# Spring 2021

Seminar time: Thursdays 10:20-11:15am.

January 14, 2021: Roger Van Peski (MIT)

Title: Limits and fluctuations of p-adic random matrix products

Abstract:  Any nonsingular matrix A over the p-adic integers Z_p may, by Smith normal form, be written as U diag(p^{\lambda_1},\ldots,p^{\lambda_n}) V for U, V in GL_n(Z_p), for some integers \lambda_i called singular numbers (also known as invariant factors or elementary divisors). The distributions of singular numbers for random A are quite well-studied, largely because the equivalent data of the abelian p-group coker(A) provides useful models for many random groups in number theory and combinatorics, collectively called Cohen-Lenstra heuristics.

By contrast, in this talk we consider a setting in which the (rescaled) fluctuations of singular numbers converge to continuous probability distributions on R. Namely, we show that the singular numbers of successive products A_k \cdots A_1, where A_i are corners of independent Haar-distributed p-adic matrices, obey a law of large numbers and their fluctuations converge dynamically to independent Brownian motions with k playing the role of time. These asymptotics rely on new exact expressions for distributions of singular numbers of products and corners of random matrices over Q_p in terms of Hall-Littlewood polynomials, which allow us to re-express the above matrix product process as a simple interacting particle system. We also indicate parallels between our results–both exact and asymptotic–and known results on singular values of complex matrices.

January 21, 2021: Vadim Gorin (Wisconsin)

Title: Infinite beta random matrix theory

Abstract: Dyson’s threefold approach suggests to deal with real/complex/quaternion random matrices as beta=1/2/4 instances of beta-ensembles. We complement this approach by the beta=\infty point, whose study reveals a number of previously unnoticed algebraic structures. Our central object is the G\inftyE ensemble, which is a counterpart of the classical Gaussian Orthogonal/Unitary/Symplectic ensembles. We encounter unusual orthogonal polynomials, random walks, and finite free polynomial convolutions.

January 28, 2021: Lisa Sauermann (IAS/MIT)

Title: On polynomials that vanish to high order on most of the hypercube

Abstract:  Motivated by higher vanishing multiplicity generalizations of Alon’s Combinatorial Nullstellensatz and its applications, we study the following problem: for fixed k and n large with respect to k, what is the minimum possible degree of a polynomial P in R[x_1,…,x_n] such that P(0,…,0) is non-zero and such that P has zeroes of multiplicity at least k at all points in {0,1}^n except the origin? For k=1, a classical theorem of Alon and Füredi states that the minimum possible degree of such a polynomial equals n. We solve the problem for all k>1, proving that the answer is n+2k−3.

Joint work with Yuval Wigderson.

February 4, 2021: Charles Bordenave (Institut de Mathématiques de Marseille)

Title: Entropy of processes on infinite trees

Abstract:  We define a natural notion of micro-state entropy associated to a random process on a unimodular random tree. This entropy is closely related to Bowen’s sofic entropy in dynamical systems. It is also connected to factor models on random graphs. We give a formula for this entropy for a large class of processes.

This is a joint work with Agnes Backhausz et Balasz Szegedy.

February 11, 2021: David Sivakoff (OSU)

Title: Neighborhood growth in heterogeneous environments

Abstract: The simplest deterministic growth dynamics on the vertices of Z^d are the bootstrap percolation, or threshold growth models, wherein an initially occupied set is enlarged by iteratively including vertices that have at least r>0 occupied neighbors. Initially occupied vertices are chosen independently with probability p, and we are interested in the time at which the origin is first occupied (which may be infinite). I will survey some of the vast literature on this problem and its generalizations, and discuss recent progress in heterogeneous environments.

February 18, 2021: Leonid Petrov (UVA)

Title: Solvable directed polymer models

Abstract: I will discuss integrable random polymers (based on gamma / inverse gamma or beta distributed weights), and explain their connection to known and new families of symmetric functions. There is some interesting combinatorics which is yet to be discovered.

