For a self-symmetric tracial von Neumann algebra $A$, we study rescalings of $A^{*n}*L\mathbb F_r$  for $n\in\mathbb N$ and $r\in (1,\infty]$ and use them to obtain an interpolation $\mathcal F_{s,r}(A)$ for all real numbers $s > $0 and $1 − s < r \leq\infty$. In this talk, I will first review the literature around this topic and explain well-definedness of the family $\mathcal F_{s,r}(A)$. I will discuss our definition of self-symmetry which includes all diffuse abelian tracial von Neumann algebras regardless of separability, and then focus on free products of infinitely many members of the family $\mathcal F_{s,r}(A_i)$. This is joint work with Ken Dykema.

The celebrated Newlander-Nirenberg theorem states that on a smooth manifold, an almost complex structure $J$ is a complex structure if and only if it is integrable, namely, the Nijenhuis tensor $N_J$ vanishes. It was known from Hill and Taylor that if $J$ has Hölder regularity above $C^{1/2}$ then $N_J$ makes sense as a tensor with distributional coefficients. However $N_J$ is undefined for generic $C^{1/2}$ tensor due to the failure of multiplication for $C^{1/2}$ functions and $C^{-1/2}$ distributions.

In the talk, we will explore the integrability condition when $J$ has regularity below $C^{1/2}$. We give a necessary and sufficient condition for $J$ being a complex structure (at least) for $J\in C^{1/3+}$ using Bony's paradifferential calculus. 

This is an in progress work joint with Gennady Uraltsev.

Recently, a class of algorithms combining classical fixed-point iterations with repeated random sparsification of approximate solution vectors has been successfully applied to eigenproblems with matrices as large as 10^108 x 10^108. So far, a complete mathematical explanation for this success has proven elusive. The family of methods has not yet been extended to the important case of linear system solves. Our recent work proposes a new scheme based on repeated random sparsification that is capable of solving sparse linear systems in arbitrarily high dimensions. We provide a complete mathematical analysis of this new algorithm. Our analysis establishes a faster-than-Monte Carlo convergence rate and justifies use of the scheme even when the solution is too large to store as a dense vector.

Jacobi's method is the oldest-known algorithm for the symmetric eigenvalue problem. It is also optimal; depending on the implementation, Jacobi can (1) compute small eigenvalues to higher relative accuracy than any other algorithm and (2) attain the arithmetic/communication complexity lower bounds of matrix multiplication (in both serial and parallel settings). This talk surveys efforts to optimize Jacobi as a one-algorithm case study into recent trends in numerical linear algebra. Based on joint work with James Demmel, Hengrui Luo, and Yifu Wang.

To achieve the greatest possible speed, practitioners regularly implement randomized algorithms for low-rank approximation and least-squares regression with structured dimension reduction maps. This talk outlines a new perspective on structured dimension reduction, based on the injectivity properties of the dimension reduction map. This approach provides sharper bounds for sparse dimension reduction maps, and it leads to exponential improvements for tensor-product dimension reduction. Empirical evidence confirms that these types of structured random matrices offer exemplary performance for a range of synthetic problems and contemporary scientific applications.

Joint work with Chris Camaño, Ethan Epperly, and Raphael Meyer; available at arXiv:2508.21189.

3d mirror symmetry predicts deep relationships between certain algebraic symplectic varieties. One such expectation is an "equivalence" between curve counts in a Higgs branch and those in the corresponding Coulomb branch. When it can be precisely formulated, this equivalence takes the form of an equality (after analytic continuation and change of variables) of meromorphic functions associated to the two branches. Bow varieties provide the largest currently known setting where the appropriate curve counts can be defined and their equivalence precisely formulated. In this talk, I will give an overview of these ideas and discuss my work with Tommaso Botta, in which we prove the duality of curve counts for finite type A bow varieties. Our proof combines geometric, combinatorial, and analytic arguments to eventually reduce to the case of the cotangent bundle of the complete flag variety. Time permitting, I will also discuss ongoing work to incorporate "descendant insertions" into the statements by using Hecke modifications of vector bundles.

The cohomology groups of tautological bundles on Grassmannians are described by the celebrated Borel-Weil-Bott theorem. Quot schemes on the projective line provide a natural generalization of Grassmannians: they parametrize rank r quotients of a vector bundle V on the projective line. In this talk, I will present formulas for Euler characteristics and for the cohomology groups of tautological bundles on these Quot schemes. Additionally, I will describe how these formulas apply to the study of the quantum K-theory of Grassmannians.

Countable Similarity Structure (CSS) groups are a class of generalized Thompson groups. I will introduce CSS$^*$ groups, a subclass, that we prove to be non-acylindrically hyperbolic, that includes the Higman-Thompson groups $V_{d,r}$, the countable R\"over-Nekrashevych groups $V_d(G)$, and the topological full groups of subshifts of finite type of Matui. I will discuss how all CSS$^*$ groups give rise to prime group von Neumann algebras, which greatly expands the class of groups satisfying a previous deformation/rigidity result. I will then discuss how CSS$^*$ groups are either C$^*$-simple with a simple commutator subgroup, or lack both properties. This extends C$^*$-simplicity results of Le Boudec and Matte Bon and recovers the simple commutator subgroup results of Bleak, Elliott, and Hyde. This is joint work with Eli Bashwinger.

This talk is devoted to combining model predictive control (MPC) and deep learning methods, specifically neural networks, to solve high-dimensional optimization and control problems. MPC is a popular method for real-life process control in various fields, but its computational requirements can often become a bottleneck. In contrast, deep learning algorithms have shown effectiveness in approximating high-dimensional systems and solving reinforcement learning problems. By leveraging the strengths of both MPC and neural networks, we aim to improve the efficiency of solving MPC problems. The talk also discusses the optimal control problem in MPC and how it can be divided into smaller time horizons to reduce computational costs. Additionally, we focus on enhancing MPC through two approaches: a machine learning–based feedback controller and a machine learning–enhanced planner, which involve implementing neural networks and iLQR. Overall, this talk provides insights into the potential of combining MPC and deep learning methods to tackle complex control problems across various fields, with applications to robotics.

