Sep 20, 2021
Monday

08:30 AM  09:15 AM


Welcome Tea

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09:15 AM  09:30 AM


Introduction

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
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09:30 AM  10:20 AM


Rare Events in RMT and Spherical Integrals
Alice Guionnet (École Normale Supérieure de Lyon)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
Estimating the probability of rare events, namely large deviations principle, for the spectrum of Wigner matrices with compactly supported entries is a challenge. In this talk we will discuss how the asymptotics of Spherical integrals can be derived and used to study such problems. This is based on recent joint works with Augeri, Belinschi, Huang and Husson.
 Supplements



10:35 AM  11:25 AM


Fluctuations of the Characteristic Polynomial of Random Jacobi Matrices
Fanny Augeri (Weizmann Institute of Science)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
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 Abstract
The characteristic polynomial of the GUE is at the heart of a conjecture from Fyodorov and Simm, where it is crucial to understand the logcorrelated structure of the field induced. As a first step in this direction, we obtain a central limit theorem for the logarithm of the characteristic polynomial of the Gaussian $\beta$ Ensemble. Relying on the tridiagonal representation of such matrix models, we will explain how the second order recursion satisfied by the characteristic polynomial allows us to give a martingale representation of its logarithm, leading to an analysis of its fluctuations. This is a joint work with R. Butez and O. Zeitouni.
 Supplements


11:40 AM  12:30 PM


The Stochastic Airy Operator and an Interesting Eigenvalue Process
Diane Holcomb (Royal Institute of Technology)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video


 Abstract
The Gaussian ensembles, originally introduced by Wigner may be generalized to an npoint ensemble called the betaHermite ensemble. As with the original ensembles we are interested in studying the local behavior of the eigenvalues. At the edges of the ensemble the rescaled eigenvalues converge to the Airy_beta process which for general beta is characterized as the eigenvalues of a certain random differential operator called the stochastic Airy operator (SAO). In this talk I will give a short introduction to the Stochastic Airy Operator and the proof of convergence of the eigenvalues, before introducing another interesting eigenvalue process. This process can be characterized as a limit of eigenvalues of minors of the tridiagonal matrix model associated to the betaHermite ensemble as well as the process formed by the eigenvalues of the SAO under a restriction of the domain. This is joint work with Angelica Gonzalez.
 Supplements



12:30 PM  01:30 PM


Lunch

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01:30 PM  02:10 PM


Random Zoom Rooms

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
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02:10 PM  03:00 PM


Mixing Times for the Simple Exclusion Process with Open Boundaries
Evita Nestoridi (Princeton University)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video


 Abstract
In the simple exclusion process on a finite segment, a particle is allowed to move to the right at rate $p$ and to the left at rate $q$, provided that the selected site is empty. In joint work with Nina Gantert and Domink Schmid, we study mixing times of the symmetric and asymmetric simple exclusion process on the segment where particles are allowed to enter and exit at the endpoints. We consider different regimes depending on $p,q$ as well as on the entering and exiting rates, and show that the process exhibits precutoff and in some special cases even cutoff.
 Supplements



03:00 PM  03:30 PM


Tea

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03:30 PM  05:00 PM


Panel Discussion on Work/Life

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
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Sep 21, 2021
Tuesday

08:40 AM  09:30 AM


Random Zoom Rooms

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
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09:30 AM  10:20 AM


Topics on Log and Coulomb Gases
Sylvia Serfaty (New York University, Courant Institute)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
We are interested in systems of points with Coulomb, logarithmic or more generally Riesz interactions (i.e. inverse powers of the distance). They arise in the study of some random matrix ensembles and socalled betaensembles. We will take a point of view based on the detailed expansion of the interaction energy to describe the microscopic behavior of the systems. In particular a Central Limit Theorem for fluctuations and a Large Deviations Principle for the microscopic point processes will be described. This allows to observe the effect of the temperature as it gets very large or very small, and to connect with crystallization questions. Based on joint works with Etienne Sandier, Nicolas Rougerie, Mircea Petrache, Thomas Leblé, Florent Bekerman and Scott Armstrong. on the statistical mechanics of systems of points with logarithmicor Coulomb interactions. After listing some motivations, we describe the “electric approach"which allows to get concentration results, Central Limit Theorems for fluctuations, and aLarge Deviations Principle expressed in terms of the microscopic state of the system.
 Supplements


