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Organizing Committee
Georgia Tech
University of Washington
Brown University
Brown University
Princeton University
Kent State University
Abstract
This workshop will explore the rich and rapidly evolving interface between high-dimensional geometry, probability, and computational methods. Recent years have witnessed major break-throughs, bringing tools from stochastic processes to resolve long-standing problems in geometry, and using geometry to investigate the limits of computation and efficiency of sampling algorithms.
The workshop will bring together researchers from the communities of geometry, probability, and computational complexity to promote further synergy and continue the progress on the major open problems of the field.
The workshop will bring together researchers from the communities of geometry, probability, and computational complexity to promote further synergy and continue the progress on the major open problems of the field.
This is a satellite event of the International Congress of Mathematicians.
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Workshop Schedule
Friday, July 31, 2026
8:30 - 8:50 AM EDT
Check In
11th Floor Collaborative Space
8:50 - 9:00 AM EDT
Welcome
11th Floor Lecture Hall
9:00 - 9:45 AM EDT
Column subset selection in sparse matrices
11th Floor Lecture Hall
Speaker
Konstantin Tikhomirov, Carnegie Mellon University
Session Chair
Orli Herscovici, St. John’s University
Abstract
Let k=o(n) and let S be a k by n deterministic matrix with d non-zero elements in every column. We prove under very mild assumptions on S that a uniform random selection of k/2 columns in S produces a poorly invertible submatrix. Joint work with Han Huang and Mark Rudelson.
10:00 - 10:30 AM EDT
Coffee Break
11th Floor Collaborative Space
10:30 - 11:15 AM EDT
Low complexity smoothed analysis
11th Floor Lecture Hall
Speaker
Mark Rudelson, University of Michigan
Session Chair
Orli Herscovici, St. John’s University
Abstract
Perturbing a deterministic ndimensional matrix with small Gaussian noise is a cornerstone of smoothed analysis of algorithms. The idea of adding the noise to a given deterministic matrix was introduced by Spielman and Teng in 2004. This noise reduces the condition number of the input matrix to O(n) and with it the complexity of many matrix algorithms. Moreover, it is oblivious to the input as it does not require any prior knowledge of the matrix besides a crude bound on its norm.
However, when deployed algorithmically, this method is expensive due to the cost of generating and storing n^2 Gaussian random variables. We will describe a random matrix construction which requires generating O(n) independent random variables and reduces the condition number of any deterministic matrix to O(n), thus matching Gaussian perturbations. Using so few independent variables requires generating a random matrix with dependent entries thus making the condition number analysis more involved.
Joint work with Shabarish Cenakkod, Michal Derezinski, and Xiaoyu Dong.
11:30 AM - 12:15 PM EDT
Illumination Bodies
11th Floor Lecture Hall
Virtual Speaker
Elisabeth Werner, Case Western Reserve University
Session Chair
Orli Herscovici, St. John’s University
Abstract
In Euclidean space, a derivative of volume involving illumination bodies leads to an affine invariant for convex bodies, the affine surface area.
We explore illumination bodies in various other settings to obtain a notion corresponding to affine surface area, namely in spaces of constant curvature, in Riemannian manifolds and the setting of ball convex bodies.
Based on joint works with R. Assouline, F. Besau, B. Oliva, C. Schuett and D. Yalikun.
12:30 - 12:35 PM EDT
Group Photo (Immediately After Talk)
11th Floor Lecture Hall
12:35 - 2:30 PM EDT
Lunch/Free Time
2:30 - 3:15 PM EDT
The Gaussian Conjugate Rogers-Shephard Inequality
11th Floor Lecture Hall
Speaker
Emanuel Milman, Technion
Session Chair
Vladyslav Yaskin, University of Alberta
Abstract
We fuse between the Rogers-Shephard inequality for the Lebesgue measure and Royen's Gaussian Correlation Inequality, simultaneously extending both into a single sharp inequality for the Gaussian measure \gamma on R^n, stating that \gamma(K) \gamma(L) <= \gamma(K \cap L) \gamma(K+L) whenever K and L are origin-symmetric convex sets. This confirms a conjecture of M. Tehranchi. In fact, we show that the inequality remains valid whenever the Gaussian barycenters of K and L are at the origin, and characterize the equality cases.
