Marco Bertola (Concordia University)
Title: The nonlinear steepest descent method in higher genus
Abstract: As we celebrate the work of Percy Deift, it seems appropriate to discuss the extension of the "nonlinear steepest descent method", or "Deift-Zhou method" to Riemann--Hilbert problems depending on large parameters but set on a Riemann surface of higher genus. While the general framework of the approach remains valid, with the sequence of transformations of a given RHP to one that can be addressed with a "small norm theorem", there are interesting novelties. These originate ultimately from the fact that when solving a matrix RHP (or scalar, for that matter), the underlying integral equation is of non-zero index. In concrete terms I will explain how one needs a matrix--valued Cauchy kernel (and corresponding integral operator), which can be constructed explicitly but requires additional data, called Tyurin parameters. To be concrete, I will focus on the study of asymptotic behaviour of "orthogonal polynomials" in higher genus, or rather, which is the same, the notion of Padé approximants. Exactly as in the study of orthogonal polynomials in the celebrated DKMVZ (and I think most people here will know what this string stands for!) another interesting novel necessity is to develop the theory of equilibrium measures in higher genus Riemann surfaces, i.e., the higher genus version of potential theory. We will try to explain how to address these questions and how the answers require some cross-pollination between the two areas of approximation theory and algebraic geometry.
Deniz Billman (University of Cincinnati)
Title: Extreme Superposition: Rogue Waves of Infinite Order, Universality, and Anomalous Temporal Decay
Abstract: Focusing nonlinear Schrödinger equation serves as a universal model for the amplitude of a wave packet in a general one-dimensional weakly-nonlinear and strongly-dispersive setting that includes water waves and nonlinear optics as special cases. Rogue waves of infinite order are a novel family of solutions of the focusing nonlinear Schrödinger equation that emerge universally in a particular asymptotic regime involving a large-amplitude and near-field limit of a broad class of solutions of the same equation. In this talk, we will present several recent results on the emergence of these special solutions along with their interesting asymptotic and exact properties. Notably, these solutions exhibit anomalously slow temporal decay and are connected to the third Painlevé equation. Finally, we will extend the emergence of rogue waves of infinite order to the first several flows of the AKNS hierarchy—allowing for arbitrarily many simultaneous flows—and report on recent work regarding their space-time asymptotic behavior under a general flow from the hierarchy.
Thomas Bothner (University of Bristol)
Title: Complex moments for characteristic polynomials in the circular unitary ensemble
Abstract: We compute joint moments of the characteristic polynomial and its derivative of a N x N unitary matrix drawn from the CUE in the case that the exponents in the moments are complex-valued. The calculation is performed for finite matrix size and in the limit N → ∞. Based on ongoing joint work with Fei Wei (Sussex).
Paul Bourgade (New York University)
Title: Analogues of Fisher Hartwig asymptotics in dimension 2
Abstract: The Liouville quantum gravity measure is a properly normalized exponential of 2d log-correlated fields, such as the Gaussian free field. It is the volume form for the scaling limit of random planar maps and numerous statistical physics models. I will explain how this random measure naturally appears in random matrix theory either in space time from random matrix dynamics, or in space from the characteristic polynomial of random normal matrices. Central to these results are joint moments of characteristic polynomials for the Dyson Brownian motion, or for Ginibre matrices, which are analogues of Fisher Hartwig asymptotics in dimension 2.
Sung-Soo Byun (Seoul National University)
Title: Large Deviation Probabilities in Last Passage Percolation
Abstract: The Deift-Zhou steepest descent method has become a powerful tool in various mathematical fields over the past decades. In this talk, I will discuss its application to the study of large deviation probabilities for the last passage time in geometric last passage percolation. Building on Johansson’s seminal work, which established the leading-order term in this expansion, we use the Deift-Zhou method to explicitly derive three further terms, up to and including the constant term. I will also explore the duality between this result and the free energy expansion of two-dimensional Coulomb gases, along with its connection to large gap probabilities in Hermitian random matrix theory. This talk is based on joint work with Christophe Charlier, Philippe Moreillon, and Nick Simm.