10:30 AM  11:30 AM


Participant Presentation Session

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 SLMath: Eisenbud Auditorium, Online/Virtual
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11:30 AM  12:20 PM


Random Matrices in Iterative Linear Solvers: Corruption Removal and Sketching
Liza Rebrova (Princeton University)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
When it is infeasible to solve a large linear system directly by inversion, light and scalable randomized iterative methods can be used instead, such as, Randomized Kaczmarz (RK) algorithm, or Stochastic Gradient Descent (SGD). I will discuss some cases when the questions from the nonasymptotic random matrix theory are inherent for proving convergence guarantees for these methods. This includes QuantileRK and QuantileSGD methods proposed for the systems that might contain large, sparse, potentially adversarial corruptions. Unlike the classical extensions of RK/SGD to noisy (inconsistent) systems, the new methods learn to avoid corruptions rather than tolerate the small noise, and result in exact convergence when up to one half of the equations can be arbitrarily corrupted. The second case is connected to the sketching techniques that considerably speed up the iterative solvers. Based on the joint work with Deanna Needell, Jamie Haddock and Will Swartworth.
 Supplements


12:30 PM  01:30 PM


Lunch

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01:30 PM  02:30 PM


Participant Presentation Session

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
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02:30 PM  03:20 PM


Fredholm Determinant Solutions of the Painlevé II Hierarchy and Gap Probabilities of Determinantal Point Processes
Manuela Girotti (St. Mary's University; Concordia University)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
We study Fredholm determinants of a class of integral operators, whose kernels can be expressed as double contour integrals of a special type. Such Fredholm determinants appear in various random matrix and statistical physics models. We show that the logarithmic derivatives of the Fredholm determinants are directly related to solutions of the Painlevé II hierarchy. This confirms and generalizes a recent conjecture by Le Doussal, Majumdar, and Schehr (2018). In addition, we obtain asymptotics at infinity for the Painlevé transcendents and large gap asymptotics for the corresponding point processes. This is a joint work with Mattia Cafasso (Univ. Angers) and Tom Claeys (UC Louvain).
 Supplements


03:30 PM  04:00 PM


Tea

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04:00 PM  04:50 PM


Principal Components of Spiked Covariance Matrices
Ke Wang (Hong Kong University of Science and Technology)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
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 Abstract
Computing the eigenvalues and eigenvectors of a large matrix is a basic task in high dimensional data analysis with many applications in computer science and statistics. In practice, however, data is often perturbed by noise. In this talk, we will focus on the spiked covariance matrix model, a popular and sophisticated model proposed by Johnstone. We will present some recent results on the limiting behavior of the extreme eigenvalues and eigenvectors of the spiked covariance matrices in the supercritical case. This talk is based on joint work with Zhigang Bao, Xiucai Ding, and Jingming Wang.
 Supplements



Sep 22, 2021
Wednesday

09:00 AM  09:50 AM


Random Forests and Nonlinear Sigma Models (and What These Have to Do with Random Matrices)
Roland Bauerschmidt (University of Cambridge)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
Given a finite graph, the arboreal gas is the measure on spanning forests of this graph in which each edge of the forest is weighted by a parameter β>0. The main question is whether the arboreal gas percolates for a given β, i.e., if there is a component of the forest covering a positive fraction of the graph. We prove that (perhaps surprisingly) the arboreal gas does not percolate in two dimension, but that it does percolate in dimensions three and higher provided β>β_0. Our analysis also implies diffusive subleading corrections to the connection probabilities in the latter case. This problem has a lot of independent motivation from probability, combinatorics, and statistical physics, but what does it have to do with random matrices? Analogous behaviour is predicted for random Schroedinger operators or random band matrices. The arguably best physics predictions for this Anderson transition rely on the supersymmetric approach to random matrices. The question of percolation of the arboreal gas is also exactly related to that of spontaneous symmetry breaking of a (supersymmetric) nonlinear sigma model (the H02 model), but one that is vastly simpler to analyse. Indeed, while understanding spontaneously broken symmetries is in general a notorious problem, in the case of the H02 model, we can construct a low temperature renormalisation group flow in dimensions three and higher (and derive the percolation results from it). This is joint work with N. Crawford, T. Helmuth, and A. Swan.
 Supplements