Our main tool is a new Gaussian Forward-Reverse Brascamp-Lieb inequality for centered log-concave functions, of independent interest, which is crucially applicable to degenerate Gaussian covariances.
Joint work with Shohei Nakamura and Hiroshi Tsuji.
3:30 - 4:00 PM EDT
Coffee Break
11th Floor Collaborative Space
4:00 - 4:45 PM EDT
On bodies with planar symmetric projections.
11th Floor Lecture Hall
Speaker
Dmitry Ryabogin, Kent State University
Session Chair
Vladyslav Yaskin, University of Alberta
Abstract
This is a joint work with Christos Saroglou and Serhii Myroshnychenko.
Let $n\ge 3$ and let \(K\subset\mathbb R^n\) be a convex body. For a two-dimensional linear subspace
\(P\subset\mathbb R^n\), let \(K|P\) be the orthogonal projection of \(K\) onto \(P\).
We prove that if for every two-dimensional subspace \(P\), the planar
convex body \(K|P\) has \(q\)-fold rotational symmetry up to translation, then
for \(q\ge4\), this forces \(K\) to be an Euclidean ball. The case
\(q=3\) is exceptional: non-spherical examples exist.
5:00 - 6:30 PM EDT
Reception
11th Floor Collaborative Space
Saturday, August 01, 2026
9:00 - 9:45 AM EDT
Duality revisited once again
11th Floor Lecture Hall
Speaker
Shiri Artstein-Avidan, Tel Aviv University
Session Chair
Artem Zvavitch, Kent State University
Abstract
We provide some new perspective on duality for bodies and functions.
10:00 - 10:30 AM EDT
Coffee Break
11th Floor Collaborative Space
10:30 - 10:35 AM EDT
Dimension-free Gaussian tail estimates for linear functionals on convex bodies
Lightning Talks - 11th Floor Lecture Hall
Speaker
Brayden Letwin, University of Washington
Session Chair
Artem Zvavitch, Kent State University
Abstract
Vitali Milman posed the following question: given a convex body $K$, does there exist a unit vector $u$ such that a random vector $X$ uniformly distributed on $K$ satisfies that its projection onto $u$ has a dimension-free subgaussian constant? In this talk, we discuss the recent affirmative resolution of this question. This is joint work with Dan Mikulincer.
10:35 - 10:40 AM EDT
A Blob Method for Kalman-Wasserstein Gradient Flows
Lightning Talks - 11th Floor Lecture Hall
Speaker
Ritvik Teegavarapu, Brown University
Session Chair
Artem Zvavitch, Kent State University
Abstract
Sampling from a target distribution is a fundamental problem in applied mathematics, arising naturally in Bayesian inference and uncertainty quantification. Classical methods such as Markov Chain Monte Carlo can converge prohibitively slowly for complex, high-dimensional targets, and typically require access to gradient information that may be unavailable when the forward model is a black-box numerical solver. The Ensemble Kalman Sampler (EKS) addresses both difficulties, as it is a gradient-free interacting particle
algorithm in which particles evolve under an empirical covariance-preconditioned force, making the dynamics affine invariant and well-adapted to the geometry of the target. In the mean field limit, the empirical distribution of the EKS converges formally to a nonlinear PDE, which can be interpreted as a gradient flow in the Kalman-Wasserstein metric. However, rigorous justification of this long-time limit has so far been confined to Gaussian or near-Gaussian targets, a restriction that precludes the majority of practical applications.
We develop a regularized variant of the EKS mean-field PDE, obtained by replacing the potential and entropy terms with mollified counterparts parameterized by a smoothing scale. This covariance-modulated blob flow inherits the gradient flow structure of EKS while admitting a rigorous analysis for general targets. We prove the existence of weak solutions via a covariance-modulated JKO variational scheme, characterize the steady states and their bias, and prove the convergence of the mollified solutions to the EKS mean-field PDE as the smoothing scale goes to 0. Numerical experiments on benchmark targets confirm that the resulting particle algorithm is competitive with EKS in practice.
