this involves a general notion of "property", it is a question of mathematical logic, and in particular suggests that we need a system of logical notation which is somehow well-adapted to the homotopical context. Such a system was introduced by Voevodsky under the name Homotopy Type Theory. I will discuss a sort of toy version of this, which is the case of "first-order homotopical logic", in which we can very thoroughly work out this question of homotopy-invariance. The proof of the resulting homotopy-invariance theorem involves some interesting ("fibrational") structures coming from categorical logic.

Many problems in finite geometry follow the following pattern: say we have a set of points in a plane, and require that some combinatorial property holds. Can we say something about the algebraic structure of this set? And what if we impose some extra symmetry conditions?  By far the most famous example of such a theorem is Segre’s beautiful characterisation of conics in a Desarguesian projective plane of odd order $q$ (1955): every oval (which is a set $C$ of $q+1$  points such that no line contains more than $2$ points of $C$) is the set of points of a conic.  In this talk, we will explore some classical results about ovals and hyperovals and present more recent results of the same flavour about KM-arcs and quasi-quadrics.

 Among the infinitely many mathematical models, which one god chooses for our universe?
What are the common features of all those models? 

We start with two of the most well-studied random matrix ensembles. The limiting eigenvalue distribution of one of which is uniform on the unit disk, and the other of which is a semicircular distribution on the real line. These two distributions have a simple relation: 2 times the real part of the uniform measure on the disk gives you the semicircular distribution. In the first part of the talk I will speak about the "heat flow conjecture" which states that this simple relation can be accomplished in the matrix level by applying the heat operator to the characteristic polynomial of one of the random matrix. Then I will move to the case where we apply the heat operator to a random polynomial which has roots distributed approximately uniform on the disk. In this random polynomial case, we can prove a version of the heat flow conjecture if we replace the characteristic polynomial by the random polynomial in the statement of the conjecture. Finally, I will speak about the case of heat flow on the plane Gaussian analytic function (GAF). These are joint work with Brian Hall and joint work with Brian Hall, Jonas Jalowy, and Zakhar Kabluchko.

For closed curves evolving by their curvature, the theorem of Gage-Hamilton and Grayson establishes that an embedded curve contracts to a round point. An efficient proof was later found by Huisken, with improvements by Andrews-Bryan, which uses multi-point maximum principle techniques. We’ll discuss the use of these techniques in other settings, particularly for the long-time behaviour of curve shortening flow with free boundary.

Perfectoid signature and local étale fundamental group Abstract: In this talk I'll talk about a (perfectoid) mixed characteristic version of F-signature and Hilbert-Kunz multiplicity by utilizing the perfectoidization functor of Bhatt-Scholze and Faltings' normalized length. These definitions coincide with the classical theory in equal characteristic. Moreover, perfectoid signature detects BCM regularity and transforms similarly to F-signature or normalized volume under quasi-étale maps. As a consequence, we can prove that BCM-regular
rings have finite local étale fundamental group and torsion part of their divisor class groups. This is joint work with Seungsu Lee, Linquan Ma, Karl Schwede and Kevin Tucker.

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Let $\Gamma < G = \operatorname{SO}(n, 1)^\circ$ be a Zariski dense torsion-free discrete subgroup for $n \geq 2$. Then the frame bundle of the hyperbolic manifold $X = \Gamma \backslash \mathbb{H}^n$ is the homogeneous space $\Gamma \backslash G$ and the frame flow is given by the right translation action by a one-parameter diagonalizable subgroup of $G$. Suppose $X$ is geometrically finite, i.e., it need not be compact but has at most finitely many ends consisting of cusps and funnels. Endow $\Gamma \backslash G$ with the unique probability measure of maximal entropy called the Bowen-Margulis-Sullivan measure. In a joint work with Jialun Li and Wenyu Pan, we prove that the frame flow is exponentially mixing.

A Kakeya set is a set of points in R^n which contains a unit line segment in every direction. The Kakeya conjecture states that the dimension of any Kakeya set is n. This conjecture remains wide open for all n \geq 3.

Together with Josh Zahl, we study a special collection of the Kakeya sets, namely the sticky Kakeya sets, where the line segments in nearby directions stay close. We prove that sticky Kakeya sets in R^3 have dimension 3.  In this talk, we will discuss background of the problem and its connection to analysis, combinatorics, and geometric measure theory. 

From a directed graph Q, called a quiver, one can construct what is known as a quiver variety Y_Q, an algebraic variety defined as a quotient of a vector space by a group defined in terms of Q.  A mutation of a quiver is an operation that produces from Q and new directed graph Q’ and a new associated quiver variety Y_{Q’}.  The mutation conjecture predicts a surprising and beautiful connection between the geometry of Y_Q and that of Y_{Q’}.  In this talk I will describe quiver varieties and mutations, and show you that you are already well acquainted with some examples of these.  Then I will discuss an interesting connection to the Gromov—Witten Theory of degeneracy loci. This is based on joint work with Nathan Priddis and Yaoxiong Wen.

We present our construction of novel continuous and discrete Type II variational principles for Hamiltonian systems on cotangent bundles of Lie groups, which allows for Type II boundary conditions, i.e., fixed initial position and terminal momenta boundary conditions. The motivation for these boundary conditions arises from the adjoint sensitivity method, which is ubiquitous in dynamically-constrained optimization and optimal control problems. Traditionally, such Type II variational principles are only defined locally. However, for dynamics on the cotangent bundle of a Lie group, left-trivialization allows us to define this variational principle globally. Our discrete variational principle leads to an intrinsic, symplectic, and momentum-preserving integrator for Lie group Hamiltonian systems that allows for Type II boundary conditions and maximally degenerate Hamiltonians. We show how this method can be used to exactly compute sensitivities for optimization problems subject to dynamics on a Lie group. We conclude with numerical examples of optimal control problems on SO(3) and a discussion of future applications of this method. 

Voiculescu developed two main candidates for the analogs of entropy in free probability: the microstates and non-microstates free entropies. In 2003, Biane-Capitaine-Guionnet established a relationship between the two: the microstates free entropy is always bounded above by the non-microstates free entropy. In this talk, we discuss the conditional versions of these notions of free entropy. Then, by connecting each of the free entropies with the asymptotics of their classical counterparts, we provide an elementary proof of the result of Biane-Capitaine-Guionnet. This is joint work with David Jekel.

Stein's restriction conjecture is one of the central topics in Fourier analysis. It is closely related to other areas of math, for example, number theory, PDEs. In this talk, I will discuss a recent improvement of this conjecture in R^3, based on the joint work with Hong Wang. Our proof is built upon the framework of polynomial partitioning, and, among other things, it uses the refined decoupling theorem, a two-ends Kakeya estimate. The two-ends estimate captures some information from the method of induction on scales.

The eigencurve is an important object in the study of p-adic families of modular forms, yet many of its geometric structures remain mysterious. I will give an elementary introduction to the theory of p-adic modular forms and the construction of eigencurve. Then we will take a glance at this beautiful conjecture.

In 2016 Abert, Gelander, and Nikolov made what they called a provocative conjecture: for lattices in higher-rank simple Lie groups, the minimum size of a generating set (rank) is sublinear in the volume. I will discuss our solution to this conjecture. It is a corollary of our main result, where we establish "fixed price one" for a more general class of "higher rank" groups. No familiarity of fixed price or cost is required for the talk. Joint work with Mikolaj Fraczyk and Amanda Wilkens.

The analysis of non-Markovian, self-interacting diffusions, which has motivations from probability, physics, biology, etc., is intimately connected with that of an associated stochastic interface. In this talk, we will look at dynamical fluctuations of this interface, and derive a KPZ-type stochastic PDE as a scaling limit. Unlike the usual KPZ equation, the geometry of the underlying manifold plays an important role in the analysis of this SPDE. Deriving the SPDE from the diffusion model is based on a novel "local-to-global" stochastic homogenization principle. Studying the SPDE itself amounts to relatively modern ideas in stochastic analysis coupled with analysis of pseudo-differential operators on manifolds. Based on joint work with Amir Dembo.

This talk is based on a joint work with Xiaojun Huang entitled “Bergman metrics as pull-backs of the Fubini-Study metric”. We study domains in Cn or Stein manifolds M such that their Bergman metrics have constant holomorphic sectional curvature κ. We prove a uniformization theorem when κ < 0 through the Calabi rigidity theorem and holomorphic extension theorems. We also discuss the case when κ ≥ 0. We provide several interesting examples of the existence of such M. Under certain conditions on M, we prove that the Bergman metric of M can not have non-negative constant holomorphic sectional curvatures.

 We prove that a sequence of multi-scaled stationary distributions of reflected Brownian motions (RBMs) has a product-form limit. Each component in the limit is an exponential distribution. The multi-scaling corresponds to the "multi-scale heavy traffic" recently advanced in Dai, Glynn and Xu (2023) for generalized Jackson networks. The proof utilizes the basic adjoint relationship (BAR) first introduced in Harrison and Williams (1987) that characterizes the stationary distribution of an RBM. This is joint work with Jin Guang and Xinyun Chen at CUHK-Shenzhen,  and Peter Glynn at Stanford.

In 1976, Ken Ribet used modular techniques to prove an important relationship between class groups of cyclotomic fields and special values of the zeta function.  Ribet’s method was generalized to prove the Iwasawa Main Conjecture for odd primes p by Mazur-Wiles over Q and by Wiles over arbitrary totally real fields.  

Central to Ribet’s technique is the construction of a nontrivial extension of one Galois character by another, given a Galois representation satisfying certain properties.  Throughout the literature, when working integrally at p, one finds the assumption that the two characters are not congruent mod p.  For instance, in Wiles’ proof of the Main Conjecture, it is assumed that p is odd precisely because the relevant characters might be congruent modulo 2, though they are necessarily distinct modulo any odd prime.

In this talk I will present a proof of Ribet’s Lemma in the case that the characters are residually indistinguishable.  As arithmetic applications, one obtains a proof of the Iwasawa Main Conjecture for totally real fields at p=2.  Moreover, we complete the proof of the Brumer-Stark conjecture by handling the localization at p=2, building on joint work with Mahesh Kakde for odd p.  Our results yield the full Equivariant Tamagawa Number conjecture for the minus part of the Tate motive associated to a CM abelian extension of a totally real field, which has many important corollaries.

This is joint work with Mahesh Kakde, Jesse Silliman, and Jiuya Wang.

Zeta functions are central objects of study in number theory, and can often be found wherever there is Galois theory. In this talk, we will discuss the Ihara zeta function of an undirected graph and compare it to the Dedekind zeta function. Then we will talk about incidence algebras and use them to describe a categorification of both types of zeta functions.

In this talk we will discuss two central problems in algebraic number theory and their interconnections: explicit class field theory and the special values of L-functions.  The goal of explicit class field theory is to describe the abelian extensions of a ground number field via analytic means intrinsic to the ground field; this question lies at the core of Hilbert's 12th Problem.  Meanwhile, there is an abundance of conjectures on the special values of L-functions at certain integer points.  Of these, Stark's Conjecture has special relevance toward explicit class field theory.  I will describe two recent joint results with Mahesh Kakde on these topics.  The first is a proof of the Brumer-Stark conjecture. This conjecture states the existence of certain canonical elements in CM abelian extensions of totally real fields.  The second is a proof of an exact formula for Brumer-Stark units that has been developed over the last 15 years.  We show that these units together with other easily written explicit elements generate the maximal abelian extension of a totally real field, thereby giving a p-adic solution to the question of explicit class field theory for these fields.