10:05 AM  10:55 AM


Counting Points in Boxes: the Riesz Family & Friends
Thomas Leblé (Université de Paris V (René Descartes))

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
Members of the Riesz family are statistical physics systems that include Coulomb and loggases. Each one gives rise to a (Gibbsian) point process, and a basic question is to study the random number of points in large ddimensional boxes. In some "solvable" cases, one finds interesting properties like hyperuniformity and (number)rigidity as well as a tripartite "large deviations" estimate. "Universality" of such features with respect to the interaction, the temperature, the external field is mostly open. I will present several questions (and few results) around this theme.
 Supplements


11:10 AM  12:00 PM


Rank One Imaginary Perturbation for Hermitian Random Matrices in the Case of Band Matrices
Mariya Shcherbina (B. Verkin Institute for Low Temperature Physics)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
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We consider a two dimensional eigenvalue density which appears on the complex plane if we add a rank one imaginary perturbation to the Hermitian random matrix. The case of band matrices is of prime interest for us.
 Supplements


12:15 PM  01:05 PM


Generalized Moments of the Classical Compact Groups
Emma Bailey (City University of New York (CUNY))

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
This talk will cover a natural generalization of moments of characteristic polynomials from the classical compact groups (unitary, symplectic, orthogonal). The results presented resolve a conjecture of Fyodorov and Keating. There are natural connections to logcorrelated fields, integrable systems (asymptotics of Toeplitz and Hankel determinants, Painlevé), symmetric function theory, and number theory.
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01:05 PM  02:00 PM


Lunch

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02:00 PM  03:00 PM


Free Discussion Time

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
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03:00 PM  03:30 PM


Tea

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03:30 PM  05:00 PM


Free Discussion Time

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 SLMath: Eisenbud Auditorium, Online/Virtual
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Sep 23, 2021
Thursday

09:00 AM  09:50 AM


Three Recipes for the Directed Landscape
Bálint Virág (University of Toronto)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
The directed landscape is the conejctured scaling limit of most natural random metrics on the plane. This random geometry seems to have its own axioms and postulates; some are like Euclid's and some entirely different. I will present three simple postulates and explain how to prove them.
 Supplements


10:05 AM  10:55 AM


Universality Results in Random Lozenge Tilings
Amol Aggarwal (Columbia University)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
The statistical behavior of random tilings of large domains has been an intense topic of mathematical research for decades, partly since they highlight a central phenomenon in physics: local behaviors of highly correlated systems can be very sensitive to boundary conditions. Indeed, a salient feature of random tiling models is that the local densities of tiles can differ considerably in different regions of the domain, depending on the shape of the domain. Thus, a question of interest is how the shape of the domain affects the statistics of a random tiling. A general prediction in this context is that, while the family of possible domains is infinitedimensional, the limiting statistics should only depend on a few parameters. In this talk, we will explain this universality phenomenon in the context of random lozenge tilings (equivalently, the dimer model on the hexagonal lattice). Various possible limiting local statistics can arise, depending on whether one probes the bulk of the domain; the edge of a facet; or the boundary of the domain. We will describe recent works that verify the universality of these statistics in each of these regimes.
These results are based on join works with Vadim Gorin and Jiaoyang Huang.
 Supplements


11:10 AM  12:00 PM


Stationary Measure for the Open KPZ Equation
Ivan Corwin (Columbia University)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
Consider the KPZ equation on an spatial interval [0,1] with mixed Neumann boundary conditions at 0 and 1. For each given pairs of boundary parameters (u,v), there should exist a unique stationary measure for the height profile differences (i.e., for the derivative of the KPZ equation). In this talk I will describe recent work in which we show that for each pair (u,v) satisfying u+v>0, certain exponentially reweighted Brownian paths measures are stationary measures for the corresponding open KPZ equation. Along the way, we will also encounter the open ASEP, as well as AskeyWilson processes and qfunction asymptotics. This is mainly based on my recent work with Alisa Knizel, though also relies on earlier work with Hao Shen as well as earlier work of Wlodzimierz Bryc, Jacek Wesolowski and Yizao Wang. I will also touch on some recent related work of Wlodzimierz Bryc, Alexey Kuznetsov, Jacek Wesolowski and Yizao Wang; as well as work of Guillaume Barraquand and Pierre Le Doussal.
 Supplements



