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.

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).

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.

^{* Optimization for Machine Learning Seminar}

TBA

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.