Oct 18, 2021
Monday

08:30 AM  09:00 AM


Welcome Tea

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09:00 AM  09:10 AM


Introduction

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
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09:10 AM  10:00 AM


Strong Szego Theorem on a Jordan Curve
Kurt Johansson (Royal Institute of Technology (KTH))

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
I will discuss certain determinants with respect to a sufficiently regular Jordan curve in the complex plane that generalize Toeplitz determinants which are obtained when the curve is the circle. This also corresponds to studying a planar Coulomb gas on the curve at inverse temperature beta =2. Under suitable assumptions on the curve we prove a strong Szego type asymptotic formula as the size of the determinant grows. The resulting formula involves the Grunsky operator built from the Grunsky coefficients of the exterior mapping function for the curve. As a consequence of our formula we obtain the asymptotics of the partition function on the curve. Interestingly, this formula involves the Fredholm determinant of the absolute value squared of the Grunsky operator which equals, up to a multiplicative constant, the so called Loewner energy of the curve.
 Supplements


10:20 AM  11:10 AM


Large Deviations for Generalized Gibbs Ensembles of the Classical Toda Chain
Alice Guionnet (École Normale Supérieure de Lyon)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
Large deviations principles for the distribution of the empirical measure of the equilibrium measure for the Generalized Gibbs ensembles of the classical Toda chain introduced by H. Spohn. We deduce its almost sure convergence and characterize its limit in terms of the limiting measure of Betaensembles. Our results apply to general smooth potentials. This is joint work with Ronan Memin.
Article discussed: https://arxiv.org/abs/2103.04858
 Supplements



11:30 AM  12:20 PM


Exact Solution of TASEP and Generalizations
Daniel Remenik (Universidad de Chile)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
I will present a general result which allows to express the multipoint distribution of the particle locations in the totally asymmetric exclusion process (TASEP) and several related processes, for general initial conditions, in terms of the Fredholm determinant of certain kernels involving the hitting time of a random walk to a curve defined by the initial data. This scheme generalizes an earlier result for the particular case of continuous time TASEP, which has been used to prove convergence of TASEP to the KPZ fixed point. The result covers processes in continuous and discrete time, with push and block dynamics, and with sequential and parallel update, as well as some extensions to processes with memory length larger than 1. Based on joint work with Konstantin Matetski.
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12:20 PM  01:30 PM


Lunch

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01:30 PM  02:20 PM


Gibbsian Line Ensembles and BetaCorners Processes
Evgeni Dimitrov (Columbia University)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
Gibbs measures are ubiquitous in statistical mechanics and probability theory. In this talk I will discuss two types of classes of Gibbs measures – random line ensembles and triangular particle arrays, which have received considerable attention due, in part, to their occurrence in integrable probability.
Gibbsian line ensembles can be thought of as collections of finite or countably infinite independent random walkers whose distribution is reweighed by the sum of local interactions between the walkers. I will discuss some recent progress in the asymptotic study of Gibbsian line ensembles, summarizing some joint works with Barraquand, Corwin, Matetski, Wu and others.
Betacorners processes are Gibbs measures on triangular arrays of interacting particles and can be thought of as analogues/extensions of multilevel spectral measures of random matrices. I will discuss some recent progress on establishing the global asymptotic behavior of betacorners processes, summarizing some joint works with Das and Knizel.
 Supplements


03:00 PM  03:30 PM


Afternoon Tea

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Oct 19, 2021
Tuesday

09:10 AM  10:00 AM


On FiniteRank NonHermitian Deformations of Random Matrix Ensembles
Yan Fyodorov (King's College London)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
I will present results, both old and new, on complex eigenvalues and left/right eigenvectors of finiterank nonHermitian deformations of classical ensembles (GUE/GOE) and  time permitting  Random Band Matrices.
 Supplements


10:20 AM  11:10 AM


Interacting Diffusions on Positive Definite Matrices
Neil O'Connell (University College Dublin)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
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 Abstract
We consider systems of Brownian particles in the space of positive definite matrices, which evolve independently apart from some simple interactions. We give examples of such processes which have an integrable structure. These are related to KBessel functions of matrix argument and multivariate generalisations of these functions. The latter are eigenfunctions of a particular quantisation of the nonAbelian Toda lattice.
 Supplements