12:15 PM  01:05 PM


Stability and Chaos in Dynamical Last Passage Percolation
Alan Hammond (University of California, Berkeley)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
Many complex statistical mechanical models have intricate energy landscapes. The ground state, or lowest energy state, lies at the base of the deepest valley. In examples such as spin glasses and Gaussian polymers, there are many valleys; the abundance of nearground states (at the base of valleys) indicates the phenomenon of chaos, under which the ground state alters profoundly when the model's disorder is slightly perturbed. Indeed, a monograph of Sourav Chatterjee from 2014 establishes that, for a class of models of Gaussian disorder, this abundance of competing minimizers is accompanied both by a rapid outset of chaos under perturbation of the system by noise, and by the effect of superconcentration, in which model statistics have lower variance than in classical scenarios, for which a central limit theorem may apply. In this talk, a recent investigation, jointly undertaken with Shirshendu Ganguly, of a natural dynamics for a model of planar last passage percolation will be discussed. Robust probabilistic and geometric technique permits a very quantified analysis of the presence of close rivals in energy to the ground state for the static version of the model; consequently, the order of the scale that heralds the transition from stability to chaos for the dynamical model is identified. The tools that drive the investigation include harmonic analytic technique present in Chatterjee's work, and the use of Brownian Gibbs resampling analysis for random ensembles of curves naturally associated to last passage percolation via the RobinsonSchenstedKnuth correspondence.
 Supplements


01:05 PM  02:00 PM


Lunch

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 Video


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02:00 PM  03:00 PM


Free Discussion Time

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
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 Abstract
 
 Supplements



03:00 PM  03:30 PM


Tea

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03:30 PM  05:00 PM


Free Discussion Time

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video


 Abstract
 
 Supplements




Sep 24, 2021
Friday

09:00 AM  09:50 AM


The FyodorovHiaryKeating Conjecture
Paul Bourgade (New York University, Courant Institute)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
FyodorovHiaryKeating proposed very precise asymptotics for the maximum of the Riemann zeta function in almost all intervals along the critical axis. After reviewing the origins of this conjecture through the random matrix analogy, I will explain a proof up to tightness, building on an underlying branching structure. This work with LouisPierre Arguin and Maksym Radziwill relies on a multiscale analysis and twisted moments of zeta.
 Supplements


10:05 AM  10:55 AM


Spectral Gap in Regular Graphs and Hypergraphs
Ioana Dumitriu (University of California, San Diego)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
Random graphs and hypergraphs have long been employed as network models for a bevy of machine learning problems, from clustering and community detection to coding theory, signal processing, and so on. The spectral properties of regular models, in particular things like the spectral gap, have often played an important role in such applications. In this wideaudience talk, I will mention some of the random matrix techniques used in the analysis of spectral gap in such regular structures, as well as some of the challenges and open problems.
 Supplements


11:10 AM  12:00 PM


Localization and Delocalization in ErdösRényi Graphs
Antti Knowles (Université de Genève)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
The ErdösRenyi graph G(N,d/N) is well known to undergo a dramatic change of behaviour around d of order log N. Below this threshold the graph develops inhomogeneities, which result in the emergence of a localized phase, in coexistence with a delocalized phase. I give an overview of the corresponding phase diagram, along with recent progress in establishing it rigorously. Joint work with Johannes Alt and Raphael Ducatez.
 Supplements


12:00 PM  12:30 PM


Free Discussion Time

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
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12:30 PM  01:30 PM


Lunch

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 Video


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01:30 PM  03:00 PM


Free Discussion Time

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
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03:00 PM  03:30 PM


Tea

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03:30 PM  05:00 PM


Free Discussion Time

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video


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