10:40 - 10:45 AM EDT
Information Computation and Quantization: Infinitely divisible privacy and beyond
Lightning Talks - 11th Floor Lecture Hall
Speaker
Aaradhya Pandey, Princeton University
Session Chair
Artem Zvavitch, Kent State University
Abstract
In this talk, I introduce a hypothesis-testing-based formulation of differential privacy in classical computation. The Gaussian Differential Privacy framework of Dong--Roth--Su (2022) established a central limit theorem for the composition of multiple private mechanisms. Building on this work, I present a Poisson extension of their result and show that both the Gaussian and Poisson limits are unified under a broader framework of infinitely divisible privacy. This perspective reveals structural connections between differential privacy, probability theory, statistics, and discrete mathematics. I conclude by discussing computational differential privacy and its connections to cryptographic constructions such as pseudorandom generators, as well as a step toward quantizing privacy in quantum computation, outlining a framework for both computational and quantum differential privacy.
10:45 - 10:50 AM EDT
An improved l_0 isoperimetric inequality via canonical paths
Lightning Talks - 11th Floor Lecture Hall
Speaker
Manuel Fernandez, Georgia Institute of Technology
Session Chair
Artem Zvavitch, Kent State University
Abstract
The l_0 boundary of a set S is a notion of boundary that captures all points that are a single coordinate change from being in S. The quantity has received recent attention by computer scientists, due to its relevance in the bounding of the mixing time of the Gibbs sampler on convex bodies. In this talk we will give new upper and lower bounds on this quantity. The proof of our main result, the lower bound, is based on an implementation of “canonical paths”, a tool for graph isoperimetric inequalities, in the continuous setting. We will sketch their construction and how they imply the lower bound.
10:50 - 10:55 AM EDT
Rigidity in the Planar Ulam Floating Body Problem
Lightning Talks - 11th Floor Lecture Hall
Speaker
Oleg Asipchuk, University of Cincinnati
Session Chair
Artem Zvavitch, Kent State University
Abstract
In this talk, I will discuss the two-dimensional Ulam floating body problem for convex domains at several rational perimetral densities. Using the framework of Zindler carousels, I will show how the floating condition can be reformulated as a system of ordinary differential equations associated with inscribed polygons. This dynamical systems approach provides a powerful tool for analyzing floating bodies and, for certain perimetral densities, leads to rigidity results showing that the disk is the unique convex domain that floats in equilibrium in every orientation. This is joint work with M. Kosmakov and P. Zatitskii.
10:55 - 11:00 AM EDT
Root Dynamics of Differentiated Polynomials with Rotationally Invariant Structure
Lightning Talks - 11th Floor Lecture Hall
Speaker
Truong Vu, UCLA
Session Chair
Artem Zvavitch, Kent State University
Abstract
We associate to an $N$-sample of a given rotationally invariant probability measure $\mu_0$ with compact support in the complex plane, a polynomial $P_N$ with roots given by the sample. Then, for $t \in (0,1)$, we consider the empirical measure $\mu_t^{N}$ associated to the root set of the $\lfloor t N\rfloor$-th derivative of $P_N$.
Under suitable conditions of regularity on $\mu_0$ and for a specific family of $N$-samplings, we show that $\mu_t^{N}$ converges to a rotationally invariant probability measure $\mu_t$ when $N$ tends to infinity, and that $\mu_t$ has a radial density satisfying a PDEs.
11:00 - 11:05 AM EDT
LDP for Tensor Forms
Lightning Talks - 11th Floor Lecture Hall
Speaker
Reihaneh Malekian, Columbia University
Session Chair
Artem Zvavitch, Kent State University
Abstract
In this talk, we study a large deviation principle (LDP) for a tensor-weighted functional of i.i.d. random variables, when the sequence of tensors converges under a variant of the \enquote{bad} cut norm. By studying an LDP in this topology, we are able to improve conditions under which the LDP holds in applications of interest. As another application, we analyze a Gibbs measure with a tensor-valued Hamiltonian, and characterize the optimizers of the limiting variational problem in terms of a functional fixed point equation. We focus on several concrete examples, which include monochromatic subgraph counts in inhomogeneous random graphs, Erd\H{o}s-R\'enyi hypergraphs, and a generalized Potts statistic of order $v\ge 2$. Studying the optimization problem, we give sufficient conditions for uniqueness of the optimizer, as well as for existence of constant optimizers (replica symmetry). Our results demonstrate universal weak laws for a large class of tensor Gibbs models with approximately regular tensors.