Recently we developed a general framework to compute asymptotic expansions of certain quantities coming from Random Matrix Theory. More precisely if one considers the expectation of the trace of a sufficiently smooth function evaluated in a random matrix, one can compute a Taylor expansion (in the dimension of our random matrix) of this quantity. This method relies notably on free stochastic calculus whom I will briefly talk about. In a previous work we studied the case of GUE random matrices, in this talk we consider polynomials in independent Haar unitary matrices. I will explain the additional difficulties that this model brings then give a few applications of this result to Random Matrix Theory as well as links with Weingarten calculus.

Given a finite set, the collection of partitions of this set forms a poset category under the coarsening relation. This category is directly related to a space of trees, which in turn has interesting connections to operads. But what if the finite set comes equipped with a group action? What is an "equivariant partition"? And what connection is there to equivariant trees? We will explore possible answers to these questions in this talk, based on joint work with J. Bergner, P. Bonventre,D. Chan, and M. Sarazola.

This talk will survey several results concerning the topology at infinity of Ricci solitons, with an emphasis on splitting theorems for solitons that have more than one end or, more generally, counting the number of ends geometrically.

Shrinking Kahler-Ricci solitons model finite-time singularities of the Kahler-Ricci flow, hence the need for their classification. I will talk about the classification of such solitons in 4 real dimensions. This is joint work with Deruelle-Sun, Cifarelli-Deruelle, and Bamler-Cifarelli-Deruelle.

Zeta and L-functions are ubiquitous in modern number theory. While some work in the past has brought homotopical methods into the theory of zeta functions, there is in fact a lesser-known zeta function that is native to homotopy theory. Namely, every suitably finite decomposition space (aka 2-Segal space) admits an abstract zeta function as an element of its incidence algebra. In this talk, I will show how many 'classical' zeta functions in number theory and algebraic geometry can be realized in this homotopical framework. I will also discuss work in progress towards a categorification of motivic zeta and L-functions.

 In dimensions four and higher, the Ricci flow may encounter singularities modelled on cones with nonnegative scalar curvature. It may be possible to resolve such singularities and continue the flow using expanding Ricci solitons asymptotic to these cones, if they exist. I will discuss joint work with Richard Bamler in which we develop a degree theory for four-dimensional asymptotically conical expanding Ricci solitons, which in particular implies the existence of expanders asymptotic to a large class of cones.
 

 We adapt some Carleman estimates from earlier joint work with L. Wang to asymptotically conical Ricci flows and prove a general backward uniqueness theorem for sections of mixed parabolic inequalities along these flows. As an application, we prove that a solution which flows smoothly into a cone on an end must be a shrinking soliton. We will also discuss other related uniqueness results for asymptotically conical shrinking solitons.

This year's symposium will celebrate the return to in-person convening and usher in the next 50 years of the symposium.  The Symposium will be held on the weekend of May 20-21, 2023 at UC San Diego. The inaugural Ronald Getoor Distinguished Lecture will be delivered on Sunday afternoon at the Symposium. Registration, which is required, for the Symposium and/or the Getoor lecture, is available via the webpage listed below. 

More information is available at the webpage here: https://sites.google.com/view/scps2023/home

Ricci flow is proved to be a powerful tool in the field of differential geometry. To obtain geometric or topological applications via continuing the Ricci flow by surgeries, it is of central importance to understand at least qualitatively the (finite-time) singularity models. In this talk, we present some recent developments mainly regarding the qualitative descriptions of the singularity models. We present two notions of blow-downs and their relations. We show an optimal scalar curvature estimate for singularity models. We then introduce some optimal qualitative and asymptotic descriptions for steady Ricci solitons, which are self-similar solutions of the Ricci flow and may arise as singularity models.

 In this talk, we will discuss some additional structure in the Kahler setting for Bamler’s limit spaces of noncollapsed Ricci flows. We will review various notions of singular set stratification, and then state an improved dimension estimate for odd dimensional strata of limits of Kahler-Ricci flows. We will also see that tangent flows of Kahler metric flows admit natural isometric actions, which are locally free away from the vertex in the case that the tangent flows are static.
 

The Morrey Conjecture pertains to the properties of quasi-convexity and rank-one convexity in functions, where the former implies the latter, but the converse relationship is not yet established. While Sverak has proven the conjecture in three dimensions, it remains unresolved in the two-dimensional case. Analyzing these properties analytically is a formidable task, particularly for vector-valued functions. Consequently, to investigate the validity of the Morrey Conjecture, we conducted numerical simulations using a set of example functions by Dacorogna and Marcellini. Based on our results, the Morrey Conjecture appears to hold true for these functions.

When Steven Strogatz wrote a 15-part series on the elements of math for the New York Times, to his surprise — and his editor's — each piece climbed the most emailed list and elicited hundreds of appreciative comments. In this talk Steve will describe his adventures in bringing math to the masses, and will reflect on what works… and what doesn’t.

Steven Strogatz is the Jacob Gould Schurman Professor of Applied Mathematics at Cornell University. He works on nonlinear dynamics with applications to physics, biology, and the social sciences. His latest book, Infinite Powers, was a New York Times bestseller and was shortlisted for the 2019 Royal Society Science Book Prize.

Presented by the UC San Diego Research Communications Program and supported by a grant from the Gordon and Betty Moore Foundation.

The classical coinvariant ring $R_n$ is obtained from the polynomial ring $\mathbb{C}[x_1, \dots, x_n]$ by quotienting by the ideal $I_n$ generated by symmetric polynomials with vanishing constant terms. The {\em superspace coinvariant ring} $SR_n$ is obtained analogously, but starting with the ring $\Omega_n$ of regular differential forms on $n$-space. We describe  the bigraded Hilbert series of $SR_n$ in terms of ordered set partitions and give an `operator theorem' which describes the harmonic space attached to $SR_n$. This proves conjectures of N. Bergeron, Li, Machacek, Sulzgruber, Swanson, Wallach, and Zabrocki. This talk is based on joint work with Andy Wilson.

The Boardman-Vogt tensor product of operads encodes the notion of interchanging algebraic structures. A classic result of Dunn tells us that the tensor product of two little cube operads is equivalent to a little cube operad with the dimensions added together. As models for $\mathbb{E}_k$-operads, this reflects a defining property of these operads.
In this talk, we will explore some equivariant generalizations to Dunn’s additivity. Along the way, we will play with little star-shaped operads, question if we really need group representations for equivariant operads, and learn to love (and hate) the tensor product.

We study a robust approximation method for solving a class of chance constrained optimization problems. The constraints are assumed to be  polynomial in the random vector. A semidefinite relaxation algorithm is proposed for solving this kind of problem. Its asymptotic and finite convergence are proven under some mild assumptions.

We consider a finite volume homogeneous space endowed with a random walk whose driving measure is Zariski-dense. In the case where jumps have finite exponential moment, Eskin-Margulis and Benoist-Quint established recurrence properties for such a walk. I will explain how their results can be extended to walks with finite first moment. The key is to make sense of the following claim: "the walk in a cusp goes down faster that some iid Markov chain on R with negative mean". Joint work with N. de Saxcé.

Imagine passing a signal $v$ through a noisy channel, where $v \in \mathbb{C}^N$ is a deterministic unit vector. We assume that the recipient observes a corrupted version of the signal in the form of $\theta vv^* + M$, where $\theta \in \mathbb{R}$ is the strength of the signal and $M$ is a random Hermitian $N \times N$ matrix representing the noise. We consider two questions:

1. (Detection) Is it possible for the recipient to conclude that a signal has been passed based on the observation?
2. (Recovery) If so, is it possible for the recipient to recover the signal

For rotationally invariant noise, Benaych-Georges and Nadakuditi answered the detection question in terms of the outlying eigenvalues and the recovery question in terms of the corresponding eigenvectors. Their proof crucially relies on the fact that the eigenvectors of a rotationally invariant ensemble are Haar distributed (in particular, delocalized). 

We consider a general class of noise that includes non mean-field models such as random band matrices in regimes where the eigenvectors are known to be localized. In contrast to the usual approach to outliers via the resolvent, our analysis relies on moment method calculations for general vector states and a seemingly innocuous isotropic global law.

On a compact manifold with boundary, the map consisting of the scalar curvature in the interior and the mean curvature on the boundary is a local surjection at generic metrics. We prove that this result may be localized to compact subdomains in an arbitrary Riemannian manifold with boundary, as motivated by an attempt to generalize the Riemannian Penrose inequality in dimension 8. This result is a generalization of Corvino's result about localized scalar curvature deformations; however, the existence part needs to be handled delicately since the problem is non-variational. For non-generic cases, we give a classification theorem for domains in space forms and Schwarzschild manifolds, and show the connection with positive mass theorems.

The classical Hele-Shaw flow describes the motion of incompressible viscous fluid, which occupies part of the space between two parallel, nearby plates. With source and drift, the equation is used in models of tumor growth where cells evolve with contact inhibition, and congested population dynamics. We consider the flow with Hölder continuous source and Lipschitz continuous drift. We show that if the free boundary of the solution is locally close to a Lipschitz graph, then it is indeed Lipschitz, given that the Lipschitz constant is small. This is joint work with Inwon Kim.

Non-unital operator systems are norm closed subspaces of B(H) that are closed under the involution map $x \mapsto x^*$. For example, C*-algebras are examples of non-unital operator systems. Much like Gelfand duality, a result of Kadison from the 80s shows that operator systems generated by commuting elements are categorically dual to a class of geometric structures, namely compact convex sets. Unlike the C*-theory, a remarkable result due to Webster-Winkler and Davidson-Kennedy shows that Kadison's duality theorem readily generalizes to the non-commutative unital setting as well. In our talk, we discuss Kadison’s original duality as well as the nc convex duality for non-commutative operator systems due to myself, Matt Kennedy, and Nick Manor. Finally, we will have a discussion of some results on the side of operator systems that this illuminates.

Applied benchmark tests for the famous `subgraph isomorphism problem' empirically discovered interesting phase transitions in random graphs. This motivates our rigorous study of two variants of the induced subgraph isomorphism problem for two independent binomial random graphs with constant edge-probabilities p_1,p_2. In particular, (i) we prove a sharp threshold result for the appearance of G_{n,p_1} as an induced subgraph of G_{N,p_2}, (ii) we show two-point concentration of the size of the maximum common induced subgraph of G_{N, p_1} and G_{N,p_2}, and (iii) we show that the number of induced copies of G_{n,p_1} in G_{N,p_2} has a `squashed lognormal' limiting distribution. These results confirm simulation-based phase transition predictions of McCreesh-Prosser-Solnon-Trimble, and resolve several open problems of Chatterjee-Diaconis.

