I am co-organizing a virtual Probability Seminar with Wei-Kuo Chen and Arnab Sen at University of Minnesota for
the Fall 2022 semester. Please email me to get the Zoom link for the seminar series.
The seminar is held every Friday 3:30-4:30pm (ET) on Zoom (unless otherwise noted below).
09/16/2022 | Gourab Ray (Math & Stat, U Victoria) Special Time: 2:30-3:30pm (ET) |
Title: Conformal invariance of dimers on Riemann surfaces, loops soups, and cycle rooted forests. | |
Abstract: We prove conformal invariance of dimers on general Temperleyan graphs embedded on Riemann surfaces of finite topological type, satisfying a certain invariance principle for simple random walk. I will describe how this problem naturally leads to a scaling limit problem of a two sided loop erased random walk, and how we solve this problem by developing a natural extension of the connection between loop soups and Uniform spanning trees to the setting of multiply connected surfaces. I will also describe certain robust estimates of the mass of large loops in a loop soup on a general graph, which seem to be new even in the simply connected setting. Joint work with N. Berestycki and B. Laslier. | |
09/23/2022 | Will Perkins (UIC/GaTech) |
Title: Potential-weighted connective constants and uniqueness of Gibbs measures. | |
Abstract: Classical gases (or Gibbs point processes) are models of gases or fluids, with particles interacting in the continuum via a density against background Poisson processes. A prominent example is the hard sphere model. The major questions about these models are about phase transitions, and while these models have been studied in mathematics for over 70 years, some of the most fundamental questions remain open. After giving some background on these models, I will describe a new method for proving absence of phase transition (uniqueness of infinite volume Gibbs measures) in the low-density regime of processes interacting via a repulsive pair potential. The method involves defining a new quantity, the "potential-weighted connective constant", and is motivated by techniques from computer science. Joint work with Marcus Michelen. | |
09/30/2022 | Perla Sousi (Cambridge) Special Time: 12:00-1:00pm (ET) |
Title: Phase transition for the late points of random walk. | |
Abstract: Let X be a simple random walk in Z_{n}^{d} with d≥3 and let t_{cov} be the expected time it takes for X to visit all vertices of the torus. In joint work with PrÃ©vost and Rodriguez we study the set L_{α} of points that have not been visited by time αt_{cov} and prove that it exhibits a phase transition: there exists α_{*} so that for all α > α_{*} and all ε>0 there exists a coupling between L_{α} and two i.i.d. Bernoulli sets B^{±} on the torus with parameters n^{-(a±ε)d} with the property that B^{-}⊆ L_{α}⊆ B^{+} with probability tending to 1 as n→∞. When α ≤ α_{*}, we prove that there is no such coupling. | |
10/07/2022 | Nima Anari (Computer Science, Stanford) |
Title: | |
Abstract: | |
10/14/2022 | Subhabrata Sen (Statistics, Harvard) |
Title: | |
Abstract: | |
10/21/2022 | TBA |
Title: | |
Abstract: | |
10/28/2022 | Ohad Feldheim (Math, Hebrew University of Jerusalem) |
Title: | |
Abstract: | |
11/04/2022 | Elliot Paquette (Math & Stat, McGill) |
Title: The homogenization of SGD in high dimensions. | |
Abstract: Stochastic gradient descent (SGD) is one of the most, if not the most, influential optimization algorithms in use today. It is the subject of extensive empirical and theoretical research, principally in justifying its performance at minimizing very large (high dimensional) nonlinear optimization. This talk is about the precise high-dimensional limit behavior (specifically generalization and training dynamics) of SGD in a highdimensional least squares problems. High dimensionality is enforced by a family of resolvent conditions on the data matrix, and data-target pair, which can be viewed as a type of eigenvector delocalization. We show that the trajectory of SGD is quantitively close to the solution of a stochastic differential equation, which we call homogenized SGD, and whose behavior is explicitly solvable using renewal theory and the spectrum of the data. Based on joint works with Courtney Paquette and Kiwon Lee (McGill), and Fabian Pedregosa, Jeffrey Pennington and Ben Adlam (Google Brain). | |
11/11/2022 | TBA |
Title: | |
Abstract: | |
12/02/2022 | Shuta Nakajima (Meiji U) |
Title: | |
Abstract: | |
12/09/2022 | Changji Xu (CMSA, Harvard) |
Title: | |
Abstract: | |