11:05 - 11:10 AM EDT
Limiting dynamics of non-reciprocally coupled spherical spin glasses
Lightning Talks - 11th Floor Lecture Hall
Speaker
Ratul Biswas, National Institute for Theory and Mathematics in Biology
Session Chair
Artem Zvavitch, Kent State University
Abstract
In this talk, I will present ongoing work on Langevin dynamics for two non-reciprocally coupled spherical-spin disordered mean-field models. I will show that, as the system size tends to infinity, the empirical state correlations and integrated response functions converge almost surely, uniformly on compact time intervals, to the unique non-random strong solution of an explicit system of nonlinear integro-differential equations. These equations are closely related to those introduced by Cugliandolo and Kurchan and later rigorously analyzed, in the single-system setting, by Guionnet, Ben Arous, Dembo, and Mazza.
11:10 - 11:15 AM EDT
The Convexity of Radial Mean Bodies
Lightning Talks - 11th Floor Lecture Hall
Speaker
Dylan Langharst, Carnegie Mellon University
Session Chair
Artem Zvavitch, Kent State University
Abstract
In this work, we extend a classical theorem of Keith Ball on integrals of log-concave functions along rays against the weight r^{p-1}, originally for p>0, to the previously inaccessible regime p in (-1,0). As a consequence of this extension, we resolve a nearly 30-year-old question of Richard Gardner and Gaoyong Zhang in the affirmative, proving that the radial pth mean bodies R_p K of a convex body K, which they introduced in 1998, are convex for all p>-1.
11:15 - 11:20 AM EDT
Multidimensional Gaussian Hypercontractivity
Lightning Talks - 11th Floor Lecture Hall
Speaker
PAVLOS KALANTZOPOULOS, University of Waterloo
Session Chair
Artem Zvavitch, Kent State University
Abstract
We study T_z-hypercontractivity for n functions, where T_z denotes the noise operator (z in R) or the Mehler transform (z in C). For jointly Gaussian random vectors xi_j in R^{k_j}, each standard normal but possibly correlated, we characterize inequalities of the form
E F(B(|T_{z_1} f_1(xi_1)|,...,|T_{z_n} f_n(xi_n)|))
<=
F(E B(|f_1(xi_1)|,...,|f_n(xi_n)|))
via a covariance-weighted matrix positivity condition involving derivatives of F and B. The proof is based on the Beckner--Janson interpolation flow, extended to a multidimensional setting. Consequences include multidimensional versions of the sharp Hausdorff--Young and log-Sobolev inequalities, as well as noisy Gaussian Jensen and Gaussian Brascamp--Lieb inequalities.
Joint work with Paata Ivanisvili.
11:20 - 11:25 AM EDT
Stochastic compression with and without permutations
Lightning Talks - 11th Floor Lecture Hall
Speaker
Anton Kutsenko, University of Hamburg
Session Chair
Artem Zvavitch, Kent State University
Abstract
We consider a simplified mass redistribution scheme, with and without permutations, that simulates stochastic compression under high and low disorder. For this case, surprisingly, the limit density distributions can be computed explicitly, and neat formulas are provided. As the dimension of space increases, the distinction between structured and turbulent compression becomes less pronounced. Already in dimension two, the density distributions are barely distinguishable.
11:25 - 11:30 AM EDT
Dirichlet heat kernel estimates on unbounded domains with finite inradius
Lightning Talks - 11th Floor Lecture Hall
Speaker
Brian Chao, Cornell University
Session Chair
Artem Zvavitch, Kent State University
Abstract
We study Brownian motion and Dirichlet heat kernels in a class of unbounded domains in Euclidean space with finite inradius. These domains have a positive first Dirichlet eigenvalue, and Brownian motion started anywhere in the domain has finite expected lifetime until hitting the boundary. We obtain sharp two-sided estimates for the Dirichlet heat kernel in this setting. A central ingredient is the construction and estimation of the ground state eigenfunction of the Dirichlet Laplacian. Joint work with Phanuel Mariano, Laurent Saloff-Coste, and Jing Wang.