The proofs are based on careful refinements of the first and second moment method, using several extra twists to (a) take the non-standard behavior into account, and (b) work around the large variance issues that prevent standard applications of the second moment method, using in particular pseudorandom properties and multi-round exposure arguments to tame the variance.

Based on joint work with Erlang Surya and Emily Zhu (both UCSD).

Variational integrators are a class of geometric structure-preserving numerical integrators for variational differential equations that are based on a discretization of Hamilton’s variational principle. We will present our construction of geometric variational integrators for multisymplectic Lagrangian and Hamiltonian PDEs, as well as our construction of geometric variational integrators for adjoint systems arising in optimization and optimal control. For multisymplectic PDEs, we will examine their continuous multisymplectic structures and variational principles, and utilize these to develop a discrete variational principle, leading to variational integrators which preserve their multisymplectic structure as well as satisfy a discrete Noether’s theorem. For adjoint systems, we will introduce novel Type II variational principles for adjoint systems on vector spaces and Lie groups. We utilize these to develop a discrete variational principle for adjoint systems, leading to symplectic and presymplectic integrators for adjoint systems which allow one to compute exact gradients of cost functions in optimization and optimal control problems. Numerical examples will be presented throughout.

Take a look at the mighty tetrahedron and its mirror image. Explore the subgroup lattice of S4 and get a geometric perspective on why some of its subgroups are normal while others have conjugates.

See the Sylow-3 and Sylow-5 subgroups of A5, and learn how counting to 20 in Greek can help you remember how many of them there are. It's quite simple, actually - but we'll finish off with the following not-entirely-trivial question: can you see S5 by holding a dodecahedron up to a mirror?

 Asymptotic counting of geometric objects has a long history. In the case of flat surfaces, various works by Masur and Veech showed the quadratic asymptotic growth of the number of saddle connections of bounded length. In this spirit, Athreya, Fairchild, and Masur showed that, for almost any flat surface, the number of pairs of saddle connection of bounded length and bounded virtual area increases quadratically with the constraint on length. In this case the « almost all » is with respect to the so-called Masur-Veech measure.

To demonstrate this, they use tools of ergodic theory (hence the result is true almost everywhere). This result can be extended in several ways, giving an error term or extending it to almost any SL(2,R)-invariant measure. We will present several useful tools for tackling these questions.

Imagine a population evolving over time, with genetic information being passed down from generation to generation, while evolution shapes it. The inherently random nature of this process makes it an ideal subject to be studied using stochastic processes, particularly Markov processes. Now, imagine if the system had memory, meaning genetic information could be inherited from many generations in the past. In such cases, we refer to a "seed bank."

Seed banks can break the Markovian nature of the process. In this presentation, we will explain how to overcome this difficulty. Furthermore, we will describe the connections between seed bank models, stochastic delay differential equations and the fractional Brownian motion.

Transportation of measure underlies many contemporary methods in machine learning and statistics. Sampling, which is a fundamental building block in computational science, can be done efficiently given an appropriate measure-transport map. We ask: what is the effect of using approximate maps in such algorithms?

We propose a new framework to analyze the approximation power of measure transport. This framework applies to existing algorithms, but also suggests new ones. At the core of our analysis is the theory of optimal transport regularity, approximation theory, and an emerging class of inequalities, previously studied in the context of uncertainty quantification (UQ).

This talk will focus on Böhm and Lafuente 2017's work on immortal homogeneous Ricci flow, where they prove that any sequence of blow-downs of such flow will subconverges to an expanding homogeneous Ricci soliton. For a \mathfrac{g}-homogeneous space M, Ricci flow of G-invariant metrics can be shown to be equivalent to a flow on Ad(H) invariant "bracket." We will show the existence of stratification of the space of brackets that induces curvature estimate on each strata, motivated by geometric invariant theory. The sharp case of the inequality corresponds to the limit case of an expanding G-invariant Ricci soliton, and we will show that as immortal homogeneous Ricci flow is of Type III, the blowdown will subconverges to such limit case. Finally if time allows, we will discuss Böhm and Lafuente recent proof of Alekseevskii's conjecture and its implication to homogeneous Ricci soliton.

 In his seminal work on modular curves and the Eisenstein ideal, Mazur studied the existence of congruences between certain Eisenstein series and newforms, proving that Eisenstein ideals associated to weight 2 cusp forms of prime level are locally principal.
In this talk, we'll explore generalizations of Mazur's result to squarefree level, focusing on recent work, joint with P. Wake and C. Wang-Erickson, about a non-optimal level N that is the product of two distinct primes and where the Galois deformation ring is not expected to be Gorenstein. First, we will outline a Galois-theoretic criterion for the deformation ring to be as small as possible, and when this criterion is satisfied, deduce an R=T theorem. Then we'll discuss some of the techniques required to computationally verify the criterion.

It is rumoured that Zariski was rather annoyed with Grothendieck, since prior to the latter's introduction of sheaves and cohomology into algebraic geometry, only Zariski could do the subject whereas subsequntly any idiot could. Equally, much of Grothendieck's enterprise is purely categorical so we could reasonably expect similar results in, for example, dynamical systems. An example in the litterature that was crying out for such an interpretation was Adam Epstein's various generalisations of Thurston's rigidity of post critically finite rational maps, which we'll put, and improve, in their natural context of a dynamical topus. En passant this also leads to some highly clarifying results about real blow ups. 

 In this paper, we consider polynomial optimization with correlative sparsity. We construct correlative sparse Lagrange multiplier expressions (CS-LMEs) and propose CS-LME reformulations for polynomial optimization problems using the Karush-Kuhn-Tucker optimality conditions. Correlative sparse sum-of-squares (CS-SOS) relaxations are applied to solve the CS-LME reformulation. We show that the CS-LME reformulation inherits the original correlative sparsity pattern, and the CS-SOS relaxation provides sharper lower bounds when applied to the CS-LME reformulation, compared with when it is applied to  the original problem. Moreover, the convergence of our approach is guaranteed under mild conditions. In numerical experiments, our new approach usually finds the global optimal value (up to a negligible error) with a low relaxation order, for cases where directly solving the problem fails to get an accurate approximation. Also, by properly exploiting the correlative sparsity, our CS-LME approach requires less computational time than the original LME approach to reach the same accuracy level.

In this survey-style talk I discuss the (old) idea of studying turbulence in stochastically-forced fluid equations. I will discuss definitions of chaos, anomalous dissipation, and various other predictions by physicists that can be phrased as mathematically precise conjectures in this context. Then, I will discuss some recent work by my collaborators and I on various aspects, namely (1) a straightforward characterization of anomalous dissipation that implies the classical Kolmogorov 4/5 law for 3d NSE (joint with Michele Coti Zelati, Sam Punshon-Smith, and Franziska Weber); (2) the study of "Lagrangian chaos" and exponential mixing of scalars and how it leads to a proof of anomalous dissipation and of the power spectrum predicted by Batchelor in 1959 for the simpler problem of Batchelor-regime passive scalar turbulence (joint with Alex Blumenthal and Sam Punshon-Smith); (3) the more recent proof of "Eulerian chaos" for Galerkin truncations of the Navier-Stokes equations (joint with Alex Blumenthal and Sam Punshon-Smith).

The foundation of equivariant stable homotopy theory is laid by Lewis-May-Steinberger in the 80's, while people's understanding of the computational aspect of the subject is very limited even until today. The reason is that the equivariant homotopy groups are $RO(G)$-graded, and even the coefficient rings of Eilenberg-Maclane spectra involve complicated combinatorics of cell structures. In this talk I'll illustrate the advantages of Tate squares in doing $RO(G)$-graded computations. Several Eilenberg-Maclane spectra of particular interest will be discussed: the Eilenberg-Maclane spectra associated with the constant Mackey functors $\mathbb{Z}$, $\mathbb{F}_2$, and the Burnside ring. Time permitting, I'll also talk about some structures of the homotopy of $HM$, for $M$ a general $C_{2^n}$-Mackey functor.

In 1995, Kuperberg introduced a collection of web bases, which combinatorially encode $SL_2$ and $SL_3$ invariant tensors. By Schur-Weyl duality, these bases are also bases for the Specht modules corresponding to partitions $(k,k) \vdash 2k$ and $(k,k,k) \vdash 3k$ respectively, and have nicer symmetry properties than the standard polytabloid basis. In 2017, Rhoades introduced a basis for the Specht module corresponding to partition $(k,k,1^{n-2k}) \vdash n$ indexed by noncrossing set partitions and with similarly nice symmetry properties. In this talk, we will explore these bases and the connections between them, and discuss how these connections might be used to create a similar basis for the Specht module corresponding to $(k,k,k,1^{n-3k}) \vdash n$.

We'll give a quick introduction to the field of random matrix theory and give partial proof of the famed semicircle law for Wigner matrices. To do so, we will incorporate tools from combinatorics, probability, and topology to understand the genus expansion of a random matrix. No background in probability is required, though it may be helpful.

The unstable foliation, that changes the horizontal components of period coordinates, plays an important role in the study of translation surfaces, including their deformation theory, and in the understanding of horocycle invariant measures.

In this talk, we show that measures of large dimension equidistribute in affine invariant manifolds and give an effective rate. An analogous result in the setting of homogeneous dynamics is crucially used in the effective equidistribution results of Lindenstrauss-Mohammadi and Lindenstrauss--Mohammadi--Wang. Background knowledge on translation surfaces and homogenous dynamics will be explained.

Mean curvature flow is the negative gradient flow of the area functional, and it has attracted a lot of interest in the past few years. In this talk, we will discuss a PDE-based, gauge theoretic, construction of codimension-two mean curvature flows based on the Yang-Mills-Higgs functionals, a natural family of energies associated to sections and metric connections of Hermitian line bundles. The underlying idea is to approximate the flow by the solution of a parabolic system of equations and study the corresponding singular limit of these solutions as the scaling parameter goes to zero. This is based on joint work with A. Pigati and D. Stern. 

While streaming PCA (also known as Oja’s algorithm) was proposed about four decades ago and has roots going back to 1949, theoretical resolution in terms of obtaining optimal convergence rates has been obtained only in the last decade. However, we are not aware of any available distributional guarantees, which can help provide confidence intervals on the quality of the solution. In this talk, I will present the problem of quantifying uncertainty for the estimation error of the leading eigenvector using Oja's algorithm for streaming PCA, where the data are generated IID from some unknown distribution. Combining classical tools from the U-statistics literature with recent results on high-dimensional central limit theorems for quadratic forms of random vectors and concentration of matrix products, we establish a distributional approximation result for the error between the population eigenvector and the output of Oja's algorithm. We also propose an online multiplier bootstrap algorithm and establish conditions under which the bootstrap distribution is close to the corresponding sampling distribution with high probability. While there are optimal rates for the streaming PCA problem, they typically apply to the IID setting, whereas in many applications like distributed optimization, the data is generated from a Markov chain and the goal is to infer parameters of the limiting stationary distribution. If time permits, I will also present our near-optimal finite sample guarantees which remove the logarithmic dependence on the sample size in previous work, where Markovian data is downsampled to get a nearly independent data stream. 

Note: The speaker will also give an overview of her latest research.