Elizabeth Collins-Woodfin (McGill University)
Title: Multi-Species Spin Glass Models at Critical Temperature
Abstract: One of the fascinating phenomena in spin glasses is the dramatic change in behavior between the high- and low-temperature regimes. In this talk, I will focus on spin glasses near the critical temperature threshold and present results on the fluctuations of the free energy, using tools from random matrix theory. I will discuss two variants of the famous Sherrington-Kirkpatrick (SK) model. For the bipartite spherical SK (BSSK) model, we show that the free energy fluctuations at critical temperature converge to a sum of Gaussian and Tracy-Widom distributions, interpolating between known results in the high- and low-temperature regimes. For the multi-species SK (MSK) model (with a positive semi-definite interaction), we provide a sharper variance bound than previously known. The techniques used to study these two models differ significantly, as BSSK has continuous spins, while MSK has discrete spins. I will highlight the key strategies for analyzing each model and discuss connections to random matrices and statistical applications (joint work with Han Le).
Ivan Corwin (Columbia University)
Title: How Yang-Baxter unravels Kardar-Parisi-Zhang
Abstract: Over the past few decades, physicists and then mathematicians have sought to uncover the (conjecturally) universal long time and large space scaling limit for the so-called Kardar-Parisi-Zhang (KPZ) class of stochastically growing interfaces in (1+1)-dimensions. Progress has been marked by several breakthroughs, starting with the identification of a few free-fermionic integrable models in this class and their single-point limiting distributions, widening the field to include non-free-fermionic integrable representatives, evaluating their asymptotics distributions at various levels of generality, constructing the conjectural full space-time scaling limit, known as the directed landscape, and checking convergence to it for a few of the free-fermion representatives. In this talk, I will describe a method that should prove convergence for all known integrable representatives of the KPZ class to this universal scaling limit. The method has been fully realized for the Asymmetric Simple Exclusion Process and the Stochastic Six Vertex Model. It relies on the Yang-Baxter equation as its only input and unravels the rich complexity of the KPZ class and its asymptotics from first principles. This is based on three works involving Amol Aggarwal, Alexei Borodin, Milind Hegde, Jiaoyang Huang and me.
Klara Courteaut (New York University)
Title: Szegő Asymptotics of the Planar Coulomb Gas on a Jordan Arc
Abstract: In this talk, I will focus on the two-dimensional Coulomb gas confined to a sufficiently regular simple arc in the complex plane, at arbitrary positive temperature. I will present a precise asymptotic formula for the Laplace transform of linear statistics as the number of particles tends to infinity. In particular, the result shows that the centered empirical measure converges to a Gaussian field, with explicit asymptotic mean and variance given by the Dirichlet energy of the test function. A key tool in our analysis is the arc-Grunsky operator associated with the arc—reminiscent of, but distinct from, the classical Grunsky operator. Based on joint work with Kurt Johansson and Fredrik Viklund.
Percy Deift (New York University)
Title: Some open problems in integrable systems, and some comments
Abstract: The speaker will discuss a variety of problems in integrable systems, some of which are of long standing. The speaker will indicate where progress has stalled and make some suggestions how to proceed. The speaker will also make some comments about the curious state of Lax Pairs.
Guillaume Dubach (Ecole Polytechnique)
Title: Powers of Ginibre matrices and cycles of commutators
Abstract: The eigenvalues of the complex Ginibre ensemble (matrices with i.i.d. complex Gaussian entries) form a highly correlated system of points; however, their high powers are distributed exactly as if they were independent. I will present a consequence of this counter-intuitive property to random permutations; more specifically, we will explicitly describe the distribution of the number of cycles in a commutator between a random (uniform) permutation and another permutation with prescribed cycle type.