11:30 AM  12:20 PM


Lecture Hall Tableaux, Non Intersecting Paths and Tilings
Sylvie Corteel (University of California, Berkeley)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
Lecture Hall tableaux were introduced by Corteel and Kim in 2018 to study the combinatorics of the multivariate Little Jacobi polynomials. They are in bijection with non intersecting paths on a lattice that is not translation invariant and with lozenge tilings of a "spiral" surface. I will present the combinatorics of these objects and asymptotic results.
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12:20 PM  01:30 PM


Lunch

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01:30 PM  02:20 PM


Random Melting Skew Young Diagram
Zhipeng Liu (University of Kansas)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
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 Abstract
We consider a model of random melting skew Young diagram whose northwest and southeast corners melt independently at two rates $\gamma_1$ and $\gamma_2$ respectively. We find an exact formula for the joint distribution of the location of the last melting box and the melting time for an arbitrary initial skew Young diagram. This formula is suitable for asymptotic analysis for some special initial skew Young diagrams. As applications, we show how this result is related to the argmax of the sum of two independent Airytype processes, such as two parabolic Airy2 processes, or a parabolic Airy2 process and an Airy1 process.
 Supplements


03:00 PM  03:30 PM


Afternoon Tea

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Oct 20, 2021
Wednesday

09:10 AM  10:00 AM


The TwoPeriodic Aztec Diamond and Matrix Valued Orthogonality
Arno Kuijlaars (Katholieke Universiteit Leuven)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
I will discuss how polynomials with a nonhermitian orthogonality on a contour in the complex plane arise in certain random tiling problems. In the case of periodic weightings the orthogonality is matrixvalued. In work with Maurice Duits (KTH Stockholm) the RiemannHilbert problem for matrix valued orthogonal polynomials was used to obtain asymptotics for domino tilings of the twoperiodic Aztec diamond. This model is remarkable since it gives rise to a gaseous phase, in addition to the more common solid and liquid phases.
 Supplements


10:20 AM  11:10 AM


Local Universality of the TimeTime Covariance and of the Geodesic Tree for Last Passage Percolation
Patrik Ferrari (Rheinische FriedrichWilhelmsUniversität Bonn)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
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 Abstract
We consider time correlation for KPZ growth in 1+1 dimensions in a neighborhood of a characteristics. We prove convergence and local universality of the covariance with droplet, flat and some random initial profiles. Furthermore, we show that also the geodesic tree is locally universal. These are joint works with Alessandra Occelli and Ofer Busani.
 Supplements


11:30 AM  12:20 PM


Lozenge Tilings and the Gaussian Free Field on a Cylinder
Marianna Russkikh (Massachusetts Institute of Technology)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
We discuss new results on lozenge tilings on an infinite cylinder, which may be analyzed using the periodic Schur process introduced by Borodin. Under one variant of the $q^{vol}$ measure, corresponding to random cylindric partitions, the height function converges to a deterministic limit shape and fluctuations around it are given by the Gaussian free field in the conformal structure predicted by the KenyonOkounkov conjecture. Under another variant, corresponding to an unrestricted tiling model on the cylinder, the fluctuations are given by the same Gaussian free field with an additional discrete Gaussian shift component. Fluctuations of the latter type have been previously conjectured for tiling models on planar domains with holes.
 Supplements


12:20 PM  01:30 PM


Lunch

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03:00 PM  03:30 PM


Afternoon Tea

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Oct 21, 2021
Thursday

09:10 AM  10:00 AM


Hankel Composition Structures in Random Matrix Theory and Beyond
Thomas Bothner (University of Bristol)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
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This talk will highlight a novel way of characterizing Fredholm determinants of Hankel composition operators via RiemannHilbert problems. Based on work in progress by the speaker.
 Supplements


10:20 AM  11:10 AM


Properties of the chGUE at the Hard Edge: Spacing Distributions and Universality with External Field
Gernot Akemann (Universität Bielefeld)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
The chiral Gaussian unitary ensemble also called Laguerre or Wishart unitary ensemble is probably one of the most studied random matrix ensembles. We will investigate its further properties at the hard edge of the spectrum at finite and largeN. Due to applications to the Dirac operator in Quantum Chromodynamics, we will add a finite number $N_f$ of characteristic polynomials to the Gaussian distribution of matrix elements. In the first part we will focus on the spacing distribution between the smallest singular values, where we give exact determinantal formulae. In contrast to the kpoint correlation functions, the spacing is almost immediately as close to the bulk spacing as the Wigner surmise. In the second part we will show that the kpoint functions are universal at the hard edge under the addition of an external, deterministic field $A$ with full rank, as long as a hard edge is present. Using recent progress in polynomial ensembles we can show that previous results from supersymmetry with and orthogonal polynomials without external field agree and can be extended to a fixed number of zero modes. In particular we show that determinantal formulae of different sizes for the kpoint functions are equivalent.
 Supplements