We overview the cost of a group action, which measures how much information is needed to generate its induced orbit equivalence relation, and the ideal Poisson-Voronoi tessellation (IPVT), a new random limit with interesting geometric features. In recent work, we use the IPVT to prove all measure preserving and free actions of a higher rank semisimple Lie group on a standard probability space have cost 1, answering Gaboriau's fixed price question for this class of groups. We sketch a proof, which relies on some simple dynamics of the group action and the definition of a Poisson point process. No prior knowledge on cost, IPVTs, or Lie groups will be assumed. This is joint work with Mikolaj Fraczyk and Sam Mellick.

In this talk, I will sketch a surprising proof due to Gyorgy Elekes on a non-trivial lower bound for the sums-versus-product problem in combinatorial number theory.   

We consider first-order methods with constant step size for minimizing locally Lipschitz coercive functions that are tame in an o-minimal structure on the real field. We prove that if the method is approximated by subgradient trajectories, then the iterates eventually remain in a neighborhood of a connected component of the set of critical points. Under suitable method-dependent regularity assumptions, this result applies to the subgradient method with momentum, the stochastic subgradient method with random reshuffling and momentum, and the random-permutations cyclic coordinate descent method.

This talk is based on ongoing work of the speaker. We will discuss the stochastic embeddability of snowflakes of finite Nagata-dimensional spaces into ultrametric spaces and the induced stochastic embeddings of their hyperbolic fillings into trees. Several results follow as applications, for example:
(1) For any uniformly concave gauge $\omega$, the Wasserstein 1-metric over $([0,1]^n,\omega(\|\cdot\|))$ biLipschitzly embeds into $\ell^1$.
(2) The Wasserstein 1-metric over any finitely generated Gromov hyperbolic group biLipschitzly embeds into $\ell^1$.

Arithmeticity of quaternionic modular forms on G_2 Abstract: Quaternionic modular forms (QMFs) on the split exceptional group G_2 are a special class of automorphic functions on this group, whose origin goes back to work of Gross-Wallach and Gan-Gross-Savin. While the group G_2 does not possess any holomorphic modular forms, the quaternionic modular forms seem to be able to be a good substitute.  In particular, QMFs on G_2 possess a semi-classical Fourier expansion and Fourier coefficients, just like holomorphic modular forms on Shimura varieties.  I will explain the proof that the cuspidal QMFs of even weight at least 6 admit an arithmetic structure: there is a basis of the space of all such cusp forms, for which every Fourier coefficient of every element of this basis lies in the cyclotomic extension of Q.

In the 90s Kollár proved non-rationality of many degrees and dimensions of complex Fano hypersurfaces by considering their degenerations modulo p and taking advantage of the surprising existence and positivity of differential forms on their reductions. In a series of papers with Chen, Church, and Ji, we have shown that these differential forms in characteristic p can give insight into many other aspects of the birational geometry of complex hypersurfaces: the degree of irrationality, rational endomorphisms, the birational automorphism group, and rational fibrations in low genus curves.

(Based on joint work with Ruofan Jiang) The Hilbert scheme of points on a variety  parametrizes -dimensional quotients of the structure sheaf. When  is a planar singular curve, its enumerative invariants have close relation to mathematical physics, knot theory and combinatorics. In this talk, we investigate two analogous moduli spaces, one being a direct generalization of the Hilbert scheme. Our results reveal their surprising relations to Hall polynomials, matrix equations, modular forms, etc.

"AI is like nuclear energy–both promising and dangerous."
Bill Gates, 2019

Data Science is central to AI and has driven most of recent advances in biomedicine and beyond. Human judgment calls are ubiquitous at every step of a data science life cycle (DSLC): problem formulation, data cleaning, EDA, modeling, and reporting. Such judgment calls are often responsible for the "dangers" of AI by creating a universe of hidden uncertainties well beyond sample-to-sample uncertainty.

To mitigate these dangers, veridical (truthful) data science is introduced based on three principles: Predictability, Computability and Stability (PCS). The PCS framework and documentation unify, streamline, and expand on the ideas and best practices of statistics and machine learning. In every step of a DSLC, PCS emphasizes reality check through predictability, considers computability up front, and takes into account expanded uncertainty sources including those from data curation/cleaning and algorithm choice to build more trust in data results. PCS will be showcased through collaborative research in finding genetic drivers of a heart disease, stress-testing a clinical decision rule, and identifying microbiome-related metabolite signatures for possible early cancer detection. 

I will explain how the Fourier transform takes us from functions on G to functions on G dual and switches convolution with pointwise operator multiplication. I’ll motivate from the case of abelian groups, where G dual has a group structure.

Partitions arise in linear algebra as Smith normal forms of finite DVR-modules. This viewpoint has given rise to several important objects in symmetric function theory; they are indexed by partitions. What if we replace the DVR by other rings? The analogous objects can no longer be indexed by partitions, but certain general properties can still be proved. They show up in discrete random matrix theory (joint with Gilyoung Cheong) and algebraic geometry (joint with Ruofan Jiang). In some special cases, these objects can still be expressed in partitions (though more convoluted), and the proven general properties give rise to new identities involving Hall polynomials that do not seem to have a direct combinatorial proof.

This presentation focuses on applying the discrete approximation method to establish necessary optimality conditions in an optimization problem for fully controlled constrained sweeping processes. Additionally, we explore its applications in various practical dynamical models. The first model deals with the dynamics of mobile robots navigating obstacles, while the second pertains to model predictive control utilizing neural networks.

We give a new short proof of the theorem due to Marquis and Sabok, which states that the orbit equivalence relation induced by the action of a finitely generated hyperbolic group on its Gromov boundary is hyperfinite. Our methods permit moreover to show that every such action has finite Borel asymptotic dimension. This is a joint work with Andrea Vaccaro.

In the presence of a large group action, even non-noetherian rings sometimes behave like noetherian rings. For example, Cohen proved that every symmetric ideal in the infinite variable polynomial ring is generated by finitely many polynomials (and their orbits under the infinite symmetric group). In this talk, I will give a brief introduction to equivariant commutative algebra where we systematically study such noetherian phenomena in infinite variable polynomial rings, and explain my work over fields of positive characteristic.

What's the general methodology behind AI for math?  AI is disrupting all industries in an unprecedented way. What might happen for mathematics? Are mathematicians ever going to be "replaced"?

Ehrhart theory—the study of lattice point enumeration in polytopes with rational vertices—can be used to study various combinatorial objects, including posets and graphs. In this talk, we explore two weighted versions of Ehrhart theory. We first ask which polynomial weights we can apply to our lattice so that the associated weighted h*-polynomials retain some of their classical properties, such as nonnegativity and monotonicity. We also study a second weighted Ehrhart theory, Chapoton’s q-analog Ehrhart theory, and discuss its relationship to the principal specialization of Stanley’s chromatic symmetric function. The first project is joint work with Robert Davis, Jesús A. De Loera, Alexey Garber, Sofía Garzón Mora, Katharina Jochemko, and Josephine Yu, and the second project is joint work with Matthias Beck and Andrés R. Vindas Meléndez.

Topological order is a notion in theoretical condensed matter physics describing new phases of matter beyond Landau's symmetry breaking paradigm. Bravyi, Hastings, and Michalakis introduced certain topological quantum order (TQO) axioms to ensure gap stability of a commuting projector local Hamiltonian and stabilize the ground state space with respect to local operators in a quantum spin system. In joint work with Corey Jones, Pieter Naaijkens, and Daniel Wallick (arXiv:2307.12552), we study nets of finite dimensional C*-algebras on a 2D $\mathbb{Z}^2$ lattice equipped with a net of projections as an abstract version of a quantum spin system equipped with a local Hamiltonian. We introduce a set of local topological order (LTO) axioms which imply the TQO conditions of Bravyi-Hastings-Michalakis in the frustration free commuting projector setting, and we show our LTO axioms are satisfied by known 2D examples, including Kitaev's toric code and Levin-Wen string net models associated to unitary fusion categories (UFCs). From the LTO axioms, we can produce a canonical net of algebras on a codimension 1 $\mathbb{Z}$ sublattice which we call the net of boundary algebras. We get a canonical state on the boundary net, and we calculate this canonical state for both the toric code and Levin-Wen string net models. Surprisingly, for the Levin-Wen model, this state is a trace on the boundary net exactly when the UFC is pointed, i.e., all quantum dimensions are equal to 1. Moreover, the boundary net for Levin-Wen is isomorphic to a fusion categorical net arising directly from the UFC. For these latter nets, Corey Jones' category of DHR bimodules recovers the Drinfeld center, leading to a bulk-boundary correspondence where the bulk topological order is described by representations of the boundary net.

I will report on recent results stemming from the analysis of Schrödinger operators on manifolds. I will first describe results dealing with isoperimetric inequalities and optimal (aka extremal) metrics on closed manifolds. These issues have been instrumental in the study of the spectrum of several classical operators, and are motivated by the understanding of the behaviour of the spectrum under changes of metrics. Then, motivated by conjectures of Yau on measures of nodal sets (but which are actually related to the first part of the talk), I will describe how eigenfunctions are concentrating in terms of Lp norms (with an explicit dependence on the eigenvalues). My goal is to emphasize on the case of Schrödinger operators with rough potentials. I will also state several open problems. 

Polynomial optimization is an important class of non-convex optimization problems, and has a powerful modelling ability for both continuous and discrete optimization. Over the past two decades, the moment-SOS hierarchy has been well developed for globally solving polynomial optimization problems. However, the rapidly growing size of SDP relaxations arising from the moment-SOS hierarchy makes it computationally intractable for large-scale problems. In this talk, I will show that there are plenty of algebraic structures to be exploited to remarkably improve the scalability of the moment-SOS hierarchy, which leads to the new active research area of structured polynomial optimization.

We study variational inequality problems (VIPs) with involved mappings and feasible sets characterized by polynomial functions (namely, polynomial VIPs). We propose a numerical algorithm for computing solutions to polynomial VIPs based on Lagrange multiplier expressions and the Moment-SOS hierarchy of semidefinite relaxations. We also extend our approach to finding more or even all solutions to polynomial VIPs. We show that the method proposed in this paper can find solutions or detect the nonexistence of solutions within finitely many steps, under some general assumptions. In addition, we show that if the VIP is given by generic polynomials, then it has finitely many Karush-Kuhn-Tucker points, and our method can solve it within finitely many steps. Numerical experiments are conducted to illustrate the efficiency of the proposed methods.

A Borel equivalence relation on a Polish space is called hyperfinite if it can be approximated by equivalence relations with finite classes. This notion has long been studied in descriptive set theory to measure complexity of Borel equivalence relations. Although group actions on hyperbolic spaces don't always induce hyperfinite orbit equivalence relations on the Gromov boundary, some natural boundary actions were recently found to be hyperfinite. Examples of such actions include actions of hyperbolic groups and relatively hyperbolic groups on their Gromov boundary and acylindrical group actions on trees. In this talk, I will show that any acylindrically hyperbolic group admits a non-elementary acylindrical action on a hyperbolic space with hyperfinite boundary action. 