The burning number of a graph is the minimal number of steps that are needed to burn all of its vertices, with the following procedure: at each step, one can choose a point to set on fire, and the fire propagates constantly at unit speed along the edges of the graph. In joint work with Alice Contat, we consider two natural random burning procedures in the discrete Euclidean torus with side-length n, in which the points that we set on fire at each step are random variables. Our main result deals with the case where at each step, the law of the new point that we set on fire conditionally on the past is the uniform distribution on the complement of the set of vertices burned by the previous points. In this case, we prove that as n tends to infinity, the corresponding random burning number (i.e, the first step at which the whole torus is burned) is asymptotic to T times n^{d/(d+1)} in probability, where T is the explosion time of a so-called generalised Blasius equation.

Consider min f(x) s.t. x>=0, where the objective function f: R→ R is smooth, and the variable is required to be nonnegative. A naive "squared variable" technique reformulates the problem to min_v f(v^2). Note that the new problem is now unconstrained, and many algorithms, e.g., gradient descent, can be applied. In this talk, we discuss the disadvantages of this approach, which have been known for decades, and the possible surprising fact of equivalence for the two problems in terms of (i) local minimizers and (ii) points satisfying the so-called second-order optimality conditions, which are keys for designing optimization algorithms. We further discuss extensions of the approach and equivalence to the vector case (where the vector variable is required to have all entries nonnegative) and the matrix case (where the matrix variable is required to be a positive semidefinite).

It is conjectured that many models of statistical mechanics have a rich, fractal-like behaviour at and near their points of phase transition, with power-law scaling governed by critical exponents that are expected to depend on the dimension but not on the small-scale details of the model such as the choice of lattice. This is now reasonably well understood in two dimensions and in high dimensions, but remains poorly understood in intermediate dimensions (e.g. d=3). I will overview the conjectures around this area and describe recent progress on related problems for models with long-range interactions.

Motivated by the Brill--Noether problems and enumerative geometry over surfaces, we study the expected behaviors of coherent sheaves. We estimate the dimension of global sections of stable sheaves. We also prove some cases of an analogue of Lange's conjecture over curves, which states that general extensions of two vector bundles are stable under some obvious conditions. These are closely related to Segre invariants of sheaves, which studies maximal subsheaves of a fixed rank. This can be understood as to determine when Grothendieck's Quot schemes are non-empty. This is based on work in progress jointly with Thomas Goller and Zhixian Zhu.

We consider ball averages on discrete groups, and associated Hardy-Littlewood maximal operator, with the balls defined by invariant metrics associated with a variety of length functions. Under natural assumptions on the rough radial structure of the group under consideration, we establish a maximal inequality of weak-type for the Hardy-Littlewood operator. These assumptions are related to a coarse radial median inequality, to almost exact polynomial-exponential growth of balls, and to the rough radial rapid decay property. We give a variety of examples where the rough radial structure assumptions hold, including any lattice in a connected semisimple Lie group with finite center, with respect to the Riemannian distance on symmetric space restricted to an orbit of the lattice. Other examples include right-angled Artin groups, Coxeter groups and braid groups, with a suitable choice of word metric. For non-elementary word-hyperbolic groups we establish that the Hardy-Littlewood maximal operator with respect to balls defined by a word metric satisfies the weak-type (1,1)-maximal inequality, which is the optimal result. This is joint work with Koji Fujiwara, Kyoto University.

Hyperplane arrangements cut space into `regions', which we like to count. Although all regions are $n$-dimensional, some are more bounded than others, captured by the `level' of a region. Can we refine our region counting by level? And how do level counts interact with other properties of the arrangement? This talk should be highly approachable, requiring only the ability to visualize high-dimensional objects.

The Ramsey number R(t, k) is the smallest n such that any red-blue edge coloring of the n-vertex complete graph has either a t-vertex red complete subgraph or a kvertex blue complete subgraph. We will investigate the history of asymptotic bounds on the extreme off-diagonal Ramsey number R(3, k), and present a new lower bound that has been conjectured to be asymptotically tight. Based on joint work with Paul Horn, Dylan King, Florian Pfender.

Our set up will consist of a countable group acting on a metric space with dense orbits. Our goal will be to develop effective gauges that measure how dense such orbits actually are, or equivalently how efficient is the approximation of a general point in the space by the points in the orbit.  We will describe several such gauges, whose definitions are motivated by classical Diophantine approximation, and are related to approximation exponents, discrepancy and equidistribution. We will then describe some of the (non-classical) examples we aim to analyze, focusing mainly on certain countable subgroups of the special linear or affine group, or of the groups of isometries of hyperbolic spaces, acting on some associated homogeneous spaces. In this set-up it is possible to establish optimal effective Diophantine approximation results in certain cases. We will very briefly indicate some ingredients of the methods involved, keeping the exposition as accessible as possible. We will also indicate some of the many challenging open problems that this circle of questions present. Based partly on previous joint work with Anish Ghosh and Alex Gorodnik, and partly on recent work with Mikolaj Fraczyk and Alex Gorodnik. 

TBA

We show that, for many choices of finite tuples of generators $\mathbf{X}=(x_1,\dots,x_d)$ of a tracial von Neumann algebra $(M,\tau)$ satisfying certain decomposition properties (non-primeness, possessing a Cartan subalgebra, or property $\Gamma$), one can find a diffuse, hyperfinite subalgebra in $W^*(\mathbf{X})^\omega$ (often in $W^*(\mathbf{X})$ itself), such that $W^*(N,\mathbf{X}+\sqrt{t}\mathbf{S})=W^*(N,\mathbf{X},\mathbf{S})$ for all $t>0$. (Here $\mathbf{S}$ is a free semicircular family, free from $\{\mathbf{X}\cup N\}$). This gives a short 'non-microstates' proof of strong 1-boundedness for such algebras.

This is joint work with Dimitri Shlyakhtenko.

In this talk, I will discuss a new physical-space approach to establishing the time decay and global asymptotics of solutions to variable-coefficient Schrödinger equation in (3+1)-dimensions. The result is applicable to possibly large, time-dependent, complex-valued coefficients under a general set of hypotheses. As an application, we are able to handle certain quasilinear cubic and Hartree-type nonlinearities, proving global existence together with global asymptotics. I will begin with a model problem and describe the construction of a good commutator. Time permitting, I will explain how to incorporate the good commutator with Ifrim--Tataru the method of testing by wave packets to obtain global asymptotics. This talk is based on upcoming work with Sung-Jin Oh and Federico Pasqualotto.

