Fall 2022
Monday 4.15  5.15 pm
Room 2147
Scheduled virtual talks will be held on Zoom, Monday 4:155:15 pm.
A link to a Zoom classroom will appear here!!
Schedule

September 12
Room: 2147
Guilherme Silva
Universidade de São Paulo (ICMC  USP)Universality for a class of statistics of Hermitian random matrices and the integrodifferential Painlevé II equation.
Abstract:
It has been known since the 1990s that fluctuations of eigenvalues of random matrices, when appropriately scaled and in the sense of onepoint distribution, converge to the Airy2 point process in the large matrix limit. In turn, the latter can be described by the celebrated TracyWidom distribution.
In this talk we discuss recent findings of Ghosal and myself, showing that certain statistics of eigenvalues also converge universality to appropriate statistics of the Airy2 point process, interpolating between a hard and soft edge of eigenvalues. Such found statistics connect also to the integrodifferential Painlevé II equation, in analogy with the celebrated TracyWidom connection between Painlevé II and the Airy2 process. 
September 19
*** Special Seminar Starting at 3pm! ***
Room: 2132
Matteo Mucciconi University of Warwick
A bijective approach to solvable KPZ models
Abstract: Explicit solutions of random growth models in the KPZ universality class have attracted, in the last two decades, significant attention in Mathematical Physics. A common approach to the problem, explored in the last 15 years, leverages remarkable relations between the KPZ equation and quantum integrable systems.
Here, I will introduce a new approach to the solutions of KPZ models, based on a bijection discovered by Imamura, Sasamoto and myself last year. This is a generalization of the celebrated RobinsonSchenstedKnuth correspondence relating at once 1) solvable growth models, 2) determinantal point processes of free fermionic origin and 3) models of Last Passage Percolation on a cylinder.
I will enumerate some of the early applications of this new approach and I will give an overview of the technical tools needed, that include Kashiwara's crystals or the inverse scattering method for solitonic systems. 
September 26
Room: 2147
Abstract:

October 3
Room: 2147
Alexandros Eskenazis CNRS
Learning lowdegree functions on the discrete hypercube
Abstract: Let f be an unknown function on the ndimensional discrete hypercube. How many values of f do we need in order to approximately reconstruct the function? In this talk we shall discuss the random query model for this fundamental problem from computational learning theory. We will explain a newly discovered connection with a family of polynomial inequalities going back to Littlewood (1930) which will in turn allow us to derive sharper estimates for the the query complexity of this model, exponentially improving those which follow from the classical LowDegree Algorithm of Linial, Mansour and Nisan (1989). Time permitting, we will also show a matching informationtheoretic lower bound. Based on joint works with Paata Ivanisvili (UC Irvine) and Lauritz Streck (Cambridge).

October 10
Room: 2147
National Indigenous Peoples Day.
Abstract: Indigenous Peoples' Day is a holiday in the United States that celebrates and honors Native American peoples and commemorates their histories and cultures. On October 8, 2021, U.S. President Joe Biden became the first U.S. President to formally recognize the holiday,[2] by signing a presidential proclamation declaring October 11, 2021, to be a national holiday.[3] It is celebrated across the United States on the second Monday in October, and is an official city and state holiday in various localities. It began as a countercelebration held on the same day as the U.S. federal holiday of Columbus Day, which honors Genoveseborn explorer Christopher Columbus. Some people reject celebrating him, saying that he represents "the violent history of the colonization in the Western Hemisphere".[4] Indigenous People’s Day was instituted in Berkeley, California, in 1992, to coincide with the 500th anniversary of the arrival of Columbus in the Americas on October 12, 1492. Two years later, Santa Cruz, California, instituted the holiday.[5] Starting in 2014, many other cities and states adopted the holiday.[6]

October 17
Room: 2147
Guillaume Remy
(IAS, Princeton)Modular transformation of conformal blocks via Liouville CFT
Abstract: Conformal blocks are objects of fundamental importance in mathematical physics. They are a key input to the conformal bootstrap program to exactly solve 2D conformal field theory (CFT) and are related to 4D supersymmetric gauge theory through the AldayGaiottoTachikawa correspondence. They are typically defined as power series via the representation theory of the Virasoro algebra but in this talk we will provide novel probabilistic expressions using the Gaussian free field. This will allow us to obtain many analytic properties such as modular transformations. Our methods are based on recent developments in the probabilistic construction of the Liouville CFT, a theory first introduced to describe random surfaces by A. Polyakov in the context of string theory. Based on joint work with Promit Ghosal, Xin Sun, and Yi Sun.

October 24
Room: 2147
Hao Shen
University of WisconsinMadisonStochastic YangMills process in 2D and 3D.
Abstract: We will discuss stochastic quantization of the YangMills model on two and three dimensional torus. In stochastic quantization we consider the Langevin dynamic for the YangMills model which is described by a stochastic PDE. We construct local solution to this SPDE and prove that the solution has a gauge invariant property in law, which then defines a Markov process on the space of gauge orbits. We will also describe the construction of this orbit space, on which we have welldefined holonomies and Wilson loop observables. Based on joint work with Ajay Chandra, Ilya Chevyrev, and Martin Hairer.