The memory capacity of a statistical model is the largest size of generic data that the model can memorize and has important implications for both training and generalization. In this talk, we will prove a tight memory capacity result for two-layer neural networks with polynomial or real analytic activations. In order to do so, we will use tools from linear algebra, combinatorics, differential topology, and the theory of real analytic functions of several variables. In particular, we will show how to get memorization if the model is a local submersion and we will show that the Jacobian has generically full rank. The perspective that is developed also opens up a path towards deeper architectures, alternative models, and training.

The work of Gromov, Lawson, Schoen, Yau on scalar curvature inspired progress in metric geometry. In turn, this progress inspired new results about geometry and topology of manifolds with positive scalar curvature. In my talk I will give some examples of how ideas travel between these two worlds. The talk will be based on joint works with Lishak-Nabutovsky-Rotman, Maximo, Chodosh-Li, Wang.

 In this talk, we study a Tannakian category of admissible pairs, which arise naturally when one is comparing etale and de Rham cohomology of p-adic formal schemes. Admissible pairs are parameterized by local Shimura varieties and their non-minuscule generalizations, which admit period mappings to de Rham affine Grassmannians. After reviewing this theory, we will state a result characterizing the basic admissible pairs that admit CM in terms of transcendence of their periods. This result can be seen as a p-adic analogue of a theorem of Cohen and Shiga-Wolfhart characterizing CM abelian varieties in terms of transcendence of their periods. All work is joint with Sean Howe.

[pre-talk at 1:20PM]

I will give an introduction to the $\overline{\partial}$-Neumann problem and the Bergman kernel, topics that are studied in several complex variables. I will conclude by discussing two open questions about the Bergman kernel which motivate research in several complex variables today.

 In prevalent cohort studies with follow-up, the time-to-event outcome is subject to left truncation leading to selection bias. For estimation of the distribution of time-to-event, conventional methods adjusting for left truncation tend to rely on the (quasi-)independence assumption that the truncation time and the event time are ``independent" on the observed region. This assumption is violated when there is dependence between the truncation time and the event time possibly induced by measured covariates. Inverse probability of truncation weighting leveraging covariate information can be used in this case, but it is sensitive to misspecification of the truncation model. In this work, we apply the semiparametric theory to find the efficient influence curve of an expected  (arbitrarily transformed) survival time in the presence of covariate-induced dependent left truncation. We then use it to construct estimators that are shown to enjoy double-robustness properties. Our work represents the first attempt to construct doubly robust estimators in the presence of left truncation, which does not fall under the established framework of coarsened data where doubly robust approaches are developed. We provide technical conditions for the asymptotic properties that appear to not have been carefully examined in the literature for time-to-event data, and study the estimators via extensive simulation. We apply the estimators to two data sets from practice, with different right-censoring patterns.

A group is said to be Frobenius stable if every function from the group to unitaries that is "almost multiplicative" in the point-Frobenius norm topology is "close" to a genuine representation of the group in the same topology. In this talk, I will summarize my proof that finitely generated nilpotent groups are Frobenius stable if and only if they are virtually cyclic. I will also explain what the same techniques say about operator norm. This generalizes explicit counterexamples developed by Voiculescu and Kazhdan. 

 In this talk, I will discuss two topics that I have been exploring at the intersection of deep learning and dynamical systems.
    I will first present my recent research on ML-based dynamics learning and surrogate modeling. To circumvent difficulties faced by dynamics models from first principles or standard neural networks, a recent research direction has been considering a hybrid approach, where physics laws and geometric properties are encoded in the design of the deep learning architectures or in the learning process. Available physics prior knowledge can be used to construct physics-constrained neural networks with improved design and efficiency and a better generalization capacity, which can take advantage of the function approximation power of deep learning methods to deal with incomplete knowledge. Here, I will introduce two different ways to incorporate prior knowledge about the structure of a dynamical system in a strong way to obtain structure-preserving deep learning architectures for dynamics learning and surrogate modeling.
    In the second part of the talk, I will discuss a specific way to leverage deep learning techniques to accelerate the computation of high-resolution solutions of parametric partial differential equations. In numerous contexts, high-resolution solutions are required to capture faithfully essential dynamics which occur at small spatiotemporal scales, but these solutions can be very difficult and slow to obtain using traditional numerical integration methods due to limited computational resources. Here, I will introduce a new approach based on the use of neural operators to obtain high-resolution solution operators.

I will introduce some interacting particle systems on finite graphs and Cayley graphs of countable groups, and discuss how sofic entropy helps understand them.

More specifically, we consider two notions of statistical equilibrium: an "equilibrium state" maximizes a functional called the pressure while a "Gibbs state" satisfies a local equilibrium condition. On amenable groups (for example, integer lattices) these notions are equivalent, under some assumptions on the interaction. Barbieri and Meyerovitch have recently shown that one direction holds for general sofic groups: equilibrium states are always Gibbs.

I will show that the converse fails in the simplest nontrivial case: the free boundary Ising state on a free group (an infinite regular tree) is Gibbs but not equilibrium. I will also discuss what this says about Gibbs states on finite locally-tree-like graphs: it is well-known that their local statistics are described by some Gibbs state on the infinite tree, but in fact they must locally look like a mixture of equilibrium states. This constraint can be used to compute local limits of finitary Gibbs states for a few interactions.

We consider general mixed $p$-spin mean field spin glass models and provide a method to prove that the spectral gap of the Dirichlet form associated with the Gibbs measure is of order one at sufficiently high temperature. Our proof is based on an iteration scheme relating the spectral gap of the $N$-spin system to that of suitably conditioned subsystems. Based on joint Work w/ C. Brennecke, C. Xu, and H-T Yau

In this talk I’ll discuss recent work studying benign overfitting in two-layer ReLU networks trained using gradient descent and hinge loss on noisy data for binary classification. Unlike logistic or exponentially tailed losses the implicit bias in this setting is poorly understood and therefore our results and techniques are distinct from other recent and concurrent works on this topic. In particular, we consider linearly separable data for which a relatively small proportion of labels are corrupted and identify conditions on the margin of the clean data which give rise to three distinct training outcomes: benign overfitting, in which zero loss is achieved and with high probability test data is classified correctly; overfitting, in which zero loss is achieved but test data is misclassified with probability lower bounded by a constant; and non-overfitting, in which clean points, but not corrupt points, achieve zero loss and again with high probability test data is classified correctly. Our analysis provides a fine-grained description of the dynamics of neurons throughout training and reveals two distinct phases: in the first phase clean points achieve close to zero loss, in the second phase clean points oscillate on the boundary of zero loss while corrupt points either converge towards zero loss or are eventually zeroed by the network. We prove these results using a combinatorial approach that involves bounding the number of clean versus corrupt updates across these phases of training. 

Contrary to what people believe, modern computers are sometimes surprisingly bad at computations. Integer division is a particular example which computers are agonizingly bad at. We will develop a little bit of the theory of continued fractions and see how these seemingly "only for pure mathematicians" - things can be used for dramatic speed-up of divisions and other types of computations with similar nature.

The effectiveness of the CDCL algorithm for SAT is complicated to understand, and so far one of the most successful tools for its analysis has been proof complexity. CDCL is easily seen to be limited by the resolution proof system, and furthermore can be thought of as being equivalent to resolution, in the sense that CDCL can reproduce a given resolution proof with only a polynomial overhead.

But the question of the power of CDCL with respect to resolution is far from closed. To begin with, the previous equivalence is subject to assumptions, only some of which are reasonable. In addition, in a setting where algorithms are expected to run in near-linear time, a polynomial overhead is too coarse a measure.

In this talk we will discuss how exactly CDCL and resolution are equivalent, what breaks when we try to make the assumptions more realistic, and how much of an overhead CDCL needs in order to simulate resolution.

This paper studies how to learn parameters in diagonal Gaussian mixture models. The problem can be formulated as computing incomplete symmetric tensor decompositions. We use generating polynomials to compute incomplete symmetric tensor decompositions and approximations. Then the tensor approximation method is used to learn diagonal Gaussian mixture models. We also do the stability analysis. When the first and third order moments are sufficiently accurate, we show that the obtained parameters for the Gaussian mixture models are also highly accurate. Numerical experiments are also provided.

A basic question in model theory is whether a theory admits any kind of quantifier reduction. One form of quantifier reduction is called model completeness, and broadly refers to when arbitrary formulas can be "replaced" by existential formulas.
Prior to the negative resolution of the Connes Embedding Problem (CEP), a result of Goldbring, Hart, and Sinclair showed that a positive solution to CEP would imply that there is no II$_1$ factor with a theory which is model-complete. In this talk, we discuss work on the question of quantifier reduction for tracial von Neumann algebras. In particular, we prove that no II$_1$ factor has a theory that is model complete by using Hastings' quantum expanders and a weaker assumption than CEP. This is joint work with Ilijas Farah and David Jekel.

For integers $s,t \geq 2$, the Ramsey number $r(s,t)$ denotes the minimum $n$ such that every $n$-vertex graph contains a clique of order $s$ or an independent set of order $t$. We prove that \[ r(4,t) = \Omega\Bigl(\frac{t^3}{\log^4 \! t}\Bigr) \quad \quad \mbox{ as }t \rightarrow \infty\] which determines $r(4,t)$ up to a factor of order $\log^2 \! t$, and solves a  conjecture of Erdős.


This is a joint work with Sam Mattheus (Accepted in the Annals of Mathematics).

A countable discrete group is exact if it has a free action on Cantor space which is measure-hyperfinite, that is, for every Borel probability measure on Cantor space, there is a conull set on which the orbit equivalence relation is hyperfinite. For an exact group, it is known that the generic action on Cantor space is measure-hyperfinite, and it is open as to whether the generic action is hyperfinite; an exact group for which the generic action is not hyperfinite would resolve a long-standing open conjecture about whether measure-hyperfiniteness and hyperfiniteness are equivalent. We show that for any countable discrete group with finite asymptotic dimension, its generic action on Cantor space is hyperfinite. This is joint work with Sumun Iyer. 

Random interface growth is all around us: tumors, bacterial colonies, infections, and propagating flame fronts. The KPZ equation is a stochastic PDE central to a class of random growth phenomena. In this talk, I will explain how to combine tools from probability, partial differential equations, and integrable systems to understand the behavior of the KPZ equation when it exhibits unusual growth.

The Fargues-Fontaine curve associated to an algebraically closed nonarchimedean field of characteristic $p$ is a fundamental geometric object in $p$-adic Hodge theory. Via the tilting equivalence it is related to the Galois theory of finite extensions of Q_p; it also occurs in Fargues's program to geometrize the local Langlands correspondence for such fields.

Recently, Peter Dillery and Alex Youcis have proposed using a related object, the "affine cone" over the aforementioned curve, to incorporate some recent insights of Kaletha into Fargues's program. I will summarize what we do and do not yet know, particularly about vector bundles on this and some related spaces (all joint work in progress with Dillery and Youcis).

Come venture into number theory in this spooky post halloween talk, where I plan on talking about some objects that are (at least tangentially) related to number theory. Which objects will show up? Maybe elliptic curves, maybe p-adic numbers, maybe Lie groups. It's a bit of a mystery, so come to the talk to find out! In order to keep the talk from being too scary, I'll try to keep the prerequisite knowledge to a minimum. 