That is the question. In this talk, I will describe several problems of varying degrees of “tubiness” (the amenability of the problem to tube technology).

The number of regions of a hyperplane arrangement is a well-understood invariant, which we can complicate by counting regions of a given \emph{level}, a statistic quantifying a region's boundedness. Rediscovering a formula of Zaslavsky, we show that the level distribution is a \emph{combinatorial invariant}, and in the process define it for all semimatroids. The formula also allows us to reprove and generalize many known results on deformations of the braid arrangement.

Gaussian multiplicative chaos (GMC) is a well-studied random measure appearing as a universal object in the study of Gaussian or approximately Gaussian log-correlated fields. On the other hand, no general framework exists for the study of multiplicative chaos associated to non-Gaussian log-correlated fields. In this talk, we examine a canonical model: the log-correlated random Fourier series, or random wave model, with i.i.d. random coefficients taken from a general class of distributions. The associated multiplicative chaos measure was shown to be non-degenerate when the inverse temperature is subcritical ($\gamma < \sqrt{2d}$) by Junnila. The resulting chaos is easily seen to not be a GMC in general, leaving open the question of what properties are shared between this non-Gaussian chaos and GMC. We answer this question through the lens of absolute continuity, showing that there exists a coupling between this chaos and a GMC such that the two are almost surely mutually absolutely continuous.

This talk will present new results on the optimal control of self-organization, motivated by a growing body of empirical work on biological pattern formation in dynamic environments. We pose a boundary control problem for the classic supercritical Turing pattern, asking the best way to reach a non-trivial steady state by controlling the boundary flux of a reactant species. Via the Pontryagin approach, first-order optimality conditions for a generic reaction-diffusion system with a suitable bifurcation structure are derived. Using formal asymptotics, we construct approximate closed-form optimal solutions in feedback law form that are valid for any Turing-unstable system near criticality, which are verified against numerical solutions for a representative reaction-diffusion model.

Algebraic curves and abelian varieties play a central role in modern algebraic geometry, with links to complex analysis, number theory, topology and others. Curves and abelian varieties are closely related: a fundamental example of an abelian variety is the Jacobian of an algebraic curve. In this talk, I will give a discussion of curves, abelian varieties and their moduli spaces. Time permitting, I will present some new tools aimed at studying geometric classes on the moduli space of abelian varieties, and conclude with a discussion of several open questions in this area. 

Linear algebra is the workhorse of modern data science and machine learning, but none of the fun applications are ever mentioned in Math 18. We remedy this by discussing Principal Component Analysis,,the best* of these applications, and show how it follows quickly from the Singular Value Decomposition, the best* theorem in linear algebra. We present a few mathematical perspectives, explain the equivalent formulations of PCA, and ultimately use PCA to build an elementary image classifier without any fancy tools from machine learning.

^{*The speaker does not necessarily believe any of these claims but will nonetheless defend them vehemently if heckled.}

The study of p-adic Lie groups and their representations is a central piece of the p-adic Langlands program.  One tool which is used to study these is the notion of a saturated pro-p group, and the famous result of Lazard which states that every p-adic Lie group contains an open saturable subgroup.  In this talk, we will demonstrate a family of open saturated subgroups of G(F) for G a reductive group over a p-adic field F, which is indexed by the semisimple Bruhat-Tits building of G, given a mild assumption on G.  We will then review some group-theoretic consequences of this result.

The exchange property, introduced by Crawley and Jonsson in 1964 in the study of direct decompositions of algebraic systems and later extended to modules and rings by Warfield, plays a central role in modern decomposition theory. One of the main open problems in the area is whether the finite exchange property implies the full exchange property. This talk surveys the development of exchange theory from its module-theoretic origins to its ring-theoretic formulation via exchange rings. The last part of the talk is based on joint work with A. Cigdem Ozcan and focuses on lifting theory, including idempotent, regular, and unit lifting ideals and morphisms, and their interaction with local morphisms.

We discuss construction of a derived Lagrangian intersection theory of moduli spaces of perfect complexes, with support on divisors on compact Calabi-Yau threefolds. Our goal is to compute deformation invariants associated to a fixed linear system of divisors in CY3. We apply a Tyurin degeneration of the CY3 into a normal-crossing singular variety composed of Fano threefolds meeting along their anti-canonical divisor. We show that the moduli space over the Fano 4 fold given by total space of the degeneration family satisfies a relative Lagrangian foliation structure which leads to realizing the moduli space as derived critical locus of a global (-1)-shifted potential function. We construct a flat Gauss-Manin connection to relate the periodic cyclic homology induced by matrix factorization category of such function to the derived Lagrangian intersection of the corresponding “Fano moduli spaces”. The latter provides one with categorification of DT invariants over the special fiber (of degenerating family). The alternating sum of dimensions of the categorical DT invariants of the special fiber induces numerical DT invariants. If there is time, we show how in terms of “non-derived” virtual intersection theory, these numerical DT invariants relate to counts of D4-D2-D0 branes which are expected to have modularity property by the S-duality conjecture. This talk is based on joint work with Jacob Krykzca.

In 2024, Hikita showed that the chromatic symmetric functions of incomparability graphs of (3+1)-free posets expand with positive coefficients in the basis of elementary symmetric functions. This result resolved the long-standing Stanley–Stembridge conjecture. Finding a combinatorial interpretation of the $e$-coefficients remains a major open problem. In this talk I will define powerful and strong $P$-tableaux and conjecture that they give upper and lower bounds for the $e$-coefficients of chromatic symmetric functions. As evidence for these conjectures, we obtain combinatorial interpretations for various e-coefficients which live in between strong and powerful $P$-tableaux. Additionally, we show how Hikita’s theorem relates to strong $P$-tableaux and the Shareshian–Wachs inversion statistic.

In this talk, I will discuss symmetrized geometric last passage percolation. This is a prototypical model in the half-space KPZ universality class that has a natural interpretation as a line ensemble (a collection of random continuous functions). The model depends on parameters q,c, which control the background noise strength away from and at the boundary of the model, respectively. Depending on whether c < 1(subcritical), c = 1(critical), or c> 1(supercritical), the model exhibits different asymptotic behavior, and I will discuss some recent progress in understanding this from a line ensemble perspective.