11:30 AM  12:20 PM


Stationary HalfSpace Last Passage Percolation
Alessandra Occelli (Instituto Superior Técnico)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
We present our result on stationary last passage percolation in halfspace geometry. We determine the limiting distribution of the last passage time in a critical window close to the origin. The result is a new twoparameter family of distributions: one parameter for the strength of the diagonal bounding the halfspace and the other for the distance of the point of observation from the origin. It should be compared with the oneparameter family giving the BaikRains distributions for fullspace geometry. Joint work with D. Betea and P. Ferrari.
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12:20 PM  01:30 PM


Lunch

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01:30 PM  02:20 PM


Fractal Geometry of the KPZ Equation
Promit Ghosal (Brandeis University)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
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 Abstract
The KardarParisiZhang (KPZ) equation is a fundamental stochastic PDE related to modeling random growth processes, Burgers turbulence, interacting particle systems, random polymers etc. In this talk, we focus on how the tall peaks and deep valleys of the KPZ height function grow as time increases. In particular, we will ask what are the appropriate scaling of the peaks and valleys of the KPZ equation and whether they converge to any limit under those scaling. These questions will be answered via the law of iterated logarithms and fractal dimensions of the level sets.
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03:00 PM  03:30 PM


Afternoon Tea

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Oct 22, 2021
Friday

09:10 AM  10:00 AM


The Riemann Hilbert Problem in Higher Genus and Some Applications
Marco Bertola (Concordia University and SISSA; Concordia University and International School for Advanced Studies (SISSA))

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
The role of (bi/multi/matrix) orthogonal polynomials in random matrices, integrable systems and combinatorics is well known. Our goal is to report on recent progress in the definition of suitable extensions of the notion of orthogonality where the polynomials are replaced by sections of appropriate line bundles on Riemann surfaces. We discuss their definition in the spirit of various generalizations of the Padé problem and the formulation of appropriate matrix Riemann Hilbert problems that allow to characterize them as well as control their asymptotic behaviour. Applications to Matrix Orthogonal Polynomials and the KP hierarchy will also be discussed.
 Supplements


10:20 AM  11:10 AM


Marked and Conditional Determinantal Point Processes
Tom Claeys (Université Catholique de Louvain)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
I will introduce a family of marked determinantal point processes and associated conditional ensembles, in which information about a random part of a point configuration is encoded. Special cases of these conditional ensembles appear in the ItsIzerginKorepinSlavnov method and in the study of number rigidity. I will discuss general properties of these ensembles, and show how they lead to a strengthened notion of number rigidity for determinantal point processes induced by a certain class of orthogonal projections, including the sine, Airy, and Bessel point processes. The talk will be based on joint work with Gabriel Glesner.
 Supplements


11:30 AM  12:20 PM


Asymptotic Analysis of the Interaction Between a Soliton and a Regular Gas of Solitons
Ken McLaughlin (Colorado State University)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
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 Abstract
We provide a detailed asymptotic description of this interaction, for the case of the modified KdV equation. Kinetic velocity equations are derived, and they will be shown to be ubiquitous in singular limits of a number of integrable systems, and in random matrix theory as well.
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12:20 PM  01:30 PM


Lunch

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01:30 PM  02:20 PM


The Edge Scaling Limit of the Characteristic Polynomial of the Gaussian βEnsembles
Gaultier Lambert (Universität Zürich)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
In this talk, I report on the asymptotics of the characteristic polynomials of the Gaussian βensembles for general β > 0. Based on the DumitriuEdelman matrix models for the Gaussian βensemble, I will present a probabilistic coupling between the characteristic polynomial, a Gaussian analytic function and a new object called the stochastic Airy function. This random entire function arise as the scaling limit of the characteristic polynomial at the spectral edge and its zero set is exactly the Airyβ point process. This is joint work with Elliot Paquette and our results are based on the study of the transfer matrix recurrence satisfied by the characteristic polynomials.
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03:00 PM  03:30 PM


Afternoon Tea

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