 It was conjectured by Milnor in 1968 that the fundamental group of a complete manifold with nonnegative Ricci curvature is finitely generated.  In this talk we will discuss a counterexample, which provides an example M^7 with Ric>= 0 such that \pi_1(M)=Q/Z is infinitely generated.

There are several new points behind the result.  The first is a new topological construction for building manifolds with infinitely generated fundamental groups, which can be interpreted as a smooth version of the fractal snowflake.  The ability to build such a fractal structure will rely on a very twisted gluing mechanism.  Thus the other new point is a careful analysis of the mapping class group \pi_0Diff(S^3\times S^3) and its relationship to Ricci curvature.  In particular, a key point will be to show that the action of \pi_0Diff(S^3\times S^3) on the standard metric g_{S^3\times S^3} lives in a path connected component of the space of metrics with Ric>0.

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. 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 is developed using the combination of techniques and knowledge from variational collision integrators and Lie group variational integrators. If time permits, we can discuss how some of these techniques can be used in optimal control and robotics.

A hyperoperation on a set M is an operation that associates to each pair of elements a subset of M. Hypergroups and hyperrings are two examples of structures defined in terms of hyperoperations. While they were respectively defined in the 1930s and 1950s, they have recently gained prominence through various appearances in number theory, combinatorics, and absolute algebraic geometry. However, to date there has been relatively little attention given to categories of hyperstructures. I will discuss several categories of hyperstructures that generalize hypergroups. A common theme is that in order for these categories to enjoy good properties like (co)completeness, we must allow for the product or sum of two elements to be the empty subset, which cannot occur in a hypergroup. In particular, I will introduce a category of hyperstructures called mosaics whose subcategory of commutative objects possess a closed monoidal structure reminiscent of the tensor product of abelian groups. This is joint work with So Nakamura.

 We present the Partition of Unity Method (PUM) and its implementation in Fraunhofer SCAI’s PUMA software framework. The fundamental idea and benefit of the PUM is to reduce the necessary number of degrees of freedom of a simulation while attaining the required accuracy of the application by using application-dependent enrichment functions which can resolve highly localized behavior of the solution instead of using mesh-refinement and standard piecewise polynomial basis functions. We discuss how such application-dependent enrichments can be constructed a-priori and on-the-fly during a simulation for e.g. laminated composites, shells and additive manufacturing problems.

Schubert polynomials are distinguished representatives of Schubert cells in the cohomology of the flag variety. Pipedreams (PD) and bumpless pipedreams (BPD) are two combinatorial models of Schubert polynomials. There are many classical perspectives to view PDs: Fomin and Stanley represented each PD as an element in the NilCoexter algebra; Lenart and Sottile converted each PD into a labeled chain in the Bruhat order. In this talk, we unravel the BPD analogues of both viewpoints. One application of our results is a simple bijection between PDs and BPDs via Lenart's growth diagram.

In the first half, I will introduce the subject of random conformal geometry. Schramm-Loewner evolution (SLE) is a random planar curve describing the scaling limits of interfaces in statistical physics models (e.g. percolation, Ising model). Liouville quantum gravity (LQG) is a random 2D surface arising as the scaling limit of random planar maps. These fractal geometries have deep connections to bosonic string theory and conformal field theories. LQG and SLE exhibit a rich interplay: cutting LQG by independent SLE gives two independent LQG surfaces [Sheffield '10, Duplantier-Miller-Sheffield '14]. In the second half, I will present extensions of these LQG/SLE theorems and give several applications.

To any group G is associated the space Ch(G) of all characters on G. After defining this space and discussing its interesting properties, I'll turn to discuss dynamics on such spaces. Our main result is that the action of any arithmetic group on the character space of its amenable/solvable radical is stiff, i.e, any probability measure which is stationary under random walks must be invariant. This generalizes a classical theorem of Furstenberg for dynamics on tori. Relying on works of Bader, Boutonnet, Houdayer, and Peterson, this stiffness result is used to deduce dichotomy statements (and 'charmenability') for higher rank arithmetic groups pertaining to their normal subgroups, dynamical systems, representation theory and more. The talk is based on a joint work with Uri Bader.

The class number formula describes the behavior of the Dedekind zeta function at s = 0 and s = 1. The Stark and Gross conjectures extend the class number formula, describing the behavior of Artin L-functions and p-adic L-functions at s = 0 and s = 1 in terms of units. The Harris–Venkatesh conjecture describes the residue of Stark units modulo p, giving a modular analogue to the Stark and Gross conjectures while also serving as the first verifiable part of the broader conjectures of Venkatesh, Prasanna, and Galatius.
In this talk, I will draw an introductory picture, formulate a unified conjecture combining Harris–Venkatesh and Stark for weight one modular forms, and describe the proof of this in the imaginary dihedral case.

Over the last few years there have been significant developments in the techniques used to understand singularities within minimal submanifolds. I will discuss this circle of ideas and explain how they enable us to reconnect the study of these geometric singularities with more classical PDE techniques, such as those used in unique continuation.

 In this talk, we consider a conjecture by Gross and Kudla that relates the derivatives of triple-product L-functions  for three modular forms and the height pairing of the Gross—Schoen cycles on Shimura curves.

Then, we sketch a proof of a generalization of this conjecture for Hilbert modular forms in the spherical case. This is a report of work in progress with Xinyi Yuan and Wei Zhang, with help from Yifeng Liu.

 In this talk, we first consider a formula proved in the 2010s that relates the height pairing of the Gross—Schoen cycles on the product of a curve over a number field and the self-product of the relative dialyzing sheaves. Then, we describe a recent application given by Xinyi Yuan on the uniform Bogomolov conjecture and the uniform Mordell—Lang conjecture.

 I will present a collection of my favorite equations in mathematical physics and discuss why they are interesting from my own perspective.

  Mean-field game (MFG) systems offer a framework for modeling multi-agent dynamics, but unknown parameters pose challenges. In this work, we tackle an inverse problem, recovering MFG parameters from limited, noisy boundary observations. Despite the problem's ill-posed nature, we aim to efficiently retrieve these parameters to understand population dynamics. Our focus is on recovering running cost and interaction energy in MFG equations from boundary measurements. We formalize the problem as an constrained optimization problem with L1 regularization. We then develop a fast and robust operator splitting algorithm to solve the optimization using techniques including harmonic extensions, three-operator splitting scheme, and primal-dual hybrid gradient method.  Numerical experiments illustrate the effectiveness and robustness of the algorithm. This is a joint work with Yat Tin Chow (UCR), Samy Wu Fung (Colorado School of Mines), Levon Nurbekyan (Emory), and Stanley J.
  Osher (
 UCLA).

 In recent years there has been a considerable interest in questions regarding approximate homomorphisms between groups, going under the name of group stability. For our setting, a group G is said to be HS-stable if any approximate finite dimensional unitary representation of G is close to a true unitary representation of G, where proximity is measured by the (normalized) Hilbert-Schmidt norm. In the situation of amenable groups, this question can be translated into a finite dimensional approximation property of the character space of G, an object originating in harmonic analysis. We will present an analysis of the character space of B. H. Neumann groups, an uncountable family of Z-by-locally finite groups, and as a result deduce they are HS-stable. The analysis involves both character bounds of finite symmetric groups, as well as character theory of infinite symmetric groups.

Totally positive matrices are matrices in which each minor is positive. Lusztig extended the notion to reductive Lie groups. He also proved that specialization of elements of the dual canonical basis in representation theory of quantum groups at $q=1$ are totally non-negative polynomials. Thus, it is important to investigate classes of functions on matrices that are positive on totally positive matrices. I will discuss several sources of such functions. One has to do with multiplicative determinantal inequalities (joint work with M. Gekhtman). Another deals with certain partial sums of Plucker relations (joint work with P. K. Vishwakarma). The third source deals with majorizing monotonicity of symmetrized Fischer's products which are a natural generalization of Hadamard-Fischer inequalities. Majorizing monotonicity of symmetrized Fischer's products was already known for hermitian positive semidefinite case which brings additional motivation to verify if they hold for totally positive matrices as well (joint work with M. Skandera). The main tools we employed are network parametrization, Temperley-Lieb and monomial trace immanants.

Let G be an algebraic group over a local field. We will show that the image of G under an arbitrary continuous homomorphism into any (Hausdorff) topological group is closed if and only if the center of G is compact. We will show how mixing properties for unitary representations follow from this topological property.

The reflection coupling method on Riemannian manifolds is a powerful tool in the study of harmonic functions and elliptic operators. In this talk, we will provide an overview of some fundamental ideas in stochastic analysis and the coupling method. We will then focus on applying these ideas to the study of the fundamental gap problem on the sphere. Based on joint work with Gang Yang and Guofang Wei.

Mirror symmetry and the Breuil-Mezard Conjecture Abstract: The Breuil-Mezard Conjecture predicts the existence of hypothetical "Breuil-Mezard cycles" that should govern congruences between mod p automorphic forms on a reductive group G. Most of the progress thus far has been concentrated on the case G = GL_2, which has several special features. I will talk about joint work with Bao Le Hung on a new approach to the Breuil-Mezard Conjecture, which applies for arbitrary groups (and in particular, in arbitrary rank). It is based on the intuition that the Breuil-Mezard conjecture is analogous to homological mirror symmetry.

Given a smooth surface in R^3, a classical question in differential geometry asks whether the surface can be continuously deformed through a smooth, nontrivial family of isometric surfaces. If such a family exists and does not arise from rigid motions of R^3, then the surface is said to be flexible. An old conjecture asserts that flexible, smooth closed surfaces do not exist. In this talk, we survey this question and the general uniqueness problem for isometric immersions. We then present new examples of flexible, smooth immersed cylinders in R^3 which are neither minimal nor developable. We conclude with a discussion of speculative approaches to the construction of flexible, smooth closed surfaces. These results are part of upcoming work with Andrew Sageman-Furnas.

Ricci flow is an important tool in geometric analysis. There have been remarkable topological applications of Ricci flow on closed manifolds, such as the Poincaré Conjecture resolved by Perelman, and the recent Generalized Smale Conjecture resolved by Bamler-Kleiner. In contrast, much less is known about the Ricci flow on open manifolds. Solitons produce self-similar Ricci flows, which often arise as singularity models. Collapsed singularities and solitons create additional difficulties for open manifolds. In this talk, I will survey some recent developments in Ricci flow on open manifolds. In particular, I will talk about the resolution of Hamilton's flying wing Conjecture, and the resulting collapsed steady solitons.

The explicit descriptions of low degree K3 surfaces lead to natural compactifications coming from geometric invariant theory (GIT) and Hodge theory. The relationship between these compactifications for degree two K3 surfaces was studied by Shah and Looijenga, and revisited by Laza and O’Grady, who also provided a conjectural description for the case of degree four K3 surfaces. I will discuss these results, as well as a verification of this conjectural picture using tools from K-moduli. This is joint work with Kristin DeVleming and Yuchen Liu.