February 25, 2021: Richard Kenyon (Yale)

Title: The multitiling model

Abstract: The study of random tilings is a cornerstone area of combinatorics and probability. Unfortunately the tiling problem is NP-hard even in quite simple-looking cases. We study a tractable variant, the multitiling model, where we tile a region with high multiplicity. In the limit of large multiplicities we compute the asymptotic growth rate of the number of multitilings: the free energy of the multitiling model. We show that the individual tile densities tend to a Gaussian field with respect to an associated discrete Laplacian. For tilings with translates of a polyomino on Z^2 we find crystallization phenomena (and accompanying phase transitions), and even naturally occurring quasicrystals.

This is joint work with Andrei Pohoata (Yale).

March 4, 2021: Anton Bernshteyn (Georgia Tech)

Title: Lower bounds for difference bases

Abstract: A difference basis with respect to $n$ is a subset $A \subseteq \mathbb{Z}$ such that $A – A \supseteq [n]$. R\'{e}dei and R\'{e}nyi showed that the minimum size of a difference basis with respect to $n$ is $(c+o(1))\sqrt{n}$ for some positive constant $c$. The precise value of $c$ is not known, but some bounds are available, and I will discuss them in this talk. This is joint work with Michael Tait (Villanova University).

March 11, 2021: Hugo Duminil-Copin (IHES)

Title: Emerging symmetries in 2D percolation

Abstract:  A great achievement of physics in the second half of the twentieth century has been the prediction of conformal symmetry of the scaling limit of critical statistical physics systems. Around the turn of the millennium, the mathematical understanding of this fact progressed tremendously in two dimensions with the introduction of the Schramm-Loewner Evolution and the proofs of conformal invariance of the Ising model and dimers. Nevertheless, the understanding is still restricted to very specific models. In this talk, we will gently introduce the notion of conformal invariance of lattice systems by taking the example of percolation models. We will also explain some recent proof of rotational invariance for a large class of such models. This represents an important progress in the direction of proving full conformal invariance.

March 18, 2021: Matthew Kwan (Stanford)

Title: Open

Abstract: Open

March 25, 2021: Gabriel Conant (University of Cambridge)

Title: Quantitative stable arithmetic regularity in arbitrary finite groups

Abstract: In 2011, Malliaris and Shelah showed that “stable” finite graphs admit a strong form of Szemeredi’s regularity lemma with polynomial bounds and no irregular pairs. Here “stable” is a combinatorial property, motivated by model theory, which is defined by omitting so-called “half-graphs”. In 2017, Terry and Wolf developed an analogue of this result for subsets of finite abelian groups, which is based on the notion of arithmetic regularity as invented by Green. Roughly speaking, they showed that any stable subset of a finite abelian group can be efficiently approximated by cosets of a subgroup whose index is bounded exponentially in the approximation error and the stability constant. Around the same time, in joint work with Pillay and Terry, we proved a version of this for arbitrary finite groups using model theoretic techniques, which resulted in stronger qualitative features, but lacked any information about quantitative bounds. In this talk, I will discuss a new effective proof of our result, which yields quantitative bounds for arbitrary finite groups, and improves the bound in Terry & Wolf’s result from exponential to polynomial.

March 25, 2021 (3-3:55pm, joint with Analysis seminar): Yen Do (UVA)

Title:

Abstract:

April 1, 2021: Sivaguru Sritharan (Air Force Institute of Technology, Dayton)

Title: Open

Abstract: Open

April 8, 2021: Marcus Michelen (UIC)

Title: Open

Abstract: Open

April 15, 2021: Michael Tate

Title: TBA

Abstract: TBA

April 22, 2021: Leonardo T. Rolla (Warwick)

Title: TBD

Abstract: TBD

April 29, 2021: Vincent Tassion (ETH)

Title: TBD

Abstract: TBD