11:30 - 11:35 AM EDT
Stolarsky-Type Inequalities and a Mixed-Alphabet Sumset Application
Lightning Talks - 11th Floor Lecture Hall
Speaker
Johannes Hosle, Massachusetts Institute of Technology
Session Chair
Artem Zvavitch, Kent State University
Abstract
For $m \in \mathbb{N}$, let $q_m := \frac{\log(2m+1)}{2\log(m+1)}$. The max-convolution inequality $$ \sum_{k=0}^{2m}\left(\max_{i+j=k} x_i y_j \right)^{q_m} \ge \left(\sum_{i=0}^{m} x_i\right)^{q_m} \left(\sum_{j=0}^{m} y_j\right)^{q_m}$$ for arbitrary sequences $x_0 \ge x_1 \ge ... \ge x_m \ge 0, y_0 \ge y_1 \ge ... \ge y_m \ge 0$ implies an affirmative answer to a question of Bourgain, Dilworth, Ford, Konyagin, and Kutzarova on the sizes of sumsets in product sets. This inequality was proven for $m = 2$ by Becker, Ivanisvili, Krachun, and Madrid by reducing the general case to the geometric block case via a max-tie analysis. We prove the geometric block case $x = (1, t, ..., t^{r}, 0, ..., 0)$ and $y = (1, t, ..., t^s, 0, ..., 0)$, $t \in [0, 1]$, for all $m \in \mathbb{N}$ via a comparison of Stolarsky means. Additionally, we prove the above inequality when one sequence has only two non-zero terms. We then characterize exactly when the endpoint $t = 1$ determines the optimal exponent in the geometric block inequalities and use this to a derive a sharp mixed-alphabet sumset bound for binary and $m$-ary product sets. Based on a solo work and a joint work with Paata Ivanisvili.
11:35 - 11:40 AM EDT
Fast Mixing of Glauber Dynamics on High Overlap Gibbs Measures
Lightning Talks - 11th Floor Lecture Hall
Speaker
Almut Rödder, ETH Zürich
Session Chair
Artem Zvavitch, Kent State University
Abstract
We give a general condition under which a sufficiently strong external field guarantees fast mixing of Glauber dynamics for quadratic Gibbs measures. More precisely, let $J$ be an interaction matrix and let the external field be $\theta \mathbf{1}_n$. Suppose that the spectral diameter of $J$ is bounded by a constant $C_1$ and that all principal submatrices of size at most $\delta n$ have spectral diameter at most $1/5$. We prove that, for $\theta$ sufficiently large, depending only on $\delta$ and $C_1$, the correlation matrices of all pinnings of the measure have uniformly bounded operator norm. The main ingredient is an overlap condition that allows us to control the correlation matrices uniformly over all pinnings. Using stochastic localization, we then obtain a lower bound on the spectral gap and consequently, polynomial-time mixing of Glauber dynamics. As an application, we consider the Sherrington–Kirkpatrick model, whose interaction matrix is a scaled GOE matrix. The required assumptions hold with high probability for every fixed finite inverse temperature $\beta$. We therefore establish polynomial-time mixing of Glauber dynamics in the presence of a sufficiently strong external field whose strength depends only on $\beta$.
This is joint work with Afonso Bandeira and Ahmed El Alaoui.
11:40 - 11:45 AM EDT
Gamma Anticoncentration and Matrix Slicing: Work in Progress
Lightning Talks - 11th Floor Lecture Hall
Speaker
Maksim Kosmakov, University of Cincinnati
Session Chair
Artem Zvavitch, Kent State University
Abstract
We will discuss work in progress on central hyperplane sections of the real spectraplex and their connection with anticoncentration for signed sums of independent gamma variables. A Wishart representation reduces the matrix problem exactly to a scalar density problem, providing a matrix analogue of probabilistic simplex slicing. We solve the extremal problem for matrix sizes two and three, including the exact maximum, minimum, and equality cases in size three, and formulate a sharp conjecture for general dimension.