Most algebraic structures can be given a lattice structure. For instance, any R-module defines a lattice where the meet and the join operations are given by the intersection and the sum of modules. Furthermore, any R-module morphism gives rise to a usual lattice morphism between the corresponding lattices. Actually, these two correspondences comprise a functor from the category of

R-modules to the category of complete modular lattices and usual lattice morphisms. However, this last category does not summon some basic algebraic properties that modules have (for example, the first theorem of isomorphism). With this in mind, we consider the category of linear modular lattices and linear morphisms, where we extend the notions of preradicals, and thus, describe the big lattice of lattice preradicals. In the process, we define some isomorphic structures to such lattice of lattice preradicals.

Surprising connections have recently emerged between two very active and previously independent research domains: random permutations and random geometry. This talk will uncover these connections, showing how random geometric objects can be directly used to reconstruct universal limits for random permutations.

We will illustrate this new general theory through concrete examples of Baxter permutations and monotone meanders, helping the audience build intuition. In the last part of the talk, we will explain how similar ideas led us to a new conjecture for the scaling limit of uniform random meanders and share progress on this long-standing open problem.

I will present two results on the formation and the interior structure of black holes in general relativity.
The first result proves that extremal (zero temperature) black holes can form dynamically in gravitational collapse. This provides a definitive disproof of the ”third law of black hole thermodynamics.” This is joint work with Ryan Unger (Princeton).
The second result concerns black hole interiors and the Strong Cosmic Censorship conjecture due to Penrose. This conjecture asserts the deterministic character of general relativity. I will present work in the context of a negative cosmological constant that shows that whether determinism holds or not surprisingly depends on the Diophantine properties of the black hole geometry.

In this work, we present an overview and comparison of spectral bundle methods for solving both primal and dual semidefinite programs (SDPs). In particular, we introduce a new family of spectral bundle methods for solving SDPs in the primal form. The algorithm developments are parallel to those by Helmberg and Rendl, mirroring the elegant duality between primal and dual SDPs. The new family of spectral bundle methods achieves linear convergence rates for primal feasibility, dual feasibility, and duality gap when the algorithm captures the rank of the dual solutions. The original spectral bundle method by Helmberg and Rendl is well-suited for SDPs with low-rank primal solutions, while on the other hand, our new spectral bundle method works well for SDPs with low-rank dual solutions. These theoretical findings are supported by a range of large-scale numerical experiments.

In this talk we will explore one useful connection between graphs and permutation groups. With a little help from A. Cayley we'll see how many spanning trees a graph with n vertices has. Then we'll use this to find the number of minimal-length factorizations of a permutation into transpositions. The material should be accessible to anyone with a rudimentary knowledge of group theory. Undergraduates welcome!

We'll be talking about recent results on the problem of splitting Brauer classes by torsors for genus one curves. In its geometric form the question to be asked is: does every Severi--Brauer variety contain a smooth and projective genus one curve? Algebraically, this question is related to the existence of certain finite Galois modules inside the linear algebraic automorphism group of the Severi--Brauer variety. Our goal will be to motivate why this is an intuitive and interesting question, giving some new results along the way.

One of the most fundamental unsolved problems in representation theory is to classify the set of irreducible unitary representations of a semisimple Lie group. In this talk, I will define a class of such representations coming from filtered quantizations of certain graded Poisson varieties. The representations I construct are expected to form the ''building blocks'' of all unitary representations.

This is joint work with Srivatsav Kunnawalkam Elayavalli. I will discuss a new viewpoint we developed on the study of II_1 factors involving sequential commutation. This gives us new insights on the elementary equivalence problem, and also reveals a new natural spectral gap type property for II_1 factors strictly strengthening fullness. 

This research presents a framework for evaluating policies in a continuous-time setting, where the dynamics are unknown and represented by Lévy processes. Initially, we estimate the model using available trajectory data, followed by solving the associated PDE to conduct the policy evaluation. Our approach encompasses not only the conventional Brownian motion but also the non-Gaussian and heavy-tailed Lévy processes. We have developed an algorithm that demonstrates enhanced performance compared to existing techniques tailored for Brownian motion. Furthermore, we provide a theoretical guarantee regarding the error in policy evaluation given the model error. Experimental results involving both light-tailed and heavy-tailed data will be presented. This research provides a first step to continuous-time model-based reinforcement learning, particularly in scenarios characterized by irregular, heavy-tailed dynamics.

The famous Caccetta-Häggkvist conjecture states that for any $n$-vertex directed graph $D$, the directed girth of $D$ (the minimum length of a directed cycle in $D$) is at most $\lceil n/k \rceil$, where $k$ is the minimum out-degree of $D$. Aharoni raised a strengthening conjecture: for any $n$-vertex graph $G$ equipped with an edge coloring (not necessarily proper) using $n$ colors, the rainbow girth of $G$ (the minimum length of a cycle in $G$ with distinctly colored edges) is at most $\lceil n/k \rceil$, where $k$ is the minimum size of the color class. We will discuss some results in the non-uniform degrees and rainbow versions of the Caccetta-Häggkvist conjecture.

Based on joint work with Ron Aharoni, Eli Berger, Maria Chudnovsky, and Shira Zerbib.

The exploratory and interactive nature of modern data analysis often introduces selection bias and poses challenges to traditional statistical inference methods. To address selection bias, a common approach is to condition on the selection event. However, this often results in a conditional distribution that is intractable and requires Markov chain Monte Carlo (MCMC) sampling for inference. Notably, some of the most widely used selection algorithms yield selection events that can be characterized as polyhedra, such as the lasso for variable selection and the $\varepsilon$-greedy algorithm for multi-armed bandit problems. This talk will present a method that is tailored for conducting inference following polyhedral selection. The proposed method transforms the variables constrained within a polyhedron into variables within a unit cube, allowing for exact sampling. Compared to MCMC, this method offers superior speed and accuracy. Furthermore, it facilitates the computation of maximum likelihood estimators based on selection-adjusted likelihoods. Numerical results demonstrate the enhanced performance of the proposed method compared to alternative approaches for selective inference.

In this talk we will introduce big mapping class groups and compare them to classical mapping class groups. The goal of the talk is to convince you that big MCGs form an interesting class of Polish groups.

In this work, we investigate applications of no-collision transportation maps introduced by Nurbekyan et al. in 2020 in manifold learning for image data. Recently, there has been a surge in applying transportation-based distances and features for data representing motion-like or deformation-like phenomena. Indeed, comparing intensities at fixed locations often does not reveal the data structure. No-collision maps and distances developed in [Nurbekyan et al., 2020] are sensitive to geometric features similar to optimal transportation (OT) maps but much cheaper to compute due to the absence of optimization. In this work, we prove that no-collision distances provide an isometry between translations (respectively dilations) of a single probability measure and the translation (respectively dilation) vectors equipped with a Euclidean distance. Furthermore, we prove that no-collision transportation maps, as well as OT and linearized OT maps, do not in general provide an isometry for rotations.  The numerical experiments confirm our theoretical findings and show that no-collision distances achieve similar or better performance on several manifold learning tasks compared to other OT and Euclidean-based methods at a fraction of a computational cost.

The first non-round Einstein metrics on spheres were described in 1973 by Jensen in dimensions 4n+3 (n > 0). For the next 25 years it remained an open problem whether the same could be done in even dimensions. This question was settled in 1998 when C. Böhm constructed infinite families of Einstein metrics on all Spheres of dimension between 5 and 9, in particular on $S^6$ and $S^8$.

In the 25 years since then, all spheres of odd dimension (at least 5) have been shown to admit non-round Einstein metrics. However, there have been no new developments in even dimensions above 8, leaving open to speculation the question of whether, if the dimension is even, non-uniqueness of the round metric is a low-dimensional phenomenon or to be expected everywhere.

I will give an overview of the methods used to construct such Einstein metrics, which we recently used to construct the first examples of non-round Einstein metrics on $S^{10}$.

This is joint work with Matthias Wink.

The classical Spezialschar is the subspace of the space of holomorphic modular forms on  $\mathrm{Sp}_4(\mathbb{Z})$ whose Fourier coefficients satisfy a particular system of linear equations. An equivalent characterization of the Spezialschar can be obtained by combining work of Maass, Andrianov, and Zagier, whose work identifies the Spezialschar in terms of a theta-lift from $\widetilde{\mathrm{SL}_2}$. Inspired by work of Gan-Gross-Savin, Weissman and Pollack have developed a theory of modular forms on the split adjoint group of type D_4. In this setting we describe an analogue of the classical Spezialschar, in which Fourier coefficients are used to characterize those modular forms which arise as theta lifts from holomorphic forms on $\mathrm{Sp}_4(\mathbb{Z})$.

It is well-known that isoperimetric hypersurfaces in a smooth, compact (n+1)-manifold are smooth up to a closed set of codimension at least 7. We prove that the dimension estimate of singularities is sharp. In this talk, we will explore an example of an 8-dimensional closed smooth Riemannian manifold, whose unique isoperimetric region, with half the volume of the manifold, displays two isolated singularities on its boundary. Furthermore, for n > 7, we utilize similar methods to construct singular isoperimetric hypersurfaces in higher dimensions.
 

Rank aggregation from pairwise and multiway comparisons has drawn considerable attention in recent years and has a variety of applications, ranging from recommendation systems to sports rankings to social choice. The existing literature on the ranking problem mainly concerns parameter estimation and algorithm implementation. However, there has been little investigation on the statistical inference theory of ranks. In this talk, I will start with a novel inference framework for ranks based on a modified Plackett-Luce model for multiway ranking with only the top choice observed. Then I will present a new methodology to construct simultaneous confidence intervals for the corresponding ranks through a sophisticated maximum pairwise difference statistic based on the MLE. Practically a valid Gaussian multiplier bootstrap procedure is developed to approximate the distribution of the proposed statistic. With the constructed simultaneous confidence intervals, we are able to study various inference problems on ranks such as testing whether an item of interest is among the top-K ranking. Our inference framework for the ranks can be widely applicable in many other ranking problems.  

The main objects of study in algebraic geometry are varieties, which are geometric objects locally defined by polynomial equations, and one goal of the subject is to classify all algebraic varieties of a given type.  We approach this problem by constructing parameter spaces, called moduli spaces, whose points correspond to the geometric objects we aim to parameterize.  Depending on the type of variety, there are several different ways to construct a compact moduli space and in this talk we will survey these different moduli spaces and stability conditions (such as GIT stability, K-stability, KSB/KSBA-stability), discuss their relationships, and give several applications.

I will discuss joint work with Kenny Ascher, Dori Bejleri, Harold Blum, Giovanni Inchiostro, Yuchen Liu, and Xiaowei Wang on construction of moduli stacks and moduli spaces of boundary polarized log Calabi Yau pairs. Unlike moduli of canonically polarized varieties (respectively, Fano varieties) in which the moduli stack of KSB stable (respectively, K semistable) objects is bounded for fixed volume, dimension, the objects here form unbounded families. Despite this unbounded behavior, we define the notion of asymptotically good moduli space, and, in the case of plane curve pairs (P2, C), we construct a projective good moduli space parameterizing S-equivalence classes of such pairs. Time permitting, I will discuss applications to the classification of special degenerations of P2, the b-semiampleness conjecture of Shokurov and Prokhorov, and the Hassett-Keel program.