October 31
Room: 2147
Robert Hough
Stony Brook UniversityCovering systems of congruences
Abstract:
A distinct covering system of congruences is a list of congruences \[ a_i \bmod m_i, \qquad i = 1, 2, ..., k \] whose union is the integers. Erdős asked if the least modulus $m_1$ of a distinct covering system of congruences can be arbitrarily large ( the minimum modulus problem for covering systems, $1000) and if there exist distinct covering systems of congruences all of whose moduli are odd (the odd problem for covering systems, $25). I'll discuss my proof of a negative answer to the minimum modulus problem, and a quantitative refinement with Pace Nielsen that proves that any distinct covering system of congruences has a modulus divisible by either 2 or 3. The proofs use the probabilistic method and in particular use a sequence of pseudorandom probability measures adapted to the covering process. Time permitting, I may briefly discuss a reformulation of our method due to Balister, Bollobás, Morris, Sahasrabudhe and Tiba which solves a conjecture of Shinzel (any distinct covering system of congruences has one modulus that divides another) and gives a negative answer to the squarefree version of the odd problem.

November 7
Room: 2147
Infinite cycles in the interchange process in five dimensions
Abstract: In the interchange process on a graph G=(V,E), there is a particle on each vertex of the graph and an independent Poisson clock on each one of the edges. Once the clock of an edge rings, the two particles on the two sides of the edge switch. After time t, the particles are permuted according to a random permutation \pi_t:V\to V. A famous conjecture of Balint Toth states that the following holds when G=\mathbb Z^d :
(1) If d=2, then the permutation \pi_t contains only finite cycles for all t>0.
(2) If d\ge 3, then there exists t_c such that for tt_c, \pi_t contains infinite cycles.
We prove the existence of infinite cycles for all d\ge 5 and all t sufficiently large. To this end, we study the cyclic time random walk obtained by exposing the cycle of the origin in \pi_t. We show that this walk is diffusive using a multiscale inductive argument.
This is a joint work with Allan Sly. 
November 14
Room: 2147
Sven Wang
MITMinimax density estimation via measure transport
Abstract: This talk is based on the preprint (joint with Y. Marzouk, MIT) https://arxiv.org/abs/2207.10231.
A natural way to represent complex probability measures is to couple them with a known reference distribution via some transport map. This principle has been used in a number of popular recent machine learning methods for highdimensional inference and generative modeling. In this talk, we study nonparametric density estimators which arise from measure transport. Given a smooth reference distribution and random samples from the unknown density, the estimators are given as the pushforward of a transport map minimizing the empirical KullbackLeibler risk, or a penalized version thereof. For triangular KnotheRosenblatt transports on the ddimensional unit cube, it is shown that both the nonpenalized and penalized versions achieve minimax optimal convergence rates over $H\"older$ classes of densities. The same is proved for sieved wavelet estimators. Our results follow from a concentration inequality for general penalized measuretransport estimators, proved using techniques from Mestimation developed in the late 90's. The minimax rates for triangular maps are then derived using natural anisotropic regularity estimates which we study. 
November 21
Room: 2147
Emma Bailey
City University of New York (CUNY)Large deviation estimates for Selberg’s central limit theorem and applications
Abstract: Selberg’s central limit theorem gives that the logarithm of the Riemann zeta function taken at a uniformly drawn height in $[T, 2T]$ behaves as a complex centered Gaussian random variable with variance $\log\log T$. A natural question is to investigate how far the Gaussian decay persists. We present results on the right tail for the real part of the logarithm, where the absolute value of zeta is `unusually large’, on the scale of the exponential of the variance. Our proof employs a recursive scheme of Arguin, Bourgade and Radziwi{\l}{\l} to inductively work with the logarithm of zeta, interpreted as a random walk. The result is in agreement with the corresponding (known) random matrix result, under the usual dictionary, and has a number of corollaries. This work is joint with LouisPierre Arguin.

November 28
Room: 2147
Sayan Das
Columbia UniversitySuperdiffusivity and localization of continuous polymers
Abstract: We study the Continuum Directed Random Polymer (CDRP) which arises as a universal scaling limit of discrete directed polymers. In this talk, I will present some of the recent progress in understanding the geometry of the CDRP. In particular, I will show CDRP exhibits pointwise localization and pathwise tightness under 2/3 scaling, confirming the existing predictions in polymer literature under continuous setting. I will explain how our results also shed light on certain properties of the KPZ equation (free energy of the CDRP) such as ergodicity and limiting Bessel behaviors around the maximum. This is based on two joint works with Weitao Zhu.

December 5
Room: 2147
Changji Xu
Harvard UniversitySpectral gap estimates for mixed pspin models at high temperature
Abstract: We consider general mixed pspin mean field spin glass models and provide a method to prove that the spectral gap of the Dirichlet form corresponding to Glauber dynamics is of order one at sufficiently high temperature. Our proof is based on an iteration scheme relating the spectral gap of the Nspin system to that of suitably conditioned subsystems.

December 12
Room: 2147
Hoi Nguyen
The Ohio State UniversityOn roots of random trigonometric polynomials and related models
Abstract: In this talk we will discuss various basic statistics of the number of real roots of random trigonometric polynomials, as well as the minimum modulus and the nearest roots statistics to the unit circle of Kac polynomials. We emphasize the universality aspects of all these problems.
Based on joint works with Cook, Do, O. Nguyen, Yakir and Zeitouni