The geometric approach to mechanics serves as the theoretical underpinning of innovative control methodologies in geometric control theory. These techniques allow the attitude of satellites to be controlled using changes in its shape, as opposed to chemical propulsion, and are the basis for understanding the ability of a falling cat to always land on its feet, even when released in an inverted orientation.

We will discuss the application of geometric structure-preserving numerical schemes to the geometric optimization control of mechanical systems, such as robots and drones. I will also describe the role of geometry and geometric structure-preservation in accelerated optimization and adjoint sensitivity analysis, and how they can be used to train neural networks derived from neural differential equations.

The classical EKR theorem states that the largest intersecting family of k-uniform subsets of an n-element set consists of all k-sets through a fixed element. More generally, it is known that large intersecting families are locally concentrated. So far such characterization results are missing for intersecting families of subspaces in vector. We will describe ongoing work which closes this gap in the literature. Joint work with Andrey Kupavskii.

The mathematical core of deep learning is function approximation by neural networks trained on data using stochastic gradient descent. I will explain an emerging geometric framework for the analysis of this process. This includes a collection of rigorous results on training dynamics for the deep linear network (DLN) as well as general principles for arbitrary neural networks. The mathematics ranges over a surprisingly broad range, including geometric invariant theory, random matrix theory, and minimal surfaces. However, little background in these areas will be assumed and the talk will be accessible to a broad audience. The talk is based on joint work with several co-authors: Yotam Alexander, Nadav Cohen (Tel Aviv), Kathryn Lindsey (Boston College), Alan Chen, Zsolt Veraszto and Tianmin Yu (Brown).

The border rank of tensors is a widely studied topic with practical applications to theoretical computer science and algebraic statistics. New lower bounds were obtained for the matrix multiplication tensor using techniques from representation theory and algebraic geometry. In this talk, we will prove non-trivial border rank lower bounds for a class of GL(V)-invariant tensors using Young flattenings. We will see how this comes down to proving results on ranks of certain maps between schur functors, the proofs of which surprisingly uses deep results in representation theory and commutative algebra.

Let $T: X\to X$ be a continuous map, where $X$ is a topological space. Fix also a family $\mathsf{I}\subseteq 𝒫(\mathbb{N})$ of "small" sets of nonnegative integers (for instance, the family $\mathrm{Fin}$ of finite sets, or the family $\mathsf{Z}$ of asymptotic density zero sets). A point $x \in X$ is said to be $\mathsf{I}$-hypercyclic if 
$$
\{n \in \mathbb{N}: T^nx \in U\}\notin \mathsf{I}
$$
for each nonempty open $U\subseteq X$. On a similar direction, a point $x \in X$ is said to be $\mathsf{I}$-strong hypercyclic if for each $y \in X$ there exists a subsequence $(T^nx: n \in A)$ of its orbit which is convergent to $y$ and, in addition, the set of indexes $A$ is \textquotedblleft not small,\textquotedblright\, that is, $A\notin \mathsf{I}$. In both cases, if $\mathsf{I}=\mathrm{Fin}$ and $X$ is a sufficiently nice topological space, then $x$ is $\mathsf{I}$-hypercyclic iff it is $\mathsf{I}$-strong hypercyclic iff its orbit is dense. 

We provide several structural relationships between the above notions and the related ones of recurrence with respect to $\mathsf{I}$. None of our results relies on the full linearity of $T$. As applications, we show that if $T$ is a homomorphism on a Fréchet space $X$ and there exists a dense set of vectors with orbits convergent to $0$, then for each $y \in X$ the set of all vectors $x \in X$ such that $\lim_{n \in A}T^nx=y$ for some $A\notin \mathsf{Z}$ is either empty or comeager. In a special case of bounded linear operators on Banach spaces, we obtain that $T$ is $\mathsf{Z}$-hypercyclic if and only if there exists a hypercyclic vector $x \in X$ for which $\lim_{n \in A}T^nx=0$ for some $A\notin \mathsf{Z}$. We conclude with several open questions. 

Reduced pipe dreams are combinatorial objects that encode some of the algebraic, enumerative, geometric, and probabilistic properties of Schubert and Grothendieck polynomials. They were introduced in 1993 by Bergeron and Billey, who showed that the set of all reduced pipe dreams for a fixed permutation w has a natural poset structure, with covering relations given by simple local operations called chute and ladder moves. In 2011, Rubey generalized chute and ladder moves on the set of reduced pipe dreams for a permutation w, and he conjectured that the induced poset on reduced pipe dreams is a lattice. We prove this conjecture and give simple recursive formulas for joins and meets in Rubey's lattice. This talk is based on joint work with Sara Billey and Clare Minnerath.

The fractional GJMS operators form a one-parameter family of conformally invariant operators defined on the conformal infinity of a Poincare-Einstein manifold. We mainly focus on operators of order between 0 and 2. First, I will show that the associated fractional Yamabe constants provide lower bounds for the relative volume of geodesic balls in the interior. Then, I will present some monotonicity properties for this family of fractional Yamabe constants. Finally, I will introduce some recent progress on the positive mass theorem for these fractional GJMS operators. This is joint work with Huihuang Zhou.

Optimizing intrathecal drug delivery procedures requires a deeper understanding of flow and transport in the spinal canal. Numerical modeling of drug dispersion is challenging because of the strong separation of time scales: dispersion occurs over approximately one hour, whereas cerebrospinal fluid pulsations driven by cardiac motion occur on a time scale of about one second. Patient-specific predictions in clinical settings therefore call for simplified descriptions that focus on dispersion time scales while bypassing the rapid concentration oscillations induced by cyclic motion. We show how asymptotic methods that exploit this separation of time scales can be used to derive a reduced transport equation in which convective transport driven by the mean Lagrangian drift governs drug dispersion. The model is validated through comparisons with MRI-informed direct numerical simulations of drug dispersion in a cervical spinal canal geometry that includes nerve rootlets and denticulate ligaments. These comparisons demonstrate that the reduced model accurately captures drug transport while enabling dispersion predictions at a fraction of the computational cost required by direct numerical simulations.

