Probability seminar

December 4, 2025: Kevin Ren (Princeton)
Title: Reconstruction of Manifold Distances from Noisy Observations

Abstract: We consider the problem of reconstructing the intrinsic geometry of a manifold from noisy pairwise distance observations. Specifically, let M denote a finite volume, diameter 1, and $d \ge 2$-dimensional manifold and $\mu$ denote the normalized volume measure on M. Suppose $X_1, X_2, \cdots, X_N$ are i.i.d. samples of $\mu$ and we observe noisy-distance random variables $d′(X_j,X_k)$ that are related (in an unknown way) to the true geodesic distances $d(X_j,X_k)$. With mild assumptions on the manifold (bounded curvature and positive injectivity radius) and noisy-distance distributions (their independence and means), we develop a new framework for recovering all true distances between points in a sufficiently dense subsample of M (the denoising problem). Our framework improves on previous work which assumed independent additive noise with known, constant mean and variance. Our key idea is to design a robust Hoeffding-type averaging estimator tailored to the inherent geometric structure of the underlying data; as a result, we are able to recover true distances up to error O(\eps \log \eps^{-1}) using a sample complexity $N \asymp \eps^{-2d-2} \log \eps^{-1}$ and runtime $o(N^3)$. We will explain which properties of a manifold are necessary for the distance recovery, which suggests further extension of our techniques to a broader class of metric probability spaces. Joint work with Charles Fefferman and Jonathan Marty.

November 20, 2025: Brian Hall (Notre Dame)
Title: Random walk approximations to (free) multiplicative Brownian motions

Abstract: Biane’s free multiplicative Brownian motion b_t is the large-N limit of the Brownian motion in the general linear group GL(N;C) and can be viewed as the solution to a free stochastic differential equation driven by a circular Brownian motion. I will consider random walk approximations to b_t, which are discrete approximations to the solution of the SDE. These approximations have the form of a product of steps, each of which is the identity plus a multiple of a circular element. We are able to compute the Brown measure of the model with a fixed number of steps using the linearization method. We are then able to let the number of steps tend to infinity and recover the previously computed Brown measure of b_t itself. 

A key step in the argument is a new freeness result for block elements. In general, matrices with freely independent entries are not freely independent in the ordinary sense but only in the “operator valued” sense . But we show that in some interesting examples, we do obtain freeness in the ordinary sense. We also show that for a fixed number of steps, the empirical eigenvalue distribution of the corresponding matrix model converges to the Brown measure of the free model.

This is joint work with Bruce Driver, Ching Wei Ho, Todd Kemp, Yuriy Nemish, Evangelos Nikitopolous, and Félix Parraud. The talk will be self-contained and have lots of pictures.

November 13, 2025: (**CANCELED: TO BE RESCHEDULED**) Tom Alberts (University of Utah)
Title: Loewner Dynamics for Real Rational Functions and the SLE(0) Process

Abstract: Recently Peltola and Wang introduced the multiple SLE(0) process as the deterministic limit of the random multiple SLE(kappa) curves as kappa goes to zero. They also showed that the limiting curves have important geometric characterizations that are independent of their relation to SLE(kappa) - they are the real locus of real rational functions, and they can be generated by a deterministic Loewner evolution driven by multiple points. We prove that the Loewner evolution is a very special family of commuting SLE(0, rho) processes (with commutation holding in a very strong sense), and use this to directly show that the curves satisfy a geodesic multichord property. We also show that our SLE(0, rho) processes lead to relatively simple solutions for the degenerate versions of the BPZ equations in terms of the poles and critical points of the rational function, and that the dynamics of these poles and critical points come from the Calogero-Moser integrable system. Although our results are purely deterministic they are again motivated by taking limits of probabilistic constructions in conformal field theory. Joint work with Sung-Soo Byun, Nam-Gyu Kang, and Nikolai Makarov.

November 6, 2025: Andres Contreras Hip (UChicago)
Title: Gaussian curvature for LQG surfaces and random planar map

Abstract: Liouville quantum gravity is a canonical model for random surfaces conjectured to be the scaling limit of various planar maps. Given that curvature is a central concept in Riemannian geometry, it is natural to ask whether this can be extended to LQG surfaces. Here, we define the Gaussian curvature for LQG surfaces (despite their low regularity) and study the relations with its discrete counterparts. We conjecture that this definition of Gaussian curvature is the scaling limit of the discrete curvature, we prove that the discrete curvature on the $\epsilon$-CRT map with a Poisson vertex set integrated with a smooth test function is of order $\epsilon^{o(1)},$ and show the convergence of the total discrete curvature on a CRT map cell when scaled by $\epsilon^{1/4}.$ Joint work with E. Gwynne.

October 30, 2025: Rob Webber (UCSD)
Title: How fast is square volume sampling Kaczmarz?

Abstract: Randomized Kaczmarz (RK) is a well-known solver for linear least-squares problems. RK iteratively processes blocks of rows in order to update an approximation to the least-squares solution. Recent work suggests that RK converges rapidly when each block of rows is sampled from the square volume distribution defined by the target matrix. Additionally, there are reports of accelerated convergence when the RK iterates produced in the tail part of the algorithm are averaged together. I will clarify the theoretical convergence guarantees for square volume sampling Kaczmarz both with and without tail-averaging.

October 16, 2025: Yubo Shuai (UCSD)
Title: The site frequency spectrum in population models

Abstract: The site frequency spectrum is a commonly used statistic to summarize the mutational data in a sample from the population. In this talk, we will consider the site frequency spectrum for populations growing exponentially or under spatial constraints. I will also briefly discuss some applications to biological data.

October 2, 2025: Benedikt Stufler (TU Vienna)
Title: Inhomogeneous scaling limits of random supertrees.

Abstract: We discuss recent results on Gibbs partitions and their application to the study of random supertrees and their novel inhomogeneous scaling limits.

September 11, 2025: Ron Nissim (MIT) (**AP&M 7321**)
Title: Area Law for Lattice Yang-Mills at Strong Coupling
Abstract: This talk is based on joint work with Scott Sheffield and Sky Cao on lattice Yang-Mills theory. Yang-Mills theory is the mathematical model for the standard model of particle physics, and the area law is the property of the Yang-Mills model said to explain the physical phenomenon of quark confinement. The lattice Yang-Mills model assigns a random NxN matrix from classical Lie groups such as U(N), SU(N), or SO(N) to each edge of a lattice. An adjustable parameter of the model, beta, sometimes referred to as "inverse temperature" describes the coupling strength of the model. It is generally believed that the lattice Yang-Mills model greatly simplifies when beta is proportional to N and N gets large, and in the N->infinity limit under this scaling, area law is known to hold. Nevertheless, for finite N the area law was only shown for beta < c_d/N for a dimensional constant c_d prior to our work (a regime of beta which gets smaller as N gets large!). In a recent preprint we use a novel surface exploration point of view to increase the range of parameters to beta < c_d independent of N, and in ongoing work we use the dynamical perspective introduced by Shen, Zhu, and Zhu to further improve the regime to  beta < c_d N which is the scaling of the previously mentioned large N limit of the model. Both of these approaches work for any dimension of the lattice, d. Introducing these two approaches to the area law question will be the goal of the talk.

Previous seminar talks from 2024-2025 are found here