11:45 - 11:50 AM EDT
Reverse metric Kwapien-Schuett inequalities and applications
Lightning Talks - 11th Floor Lecture Hall
Speaker
Mira Gordin, Brown University
Session Chair
Artem Zvavitch, Kent State University
Abstract
Functional inequalities acheiving optimal bi-Lipschitz distortion bounds are powerful tools in the Ribe program: formulated by Bourgain in 1985, the program aims to characterize Banach spaces via metric properties only, not linear ones. We present a reverse metric version of the celebrated combinatorial linear inequality of Kwapien-Schuett (1989) which achieves optimal distortion lower bounds for l_1(l_infty) grids. time permitting we will discuss some further applications. (joint work with Alexandros Eskenazis)
11:50 - 11:55 AM EDT
Analysis of an Interacting Particle System with State-Dependent Common and Idiosyncratic Noises
Lightning Talks - 11th Floor Lecture Hall
Speaker
Vinh Nguyen, University of California, Berkeley
Session Chair
Artem Zvavitch, Kent State University
Abstract
We study an interacting particle system driven by common and idiosyncratic noises, where the noise intensity depends on the empirical variance via a nonlinear feedback law. We derive the associated stochastic McKean--Vlasov equation and analyze its long-time behavior, showing the emergence of random consensus induced by the interaction between alignment and state-dependent noise.
11:55 AM - 12:00 PM EDT
Grünbaum and Loewner’s problem on centroids of hyperplane sections
Lightning Talks - 11th Floor Lecture Hall
Speaker
Shouda Wang, princeton university
Session Chair
Artem Zvavitch, Kent State University
Abstract
Let s(n) be the largest integer such that every n-dimensional convex body K has at least s(n) hyperplane sections whose centroid coincides with the centroid of K itself. Loewner posed the problem of determining s(n) for every n, and Grünbaum conjectured that s(n)=n+1 in every dimension. A recent result of Myroshnychenko, Tatarko, and Yaskin showed that s(n)=1 in dimensions 5 and higher, thereby disproving Grünbaum’s conjecture in those dimensions. In this talk, I will present a proof that, in fact, s(n)=1 for every n ≥ 3, thereby completely resolving Grünbaum and Loewner’s problem in all dimensions. This is joint work with Xiong Ge and Kaiwen Yang.
12:00 - 12:05 PM EDT
How abundant are good interpolators?
Lightning Talks - 11th Floor Lecture Hall
Speaker
August Chen, Cornell University
Session Chair
Artem Zvavitch, Kent State University
Abstract
We consider the fundamental problem of correctly classifying labelled Gaussian data to a fixed margin in the overparametrized regime. Under two natural data-generating distributions, we establish a large deviation principle on the event that a uniformly random classifier achieves a given generalization error, with high probability over the data. The associated large deviation rate function describes the proportion, at the exponential scale in dimension, of interpolating classifiers having a given desired performance. As a consequence, we establish the following concentration phenomenon: all but an exponentially small fraction of interpolating classifiers have approximately the same generalization performance given by the unique maximizer of this rate function.
We numerically compare this maximizer to the performance of empirical risk minimization by gradient descent and a natural linear program, both finding an interpolating classifier. We deduce that in the overparametrized regime, these efficient procedures outperform the vast majority of interpolators, pointing to their nontrivial benign overfitting in this setting.
12:05 - 12:10 PM EDT
Finite Dissociability: A Geometric Route to Well-Posed Particle Systems on Sparse Networks
Lightning Talks - 11th Floor Lecture Hall
Speaker
Rounak Ray, Brown University
Session Chair
Artem Zvavitch, Kent State University
Abstract
Interacting particle systems on large sparse networks are governed by the network's local geometry: via local weak convergence, the empirical measure of the dynamics converges to the law of the root of a limiting, typically infinite, rooted graph—provided the dynamics on that limit is well-posed. Unlike the finite setting, well-posedness can fail on infinite graphs, where influence may propagate across unboundedly many vertices in bounded time. A sufficient geometric condition ruling this out is that the limit be finitely dissociable, a degree-dependent percolation-type property. We give a sufficient criterion for branching processes with general type space, and deduce that finite-type Galton–Watson trees, the Pólya point tree, and a class of random connection models on $\mathbb{R}^d$ are finitely dissociable—justifying the hydrodynamic limit for canonical sparse random graph models. Conversely, branching processes with infinite branching number are not finitely dissociable.