Graph rigidity is one of the most classical subjects in graph theory, studying geometric properties of graphs. Formally, a graph $G=(V,E)$ is $d$-rigid if a generic embedding of its vertex set $V$ into $R^d$ is rigid, namely, every continuous motion of its vertices preserving the lengths of the edges of $G$ necessarily preserves all pairwise distances between the vertices of $G$.

We develop a new sufficient condition for d-rigidity, formulated in graph theoretic terms. This condition allows us to obtain several newresults about rigidity of random graphs. In particular, we argue that for edge probability $p>2\ln(n)/n$, a random graph $G(n,p)$ is with high probability (whp) $cnp$-rigid, for $c>0$ being an absolute constant. We also show that a random r-regular graph $G_{n,r}$, $r>=3$, is whp $cr$-rigid. Another consequence is a sufficient condition for $d$-rigidity based on the minimum co-degree of the graph.   

The talk should be accessible to a general graph theoretic audience, no previous experience (whether positive or negative) with graph rigidity will be assumed.
 

A joint work with Alan Lew and Peleg Michaeli.

Random matrix theory is a very broad and well-developed research area at the intersection of physics, statistics, probability and combinatorics (and arguably others). Applications range from numerical analysis to engineering, to biology, economics, signal processing and machine learning. Classically, the type of matrices that have been studied have certain invariance properties (e.g., orthogonal), and therefore are mostly dense; however, the last decade has been marked by a tremendous increase in sparse applications, particularly related to the ubiquity of sparse networks and graphs. This, in turn, has led to rapid development of sparse random matrix theory. Some spectral properties (eigenstatistics) of sparse matrices turn out to match the dense ones, but others generate new and interesting phenomena. I will provide a high-level perspective on this rapidly evolving field, and describe some applications of interest.

Quantum information is encoded in a state vector of a tensor product of Hilbert spaces. Quantum error correction codes are useful for correcting errors when transporting quantum information through a noisy channel. In this talk, we will discuss a family of 2D "local topological" quantum error correction codes which use the robustness of topology to deformation to protect quantum information. We will then explain how operator algebra and tensor category techniques can be used to analyze quasi-particle excitations called anyons.

The proximal bundle method (PBM) is a powerful and widely used approach for minimizing nonsmooth convex functions. For smooth objectives, its best-known convergence rate has remained suboptimal. We present the first accelerated proximal bundle method to achieve the optimal O(1/sqrt(epsilon)) iteration complexity. The proposed method is conceptually simple, differing from Nesterov's accelerated gradient descent by a single line, and preserving the key structural properties of the classical PBM. As a result, we obtain an accelerated algorithm with much broader stepsize selection than what is allotted by accelerated gradient descent. The talk will include many pictures and numerical simulations to motivate algorithm design and illustrate fast convergence, respectively.

Many classical computer algorithms can be paralyzed efficiently; what about quantum computers? An algorithm can be described as having layers, one composed with another, with  the depth n of the circuit being the number of layers. An algorithm might be presented as having n simple layers, but if we are able to build more complicated layers, can we construct an equivalent algorithm with a few layers? This  is an issue, which goes back to the early days when people became enthusiastic about the possibility of quantum computers.

One of the most straightforward test cases is called the quantum waterfall or quantum staircase. It is a tensor product analog of a matrix of 2 x 2 blocks supported on the diagonal and the first diagonal below it. It was conjectured in the late 90s that an n layer  quantum waterfall cannot be produced with an algorithm having fewer than order n layers.

This conjecture (Moore-Nillson 1998) turns out to be way too pessimistic and the talk describes recent work with Adam Bene  Watts, Joe Slote, Charlie Chen on a theorem constructing a parallelization of any n layer quantum waterfall which yields  (asymptotically) log n layers.  Gratifying to  operator theorists is that a substantial ingredient is a matrix decomposition originating with Chandler Davis.

Development reliably produces complex organisms despite external perturbations and intrinsic stochasticity. It remains a central challenge not only to understand specific examples of development in vivo, but also to infer underlying principles that extend beyond any particular model system. In this talk, we will first introduce the formation of dorsal branches in the Drosophila larval trachea as a model for structural developmental defects. In each branch, progenitor cells robustly organize themselves into distinct cell fates, driven by an external morphogen concentration. By perturbing the external signal, partially penetrant stochastic phenotypes emerge in which a variable number of "terminal" cells are specified. Using live imaging to capture both morphology and expression of key genes, we observe dynamically how successful fate patterning occurs and how it fails. Partially penetrant phenotypes are modeled by geneticists using "threshold-liability", a phenomenological model with unspecified molecular details. Here, we are able to connect the abstract model to the molecular implementation by directly measuring receptor activation. Next, we consider self-organization theoretically by introducing a minimal model of cell patterning via local cell-cell communication. Recent advances have clarified how isolated cells can respond to an exogenous signal, but cells often interact and act collectively. In our framework we prove that a trade-off between speed and accuracy of collective pattern formation exists. Moreover, for the first time we are able to quantify how information flows between interacting cells during patterning. Our analysis reveals counterintuitive features of collective patterning: globally optimized solutions do not necessarily maximize intercellular information transfer and individual cells may appear suboptimal in isolation.

The combinatorial nullstellensatz was discovered by Noga Alon in 1995, and has since become an important tool in a variety of areas of mathematics. I will discuss this theorem and some of its numerous applications to Additive Number Theory, Geometry, and Combinatorics.

I will discuss the rational cohomology of $SL_n(R), Sp_{2n}(R)$, and their principal congruence subgroups for $R$ a number ring. Borel--Serre showed that these groups satisfy a (co)homological duality that lets us study their cohomology groups via certain representations called the `Steinberg modules’, which have a beautiful combinatorial description in terms of Tits buildings. I will describe a conjecture of Lee--Szczarba on the top cohomology of principal congruence subgroups of $SL_n(Z)$, and its resolution due to Miller--Patzt--Putman. I will then discuss forthcoming work on generalizations of this to other Euclidean rings, and also to symplectic groups.

Hyperrigidity is an interesting and important approximation-theoretic property of generating sets of C*-algebras. It plays a key role in, for example, the theory of strong convergence. In this talk, I will discuss a new characterization of hyperrigid generating sets in terms of the solvability of a certain noncommutative Dirichlet problem. I will also demonstrate how this result can be applied in practice.

Classical Choquet theory plays a key role in the study of classical Dirichlet problems, so it is perhaps not surprising that our results utilize noncommutative Choquet theory. I will provide a brief overview of some of these ideas. 