Ioana Dumitriu (New York University)
Title: Extremal singular values of sparse random matrices at and above criticality
Abstract: We developed a unified approach to study the largest and smallest singular values of sparse rectangular random matrices, both above the critical log(n)/n sparsity regime and at criticality. The approach is based on the non-backtracking operator and the Ihara-Bass formula for general Hermitian matrices with bipartite block structure, as well as on a graph-based approximation scheme developed by Alt, Ducatez, and Knowles (2019). Above criticality, the bounds are given in terms of the maximum and, respectively, minimum l_2 norms of the rows and columns of the variance profile matrix, and they work for the inhomogeneous case. At criticality, we study the centered adjacency matrices of homogeneous Erdős–Rényi random graphs and find outlier thresholds depending on the aspect ratio. This is joint work with Yizhe Zhu, Haixiao Wang, and Zhichao Wang.
Iryna Egorova (B. Verkin Institute for Low Temperature Physics and Engineering)
Title: The Toda shock problem via Riemann-Hilbert approach
Abstract: In the talk, some results of the long-time asymptotical analysis of steplike solutions for the Toda lattice will be presented. The Toda shock problem is associated with such a mutual location of the background spectra when the spectrum of the left background is located below the lower boundary of the right background spectrum. We will discuss in detail the case of constant backgrounds which was studied by use of the Deift-Zhou Nonlinear Steepest Descent method (joint works with J. Michor and G. Teschl). We will also discuss long-time asymptotical behavior of steplike solutions on finite gap (two-band) backgrounds in some DSW region.
Tamara Grava (SISSA and University of Bristol)
Title: Soliton synchronisation for the nonlinear Schroedinger equation
Abstract: We consider a N soliton solution of the focusing nonlinear Schrodinger equation. Generically solitons collide pairwise. We give conditions for the synchronous collision of these N solitons. When the solitons velocities are well separated and the solitons have equal amplitude, we show that the local wave profile at the collision point scales as the sinc(x) function. We show that this behaviour persists when the amplitudes of the solitons are i.i.d. sub-exponential random variables. We derive Central Limit Theorems for the fluctuations of the profile in the near-field regime (near the collision point) and in the far-regime.
Alexander Its (Indiana University Indianapolis)
Title: On Determinants of Integrable Operators with Shifts
Abstract: In the talk, some recent results of the asymptotic analysis of the Fredholm determinants of integral operators appearing in the study of the correlation functions in non-free fermion exactly solvable quantum field models will be presented. Specifically, we shall consider the shifted sine-kernel as a case study. Our analysis is based on the Deift-Zhou nonlinear steepest descent method. The history and some of the still open questions in the field will be presented as well. The talk is based on the earlier works of the speaker with V. Korepin, A. Izergin, N. Slavnov and K. Koslowski and on the ongoing project with T. Bothner, A. Simon and K. Kozlowski.
Kurt Johansson (KTH)
Title: Coulomb gas on a Jordan curve and a Jordan arc
Abstract: I will discuss the asymptotics of the partition function of a planar Coulomb gas at inverse temperature beta confined to a Jordan curve or a Jordan arc in the plane. There are both similarities and interesting differences between these two cases. I will also discuss its relation to Loewner energy a quantity that has been studied recently in other contexts. This is joint work with Fredrik Viklund and Klara Courteaut.
Igor Krasovsky (Imperial College)
Title: Strong and weak confinement in the Freud ensemble of random matrices
Abstract: We discuss the Freud ensemble of random matrices (the unitary ensemble associated with the Freud orthogonal polynomials with weight exp(-|x|^beta), beta>0). At the point x=0, a curious transition takes place when beta crosses the value 1. We obtain asymptotic formulas for the probability of a large gap around x=0 in the local scaling regime and present the 2 distinct cases for beta<1 and beta>1. This is a joint work with Tom Claeys and Olexander Minakov.
Ji Oon Lee (KAIST)
Title: Detection problems in spiked Wigner matrices
Abstract: The spiked Wigner matrix model is one of the most basic yet fundamental models for the signal-plus-noise data, where the signal is a vector and the noise is a symmetric random matrix. One of the main questions in the study of the spiked Wigner matrices is the detection problem, where the main goal is to detect the presence of the signal in a given data matrix. In this talk, I will explain various results on the detection problem, such as the fundamental limit and detecting algorithms, which are based on study of random matrices and spin glass models.