12:15 - 2:00 PM EDT
Lunch/Free Time
2:00 - 2:45 PM EDT
Sampling from Gibbs measures via diffusion processes
11th Floor Lecture Hall
Speaker
Andrea Montanari, Stanford University
Session Chair
Dan Mikulincer, University of Washington
Abstract
Sampling from high-dimensional distributions is a notoriously difficult problem, especially when the distribution isn't logconcave or has multiple modes. While Markov Chain Monte Carlo is a powerful approach, new tools are much needed. I will present a different class of algorithms related to diffusions method in generative AI, and to the stochastic localization technique in probability theory.
I will use this approach to establish new sampling guarantees in a problems in Bayesian estimation and statistical physics.
3:00 - 3:45 PM EDT
A free-probabilistic state evolution
11th Floor Lecture Hall
Speaker
Ahmed El Alaoui, Cornell University
Session Chair
Dan Mikulincer, University of Washington
Abstract
We propose an iterative algorithm taking as input a random matrix-valued linear operator, and characterize its behavior in terms of a limiting correlated Gaussian/semicircular process in a free probability space. We apply this iteration to a random version of the matrix discrepancy problem where given independent GOE matrices, one is asked to find signings so that the signed sum has as small of an operator norm as possible.
4:00 - 4:15 PM EDT
Coffee Break
11th Floor Collaborative Space
4:15 - 5:00 PM EDT
Stahl's theorem and random matrices
11th Floor Lecture Hall
Speaker
Joel Tropp, California Institute of Technology
Session Chair
Dan Mikulincer, University of Washington
Abstract
Stahl's theorem, formerly the BMV Conjecture, is a beautiful and deep fact about the monotonicity properties of the trace exponential function. This talk describes some elegant new applications of Stahl's theorem in the study of random matrices. Preprints: arXiv:2501.16578 and arXiv:2603.04365.
Sunday, August 02, 2026
9:00 - 9:45 AM EDT
Four-function theorems, log-supermodularity, and entropy power inequalities
11th Floor Lecture Hall
Speaker
Mokshay Madiman, University of Delaware
Session Chair
Rachel Ward, University of Texas at Austin
Abstract
We show log-supermodularity of densities on a Euclidean space is preserved under convolution with log-concave product densities, confirming a generalized form of a conjecture of Zartash and Robeva. Additionally, this stability yields a "conditional entropy power inequality" for log-supermodular random vectors. The proof relies on a known continuous form of the Ahlswede-Daykin four-function theorem, for which we give a new proof. The talk is based on joint work with James Melbourne and Cyril Roberto.
10:00 - 10:45 AM EDT
On estimating the mean width of a convex body
11th Floor Lecture Hall
Speaker
Grigoris Paouris, Texas A&M University
Session Chair
Rachel Ward, University of Texas at Austin
Abstract
The Gaussian width of a set is a fundamental quantity in high-dimensional phenomena. I will present some new bounds for the width as well as for the quermassintegrals of a convex body. This is joint work with Reese Pathak.
11:00 - 11:15 AM EDT
Coffee Break
11th Floor Collaborative Space
11:15 AM - 12:00 PM EDT
Finite free information theory: roots of polynomials under differential operators
11th Floor Lecture Hall
Speaker
Jorge Vargas, California Institute of Technology
Session Chair
Rachel Ward, University of Texas at Austin
Abstract
I will discuss recent results at the intersection of free probability and geometry of polynomials. Concretely, we establish finite free analogues of the Stam and entropy power inequalities for real-rooted polynomials, which, in the large-degree limit, provide fundamentally new proofs of their known free probability counterparts.
Equivalently, our results may be interpreted as potential-theoretic inequalities for the zeros of real-rooted polynomials under differential operators that preserve real-rootedness.
This is joint work with Nikhil Srivastava and Zachary Stier.
12:15 - 1:00 PM EDT
Recent Advances in Probabilistically Checkable Proofs
11th Floor Lecture Hall
Speaker
Dor Minzer, Massachusetts Institute of Technology
Session Chair
Rachel Ward, University of Texas at Austin
Abstract
We discuss recent developments in probabilistically checkable proofs (PCPs) that use high-dimensional expansion. We will focus on the construction of low-soundness size efficient PCPs, and their relation to fault-tolerant distributed protocols.
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