Spring 2025
Time & Location: All talks are on Wednesday in different places, at 3:00 PM .
Organizer: Katerina Gkogkou and Ken McLaughlin
January 27,
Topic: Random Matrices: Fredholm determinant representations for (a) spectral gaps, (b) largest eigenvalues, and (c) number statistics.
Ken McLaughlin - Tulane University
Abstract: How to compute interesting statistical quantities in random matrices: Fredholm determinant representations for (a) spectral gaps, (b) largest eigenvalues, and (c) number statistics.
With the assistance of the whole group, this should be very introductory.
Speakers: to be determined, starting with Ken McLaughlin on January 27.
Location: Gibson Hall 310
Time: 3:00 PM
______________
February 3,
Topic: Random Matrices: Fredholm determinant representations for (a) spectral gaps, (b) largest eigenvalues, and (c) number statistics.
Ken McLaughlin - Tulane University
Abstract: We will start calculating some basic number statistics for finite sized random matrices using Hermite polynomials, and compare to numerical experiments. In the second half, I will explain the connection between random matrices and Hermite polynomials.
Location: Gibson Hall 310
Time: 3:00 PM
______________
February 10,
Topic: Random Matrices: Fredholm determinant representations for (a) spectral gaps, (b) largest eigenvalues, and (c) number statistics.
Ken McLaughlin - Tulane University
Abstract: We will continue exploring the connection between random matrices and Hermite polynomials, and start the proof of the relation between eigenvalue probabilities and Fredholm determinants. Time permitting, we will return to the numerical experiments, to carefully develop intuition.
Location: Gibson Hall 310
Time: 3:00 PM
______________
February 17,
Topic: Random Matrices: Fredholm determinant representations for (a) spectral gaps, (b) largest eigenvalues, and (c) number statistics.
TBD - Tulane University
Abstract: We will return to the numerical experiments, to carefully develop intuition. And explore some more precise open problems. Then we will return to complete the proof of the fundamental relation between eigenvalues and Fredholm determinants.
Location: Gibson Hall 310
Time: 3:00 PM
______________
February 24,
Topic: A Vision of Integrability: McKean’s Unimodularity Conjecture
Nick Ercolani - University of Arizona (Host: Ken McLaughlin)
Abstract: In a striking series of papers, titled Geometry of KdV(1) - Geometry of KdV(5), Henry Mckean formulated a precise notion of what should be the function space foliation by invariant sets for the Korteweg-deVries evolution. This is meant to pertain to initial data that is smooth but otherwise only required to be bounded below. This foliation should generalize the picture of (typically infinite dimensional) Arnold-Liouville torii familiar from the particular case of periodic initial data. The proposed answer is phrased in terms Kodaira’s elegant extension of the classical Weyl-Titchmarsh theory for spectral weights of Schrodinger operators.
The goal of the talk will be to first present an overview of McKean’s conjecture and then to describe some recent work, joint with Dylan Murphy, on analogous investigations for the Toda lattice and Jacobi operators.
Location: Gibson Hall 310
Time: 3:30 PM
______________
March 10,
Topic: How to compute interesting statistical quantities in random matrices: Fredholm determinant representations for (a) spectral gaps, (b) largest eigenvalues, and (c) number statistics, Part V.
Ken McLaughlin - Tulane University
Abstract: The plan: (1) Executive summary of the connection between eigenvalues of random matrices and Fredholm determinants. (2) T random matrix theory laboratory – testing the theory. (3) Behavior of eigenvalues when the size of the matrices grows to $\infty$.
Location: Gibson Hall 310
Time: 3:00 PM
______________
March 17,
Topic: How to compute interesting statistical quantities in random matrices: Fredholm determinant representations for (a) spectral gaps, (b) largest eigenvalues, and (c) number statistics, Part 6.
Ken McLaughlin - Tulane University
Abstract: The plan: investigate the behavior of eigenvalues of random matrices when the size of the matrix grows to infinity, starting with more numerical experiments, and then the asymptotic behavior of the kernel, and the Fredholm determinant. The aim is to understand known asymptotic behavior and identify some interesting but tractable challenging problems.
Location: Gibson Hall 310
Time: 3:00 PM
______________
April 7,
Topic: Estimates for the largest eigenvalue of GUE random matrices
John Lopez Santander - Tulane University
Abstract: This is a continuation of the lecture from 2 weeks ago. The speaker will present some estimates regarding the probability that the largest eigenvalue is greater than (2 + \delta) \sqrt{N}.
Location: Gibson Hall 310
Time: 3:00 PM