This is joint work with Eli Shamovich.

Regular multitypes are CR invariants used to exhibit (non-)degeneracies in the Levi form of a pseudoconvex hypersurface in $\mathbb{C}^n$. We study the properties of the regular multitypes as defined by Bloom, and understand how the notion of finite type implies global regularity in the $\bar{\partial}$-Neumann problem.

Lam, Lee, and Shimozono introduced the double Stanley symmetric functions in their study of the equivariant geometry of the affine Grassmannian. They proved that the assocaited double Edelman– Greene coefficients, the double Schur expansion of these functions, are positive, a result later refined by Anderson. They further asked for a combinatorial proof of this positivity. We provide the first such proof, together with a combinatorial formula that manifests the finer positivity established by Anderson. Our formula is built from two combinatorial models: bumpless pipedreams and increasing chains in the Bruhat order. The proof relies on three key ingredients: a correspondence between these two models, a natural subdivision of bumpless pipedreams, and a symmetry property of increasing chains. This talk is based on joint work with Jack Chou.
 

Convex subsets in higher-rank symmetric spaces are pretty rigid compared to rank 1 symmetric spaces, as proved by Kleiner and Leeb. In this talk, I will talk about convex subsets in products of negatively curved Hadamard manifolds. In particular, we show that the limit cone is 1-dimensional if the diagonal action is convex cocompact, which induces some rigidity-type results of the diagonal representation. This is joint work with Subhadip Dey.

What makes a game mathematical? In this talk I will not attempt to answer that question, but teach you to play a few games that are simple to learn, surprisingly thought-provoking, and feel somewhat mathematical. Come prepared to learn and play!

In physical models of fluids, a singularity occurs when a quantity of interest (velocity, pressure, vorticity…) becomes infinite at some finite time, starting from a "nice" initial configuration. In this talk, we will first introduce some classical models of compressible and incompressible fluids. We will then describe several mechanisms by which fluids can form singularities in finite time, allowing us to discuss the singularity formation problem for the incompressible three-dimensional Euler equations and the Navier--Stokes equations. We will finally touch upon computer-assisted methods and their applications to singularity formation in fluids.
 

The theory of majorization was introduced by Hardy, Littlewood and Pólya in order to formalize the intuitive idea of one set of numbers being more "spread out" than another. They established a surprising characterization of this property in terms convex functions, which allowed them to provide a unified approach to a number of seemingly disparate inequalities from in the literature from that era. The theory of majorization has subsequently found important applications throughout mathematics, mathematical economics and, more recently, quantum information theory. In this talk, I will discuss these developments and introduce a generalized theory of majorization, where numbers are replaced by (not necessarily commuting) matrices. This is joint work with Paul Skoufranis.

Dynamical sampling of probability distributions based on models or data (i.e., generative modeling) is a central task in scientific computing and machine learning. I will present some recent work on understanding and improving algorithms in high-dimensional settings. This includes a novel "delocalization of bias" phenomenon in Langevin dynamics, where biased methods are shown to achieve dimension-free scaling for low-dimensional marginals while unbiased methods cannot—a finding motivated by molecular dynamics simulations. I will also briefly discuss a new unbiased affine-invariant Hamiltonian sampler that outperforms popular samplers in the emcee package in high dimensions, and introduce a design of optimal Lipschitz energy for measure transport in generative modeling that leads to dimension-robust numerical performance with respect to resolution, offering an alternative to the optimal kinetic energy used in optimal transport. These examples demonstrate how dimensional scaling may be flattened, enabling efficient stochastic algorithms for high-dimensional sampling and generative modeling in relevant scientific applications.

Consider a finite modular tensor category $\mathcal A$. In [DGGPR] the authors exhibit a 3-dimensional topological field theory  $Z_{\mathcal A}: \operatorname{Bord}_{\mathcal A} \to \operatorname{Vect}$, which, in the case where $\mathcal A$ is semisimple, recovers the usual Reshetikhin-Turaev TQFT. In the present work we show that this extends naturally to a TQFT $Z_{\operatorname{Ch}(\mathcal A)}$, which takes values in the symmetric tensor category $\operatorname{Ch(Vect)}$ of linear cochains. This cochain valued theory furthermore respects (certain classes of) homotopies.

[DGGPR] M. De Renzi, A. M. Gainutdinov, N. Geer, B. Patureau-Mirand, and I. Runkel. 3-dimensional TQFTs from non-semisimple modular categories. Sel. Math. New Ser., 28(2):42, 2022.

In 1944, von Neumann and Morgenstern raised a question in their famous book Theory of Games and Economic Behavior: For a rational agent with preferences over all probabilistic mixtures of finitely many deterministic outcomes, is there always a unique utility function on deterministic outcomes whose expected value on probabilistic mixtures represents the preferences? Under natural assumptions on the preference order, they answered the question positively. We attempt to generalize this result to the case when the outcomes are infinite. We first identify the outcomes with the extreme points of a Choquet simplex, a natural generalization of the classical simplex to infinite-dimensional spaces. We then prove a similar result in the setting of Choquet simplex.

In this talk, we explore how data scientists in industry are using modern AI tools such as ChatGPT to write code and perform mathematical reasoning. Chris Deotte is a Senior Data Scientist at NVIDIA, a seven-time Kaggle Grandmaster, and holds a PhD in mathematics.

In recent years, data scientists and mathematicians have increasingly shifted from writing all code and derivations by hand to collaborating with AI assistants such as ChatGPT, Claude, and Gemini. These tools are now capable of generating high-quality code, solving mathematical problems, and accelerating research and development workflows.

We will examine concrete examples of how these AI tools perform on real-world coding and mathematical tasks. In particular, we will demonstrate how ChatGPT recently wrote over 99% of the code for a gold-medal-winning solution in an online competition focused on predicting mouse behavior from keypoint time-series data.

We will discuss recent progress on the corners problem from additive combinatorics and its connection to communication complexity. In particular, we will sketch a proof that every set $A \subseteq [N]^2$ of density $\gg exp(- polylog N)$ must contain three points of the form $(x,y)$, $(x+d,y)$, $(x,y+d)$ for some nonzero $d$. We will also show how this result implies strong lower bounds for the multiparty communication complexity of the Exactly-$N$ function.