Seung-Yeop Lee (University of South Florida)
Title: Asymptotics of Planar Orthogonal polynomials via Riemann-Hilbert Method
Abstract: The Riemann-Hilbert nonlinear steepest descent method, developed by Deift, Kriecherbauer, McLaughlin, Venakides, and Zhou, has proven to be a powerful tool in the study of orthogonal and multiple orthogonal polynomials supported on one-dimensional contours with complex weights. However, for planar orthogonal polynomials, where the orthogonality is defined over two-dimensional supports, the Riemann-Hilbert method generally does not apply. In certain cases where the weight function has an algebraic structure, planar orthogonality can be reformulated as multiple orthogonality on contours. This reformulation allows the application of the Riemann-Hilbert method to derive asymptotic behavior of the orthogonal polynomials, shedding light on two-dimensional Coulomb gas models. In this talk, we will review several such results and explore how they contribute to our understanding of the 2D Coulomb gas. We will also present preliminary findings on two topics: the averaged characteristic polynomials of the Ginibre ensembles (joint work with Meng Yang), and the Riemann-Hilbert formulation of certain planar orthogonal polynomials with non-Hele-Shaw type weights (joint work with Abril Arenas).
Zhipeng Liu (University of Kansas)
Title: An upper tail field of the KPZ fixed point
Abstract: The KPZ fixed point is a (1+1)-dimensional space-time random field conjectured to be the universal limit for models within the Kardar-Parisi-Zhang (KPZ) universality class. We consider the KPZ fixed point with the narrow-wedge initial condition, conditioning on a large value at a specific point. By zooming in the neighborhood of this high point appropriately, we obtain a limiting random field, which we call an upper tail field of the KPZ fixed point. Different from the KPZ fixed point, where the time parameter has to be nonnegative, the upper tail field is defined in the full 2-dimensional space. Especially, if we zoom out the upper tail field appropriately, it behaves like a Brownian-type field in the negative time regime, and the KPZ fixed point in the positive time regime. One main ingredient of the proof is an upper tail estimate of the joint tail probability functions of the KPZ fixed point near the given point, which generalizes the well known one-point upper tail estimate of the GUE Tracy-Widom distribution. This is a joint work with Ruixuan Zhang.
Mateusz Piorkowski (KTH Royal Institute of Technology)
Title: Orthogonal polynomials with weights having logarithmic singularities: recent results via the Riemann-Hilbert method
Abstract: In this talk I will report on recent results obtained in a collaboration with P. Deift on the asymptotics for the recurrence coefficients of orthogonal polynomials with a logarithmic weight function. More precisely, we consider the weight log((1 - x)/2), for x in (-1,1), which has a logarithmic singularity at x = 1 and a simple zero at x = -1. The Riemann-Hilbert analysis is surprisingly intricate due to the logarithmic singularity of the weight at x = 1. The main technical obstacle is the lack of an explicit local parametrix around the logarithmic singularity, implying that the nonlinear steepest descent analysis cannot be performed in the usual manner. An asymptotic formula of the recurrence coefficients for orthogonal polynomials with such (and more general) logarithmic weights was conjectured by Magnus '18. Our proof of this conjecture for the weight stated above is based on the Riemann-Hilbert analysis performed by Conway & Deift '18, where the case of logarithmic singularity without a simple zero of the weight was treated. We will discuss in detail the new difficulties that arise from the simple zero at x = -1.
Carlos Tomei (PUC-Rio)
Title: Seeing functions
Abstract: Knowledge of the critical set of a function yields substantial information about a large class of nonlinear functions. After some basic examples, we sketch the proof of a conjecture of Henry McKean, which implies the following result. Let n be a natural number, r > 0. Then, there is a function g in C([0,1]) and a ball B of radius r in CD2([0,1]) containing exactly n solutions of the equation -u'' + u2/2 = g, u(0) = u(1) = 0.