On a two dimensional Stein space with isolated, normal singularities, finite type boundary and locally algebraic Bergman kernel, we find an estimate of the local algebraic degree of the Bergman kernel in terms of the type of the boundary. As an application, we characterize two dimensional ball quotients as finite type Stein spaces with a rational Bergman kernel.

In this talk, I will explore the interesting geometries that emerge in high-dimensional attraction basins, which are important in applications such as protein folding, cell differentiation, and neural networks. As a paradigmatic model, I will consider networks of coupled Kuramoto oscillators and show that high-dimensional basins generally cannot be approximated by simple convex shapes. Instead, they have tentacle-like structures where most of the basin volume is concentrated. Next, I will show that introducing non-pairwise couplings among Kuramoto oscillators can make basins deeper but smaller—the attractors become linearly more stable but much harder to find due to basins shrinking dramatically. Time allowing, I will also briefly mention a few related projects, including learning basins with reservoir computing and modeling circadian clocks with Kuramoto oscillators.

The class of SQP methods solve nonlinear constrained optimization problems by solving a related sequence of simpler problems. These SQP subproblems involve minimization of a quadratic model of the Lagrangian function subject to linearized constraints. In contrast to the quasi-Newton approach, which maintains a positive definite Hessian approximation, Second-derivative SQP methods use the exact Hessian of the Lagrangian. In this context, we will discuss an adaptive convexification strategy that makes minimal matrix modifications while ensuring the subproblem iterates are bounded and the solution defines a descent direction for the relevant Lagrangian. This talk will focus on adaptive convexification of stabilized SQP methods, as well as their connection with primal-dual interior methods.

The notion of biexactness for groups was introduced by Ozawa in 2004 and has since become a major tool used for studying solidity of von Neumann algebras. We introduce the notion of biexactness for von Neumann algebras, which allows us to place many previous solidity results in a more systematic context, and naturally leads to extensions of these results. We will also discuss examples of solid factors that are not biexact. This is a joint work with Jesse Peterson.

We prove a conjecture of K. Marton, widely known as the polynomial Freiman– Ruzsa conjecture, in characteristic 2. The argument extends to odd characteristic, with details to follow in a subsequent paper. This is a joint work with Timothy Gowers, Ben Green and Terence Tao.  

The singularity formation problem is a central question in fluid dynamics, and it is still widely open for fundamental models such as the 3d incompressible Euler equations and the Navier-Stokes equations. In this talk, I will first review the singularity formation problem, and I will describe how particle transport poses the main challenge in constructing blow-up solutions — an effect known as “regularization by transport.” I will then outline a new mechanism, arising from the classical Taylor-Couette instability, allowing us to overcome regularization by transport in the 3d Euler equations, thereby constructing the first swirl-driven singularity in R^3. This is joint work with Tarek Elgindi (Duke University).

Topology plays a fundamental role in controlling the dynamics of adaptive biological and physical systems, from chromosomal DNA and biofilms to cilia carpets and worm collectives. Despite their long history, the subtle interplay between topology, geometry and mechanics in tangled elastic filaments remains poorly understood. To uncover the topological principles underlying the dynamics of knotted and tangled matter, we first develop a mapping between human-designed elastic knots and long-range ferromagnetic spin systems. This mapping gives rise to topological counting rules that predict the relative mechanical stability of commonly used climbing and sailing knots. Building upon this framework, we then examine the adaptive topological dynamics exhibited by California blackworms, which form living tangled structures in minutes but can rapidly untangle in milliseconds. Using blackworm locomotion datasets, we construct stochastic trajectory equations that explain how the dynamics of individual active filaments control their emergent topological state. By identifying the principles behind stability and adaptivity in living tangled matter, our results have applications in understanding broad classes of adaptive, self-optimizing biological systems.

Let $H < G$ both be noncompact connected semisimple real algebraic groups and $\Gamma < G$ be a lattice. In the work of Gorodnik--Weiss, they showed that the distribution of dense $\Gamma$-orbit on homogeneous space $G/H$ is asymptotically the same as $G$-orbit on $G/H$. One key ingredient in their proof is Shah's theorem derived from the famous Ratner's theorem. In this talk, we report an effective version of this result in the case $(G, H, \Gamma) = (\mathrm{SL}_3(\mathbb{R}), \mathrm{SO}(2, 1), \mathrm{SL}_3(\mathbb{Z}))$. The proof uses recent progress by Lindenstrauss--Mohammadi--Wang--Yang towards an effective version of Ratner's theorem. We also prove the general case if an effective version of Ratner's theorem is provided. The talk is based on an ongoing work with Pratyush Sarkar.

In this talk, I will describe the regularity theory for surfaces minimizing the prescribed mean curvature functional over isotopies in a closed Riemannian 3-manifold, which is a prescribed mean curvature counterpart of the celebrated regularity result of Meeks, Simon and Yau about minimizers of the area functional over isotopies. 

Whereas for the area functional minimizers over isotopies are smooth embedded minimal surfaces, minimizers of the prescribed mean curvature functional turn out to be C^{1,1} immersions which can have a large self-touching set where the mean curvature vanishes. 

Even though the proof broadly follows the same general strategy as in the case of the area functional, several new ideas are needed to deal with the lower regularity setting. This regularity theory plays an important role in Z. Wang-X. Zhou’s recent proof of the existence of 4 embedded minimal spheres in a generic metric on the 3-sphere.

The results in this talk are joint work with Douglas Stryker (Princeton). 

Sharp geometric and functional inequalities play an important role in analysis,  PDEs and differential geometry. In this talk, we will describe our works in recent years on sharp higher order Poincare-Sobolev and Hardy-Sobolev-Maz'ya inequalities on real and complex hyperbolic spaces and noncompact symmetric spaces of rank one. The approach we have developed crucially relies on the Helgason-Fourier analysis on hyperbolic spaces and establishing such inequalities for the GJMS operators.  Best constants for such inequalities will be compared with the classical higher order Sobolev inequalities in Euclidean spaces. The borderline case of such inequalities, such as the Moser-Trudinger and Adams inequalities will be also considered. 

The concept of p-adic families of automorphic forms has far reaching applications in number theory. In this talk, we will discuss one of the first examples of such a family, built from the Eisenstein series, before allowing this to inform a construction of a family on an exceptional group of type G_2.

[pre-talk at 1:20PM in APM 6402]

Modern statistical learning is featured by the high-dimensional nature of data and over-parameterization of models. In this regime, analyzing the dynamics of the used algorithms is challenging but crucial for understanding the performance of learned models. This talk will present recent results on the dynamics of two pivotal algorithms: Approximate Message Passing (AMP) and Stochastic Gradient Descent (SGD). Specifically, AMP refers to a class of iterative algorithms for solving large-scale statistical problems, whose dynamics admit asymptotically a simple but exact description known as state evolution. We will demonstrate the universality of AMP's state evolution over large classes of random matrices, and provide illustrative examples of applications of our universality results. Secondly, for SGD, a workhorse for training deep neural networks, we will introduce a novel mathematical framework for analyzing its implicit regularization. This is essential for SGD's ability to find solutions with strong generalization performance, particularly in the case of over-parameterization. Our framework offers a general method to characterize the implicit regularization induced by gradient noise. Finally, in the context of underdetermined linear regression, we will show that both AMP and SGD can provably achieve sparse recovery, yet they do so from markedly different perspectives.

Integrable lattice models play a pivotal role in the investigation of microscopic multi-particle systems, with their continuum limits forming the foundation of the macroscopic effective theory. These models have found broad applications in condensed matter physics and numerical analysis. In this talk, I will discuss our recent work on the continuum limit of some differential-difference equations. Using the Ablowitz--Ladik system (AL) as our prototypical example, we establish that solutions to this discrete model converge to solutions of the cubic nonlinear Schr\"odinger equations (NLS). Notably, we consider merely $L^2$ initial data which combines both slowly varying and rapidly oscillating components, and demonstrate convergence to a decoupled system of NLS. This surprising result highlights that a sole NLS does not suffice to encapsulate the AL evolution in such a low-regularity setting reminiscent of the thermal equilibrium state. I will also explain the framework of our proof and how it has been successfully extended to address more general lattice approximations to NLS and mKdV.

F-bundle is a formal version of variation of non-commutative Hodge structures. I will explain basic ideas and properties of F-bundles, present an explicit construction of F-bundles associated to blowups, and discuss its relation with Iritani’s work on quantum cohomology. Joint work with Katzarkov, Kontsevich and Pantev.

Reaction-diffusion equations with nonlocal constraints naturally arise as limiting cases of mathematical models of intracellular signaling. Among the interesting behaviors of these models, much has been made of their 'geometry-sensing' properties: the strong sensitivity of steady-state solutions to domain geometry is widely seen as illustrative of how a cell establishes an internal coordinate axis. In this talk, I describe recent efforts to formally clarify this geometry dependence through careful study of the long-time behavior of a popular model of biochemical symmetry breaking.  Using the tools of formal asymptotics, calculus of variations, and a new fast solver for surface-bound PDEs, we study the formation and motion of interfaces on a curved domain across three dynamical timescales. Our results allow us to construct several analytical steady-state solutions that serve as counter-examples to received wisdom regarding the geometry-dependence of this class of model. 

From social media trends to family dynamics, social interactions shape our daily lives. In this talk, I will present tools I have developed for statistical inference and decision-making in light of these social interactions.

(1) Inference: I will talk about estimation of causal effects in the presence of interference. In causal inference, the term “interference” refers to a situation where, due to interactions between units, the treatment assigned to one unit affects the observed outcomes of others. I will discuss large-sample asymptotics for treatment effect estimation under network interference where the interference graph is a random draw from a graphon. When targeting the direct effect, we show that popular estimators in our setting are considerably more accurate than existing results suggest. Meanwhile, when targeting the indirect effect, we propose a consistent estimator in a setting where no other consistent estimators are currently available.

(2) Decision-Making: Turning to reinforcement learning amid social interactions, I will focus on a problem inspired by a specific class of mobile health trials involving both target individuals and their care partners. These trials feature two types of interventions: those targeting individuals directly and those aimed at improving the relationship between the individual and their care partner. I will present an online reinforcement learning algorithm designed to personalize the delivery of these interventions. The algorithm's effectiveness is demonstrated through simulation studies conducted on a realistic test bed, which was constructed using data from a prior mobile health study. The proposed algorithm will be implemented in the ADAPTS HCT clinical trial, which seeks to improve medication adherence among adolescents undergoing allogeneic hematopoietic stem cell transplantation.

Many statistical learning problems, where one aims to recover an underlying low-dimensional signal, are based on optimization, e.g., the linear programming approach for recovering a sparse vector. Existing work often either overlooked the high computational cost in solving the optimization problem, or required case-specific algorithm and analysis -- especially for nonconvex problems. This talk addresses the above two issues from a unified perspective of conditioning. In particular, we show that once the sample size exceeds the intrinsic dimension of the signal, (1) a broad range of convex problems and a set of key nonsmooth nonconvex problems are well-conditioned, (2) well-conditioning, in turn, inspires new algorithm designs and ensures the efficiency of many off-the-shelf optimization methods.