Based on joint work with Michael Jaber, Yang P. Liu, Shachar Lovett, and Mehtaab Sawhney.

An IRS is a conjugacy-invariant probability measure on the space of subgroups of a locally compact group. We are interested in those IRS that give full measure to the set of compact subgroups. This talk is about what we know about those, how it connects to the structure theory of locally compact groups, and what we would still like to figure out.

Joint with Tal Cohen, Helge Glöckner and Gil Goffer.

This study considers simultaneous inference of economic time series models for potential policy implications. The process of interest is an unknown function of either time or an observable random vector in macro-finance. To overcome the well-known slow convergence issue with the traditional asymptotic-based approach, we utilize a Gaussian approximation for our time-dependent processes. Relevant theories and finite-sample simulations justify our approach. The empirical applications include the U.S. Phillips curve in macroeconomics and the forward premium puzzle in international finance.

Let R be a polynomial ring, J ⊆ R an ideal, and m a maximal ideal containing J. We consider invariants of the pair (R, J) which measure the singularities of the embedding Spec(R/J) ⊆ Spec(R) at m: the log canonical threshold (lct) in characteristic zero and the F-pure threshold (fpt) in positive characteristic. A smaller value of the lct/fpt means that the embedding is "more singular;" we seek to classify pairs which are as singular as possible.

In 1972, Skoda showed that lct_m(R, J) >= 1/ord_m(J), where ord_m denotes the order of vanishing at m. Skoda's bound has been generalized and refined many times since; among these improvements is a 2014 result by Demailly and Pham using mixed multiplicities of J and m. We extend Demailly and Pham's lower bound to positive characteristic and study the pairs (R, J) for which lct_m(R, J) (or fpt_m(R, J)) equals the lower bound. We classify these "extremal pairs" in the standard graded case, the codimension 1 case, and the dimension 2 case, confirming special cases of a conjecture by Bivià-Ausina.

We consider the three-dimensional ideal MHD system on a domain in  with a controllable part of the boundary where we prescribe the boundary data. The basic question of boundary controllability is whether, given two states, one can by means of the control on the boundary drive one state to another. We will review the existing literature on this problem and provide a positive result for domains with only Sobolev regularity. The results are based on works with Matthew Novack, Wojciech Ozanski, and Vlad Vicol.
 

Hiai’s construction of q-Araki-Woods factors generalizes both Shlyakhtenko’s free Araki-Woods factors and Bozejko-Speicher’s q-Gaussian algebras. I will discuss joint work with Changying Ding where we show the q -Araki-Woods factors are strongly solid if the associated representation U is almost periodic, and the non-isomorphism between q-Araki-Woods factors and free Araki-Woods factors for certain classes of representations, contrasting the case for q-Gaussians.

Low-rank representation-based matrix or tensor completion algorithms have been developed over the past two decades for various scientific and data-science applications. Given an incomplete data matrix/tensor with missing or noisy entries but certain underlying algebraic structures, completion algorithms rely on optimization techniques to recover the full data matrix/tensor directly in a compressed representation. In the past, various compression formats have been considered such as low-rank matrix format, and Tucker, CP or tensor-train-based tensor formats. Moreover, different optimization algorithms have been exploited including alternating least squares (ALS), alternating direction filtering (ADF), nuclear norm-based optimization, Riemannian optimization and adaptive moment estimation (ADAM). Despite the success of these completion algorithms, they become less effective when dealing with highly oscillatory operators, rising from e.g., large-scale seismic or tomographic applications where physical or cost constraints limit the amount of data acquisition. This is largely due to the incapability of the abovementioned compression formats for representing non-smooth operators. Therefore, a more effective completion algorithm is called for, as the successful completion of the data matrix can significantly improve the quality of downstream algorithm pipelines for these inverse or imaging problems.   

In this talk, I will present our recent work on new completion algorithms for highly oscillatory operators (arXiv:2510.17734). In a nutshell, we consider a different compression format called butterfly for the incomplete data matrix. Butterfly formats have been proven highly effective for compressing highly oscillatory operators such as Green’s functions for high-frequency wave equations, Fourier integral operators and special function transforms, but haven’t been investigated in the matrix completion context. Our work relies on a tensor reformulation of the butterfly format into a tensor network, and we consider a variety of optimization algorithms including ALS, ADF and ADAM. Numerical results demonstrate that our butterfly completion algorithms can efficiently recover a n×n matrix representing Green’s functions or Radon transforms with only O(nlogn) observed entries in O(nlogn) operation counts. I will also discuss about the limitation and future work regarding our proposed algorithm.

Biography: Yang Liu is a staff scientist in the Scalable Solvers Group of the Applied Mathematics and Computational Research Division at the Lawrence Berkeley National Laboratory, in Berkeley, California. Dr. Liu received the Ph.D. degree in electrical engineering from the University of Michigan in 2015. From 2015 to 2017, he worked as a postdoctoral fellow at the Radiation Laboratory, University of Michigan. From 2017 to 2019, he worked as a postdoctoral fellow at the Lawrence Berkeley National Laboratory. His main research interest is in numerical linear and multi-linear algebras, computational electromagnetics and plasma, scalable machine learning algorithms, and high performance scientific computing. Dr. Liu is the lead developer of the linear solver package ButterflyPACK and autotuning package GPTune, and is a core developer for linear solver packages SuperLU_DIST and STRUMPACK. Dr. Liu is the recipient and co-recipient of the ACES Early Career Award 2025, PDSEC Best Paper Award 2025, AT-AP RASC Young Scientists Award 2022, the APS Sergei A. Schelkunoff Transactions Prize 2018, FEM first place student paper award, 2014, and the ACES second place student paper award, 2012.

Topology and geometry play fundamental roles in controlling the dynamics of biological and physical systems, from chromosomal DNA and biofilms to cilia carpets and worm collectives. How topological rules give rise to adaptive, self-optimizing dynamics in soft and living matter remains poorly understood. Here we investigate the interplay between topology, geometry and mechanics in knotted and tangled matter. We first examine the adaptive topological dynamics exhibited by California blackworms, which form disordered living tangles in minutes but can rapidly untangle in milliseconds. By combining link-based tangling metrics with stochastic trajectory equations, we explain how the dynamics of individual active filaments controls their emergent topological state. Building on this framework, we then investigate tangled structures with local alignment. We demonstrate how the algebra of braids governs the mechanics and stability of braided filamentous networks in a range of biological systems. By identifying how topology and adaptivity produce stable yet responsive structures, these results have applications in understanding broad classes of adaptive living systems.

In 2000, the Clay Institute offered one million dollars for a mathematical construction of 4D Yang-Mills gauge theory. That problem remains unsolved, but there has been spectacular progress in recent years on many related 2D and 4D problems.

It all starts with 1+1=2. The fact that 1+1=2 implies that two non-parallel lines in the plane (co-dimension 1) meet at a point (co-dimension 2). Less trivially, any two paths through a square (one top to bottom, one left to right) intersect somewhere. Similarly, 2+2=4 implies that two fully-non-parallel 2D planes in 4D meet at a point (interpret one dimension as time and imagine moving lines in 3D colliding like light sabers) and that knotted loops in 3D cannot be disentangled without tearing rope.

Further implications include the self-duality of 1-forms (in 2D) and 2-forms (in 4D), the conformal invariance of special Gaussian fields in 2D and 4D, and the self-duality of cellular spanning trees, along with other fundamental results about random curves and surfaces, spin systems and connections. How will this help with our remaining open problems?

The goal of this talk is to explore what symmetries can say about an object. We will then focus on the case of algebraic varieties, where the symmetries are birational self-maps. This is joint work with L. Esser, A. Regeta, C. Urech, and I. van Santen.

A key problem in computer proofs based on solutions from copositive optimization, is checking whether or not a given quadratic form is completely positive or not. In this talk we describe the first known algorithm for arbitrary rational input. It is based on a suitable adaption of Voronoi's Algorithm and the underlying theory from positive definite to copositive quadratic forms. We observe several similarities with the classical theory, but also some differences, in particular for three and more variables. A key element and currently the main bottleneck in our algorithm is an adapted shortest vector computation, asking for all nonnegative integer vectors attaining the copositive minimum of a given copositive quadratic form. (This is based on joint work with Valentin Dannenberg, Alexander Oertel, Mathieu Dutour Sikiric and Frank Vallentin).

The rank $n$ superspace $\Omega_n$ is the algebra of polynomial-valued differential forms on affine $n$-space. This carries an $G$-action for any pseudo-reflection group $G$ -- two important examples being the symmetric group $\mathfrak S_n$ and the hyperoctahedral group $\mathfrak B_n$. The superspace coinvariant ring for $G$, defined as the quotient of $\Omega_n$ cut out by $G$-invariants of $\Omega_n$ with vanishing constant term, has received increased attention in recent years. In this talk, we explore some recent results on the superspace coinvariant rings for $\mathfrak S_n$ and $\mathfrak B_n$, including their Hilbert series, explicit monomial bases, and their representation-theoretic structures.

We will discuss new estimates for the size and structure of the nodal set $\{u=0\}$ and the singular set $\{u=|\nabla u|=0\}$ of solutions to parabolic inequalities with parabolic Lipschitz coefficients. In particular, we show that almost all of these sets are covered by regular parabolic Lipschitz graphs, with quantitative control, and that both satisfy parabolic Minkowski bounds depending only on a doubling quantity at a point. Many of these results are new even in the case of the heat equation on $\mathbb{R}^n \times \mathbb{R}$. This is joint work with Max Hallgren and Zilu Ma.

In this talk, I will construct a $p$-adic height pairing for curves with split degenerate stable reduction over a prime $p$ using the higher class field theory of Kato-Saito. This pairing can be shown to coincide with the standard Coleman-Gross height pairing when extended to the semistable reduction case using methods by Besser and Vologodsky. At the end, I will briefly mention how this new method inspires the definition of a height pairing valued in certain Galois groups related to the function field of the original curve.

[pre-talk at 3pm]

I will discuss various problems related to the study of the incompressible Euler equation. The main questions that we will look at have to do with the construction and classification of steady solutions, their stability properties, and the dynamics of nearby unsteady solutions.

Ricci solitons are generalizations of the Einstein manifolds and are self similar solutions to the Ricci flow. In particular, steady Ricci solitons are eternal solutions to the Ricci flow. In this talk, we will discuss the flying wing construction of some Kahler and Riemannian steady Ricci solitons of nonnegative curvature. This is based on joint work with Ronan Conlon and Yi Lai, as well as with Yi Lai and Man Chun Lee.

We review some recent results on the problem of reconstructing a variety from its topology.  This includes some recent work with Fanjun Meng and Lingyao Xie.

I will review recent constructions of oligomorphic tensor categories generalizing Deligne's Rep(S_t). Then, I will lean into the model theoretic part of the question. Specifically, I will explain where there are no continuous families like the original Rep(S_t) and where you should look for n-parameter families, i.e., depending on n free variables. Ultimately, these questions are closely related to classes of structures in model theory.

In this work, we consider an inverse problem in mean-field games (MFG). We aim to recover the MFG model parameters that govern the underlying interactions among the population based on a limited set of noisy partial observations of the population dynamics under the limited aperture.  Due to its severe ill-posedness, obtaining a good quality reconstruction is very difficult.   Nonetheless, it is vital to recover the model parameters stably and efficiently in order to uncover the underlying causes for population dynamics for practical needs.

Our work focuses on the simultaneous recovery of running cost and interaction energy in the MFG equations from a finite number of boundary measurements of population profile and boundary movement.  To achieve this goal, we formalize the inverse problem as a constrained optimization problem of a least squares residual functional under suitable norms 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 Samy W. Fung (Colorado School of Mines), Siting Liu (UCR), Levon Nurbekyan (Emory University), and Stanley J. Osher (UCLA).

In 1966, Mark Kac asked the famous question “Can one hear the shape of a drum?”
In his article with this question as the title, he translated it into eigenvalue problems for planar domains.
This question highlighted the relationship between eigenvalues and geometry.
One can then ask how eigenvalues are related to the geometry of the boundary.
In this talk, we consider a special type of eigenvalues, called Steklov eigenvalues, that are closely tied to boundary geometry.
We will introduce Steklov eigenvalues and explain their basic background and applications.
Then we will discuss our recent results on inequalities relating Steklov eigenvalues to the boundary area of compact manifolds.