Math Courses

The mystery of universality in Random Matrices

Name: The mystery of universality in Random Matrices
Start: 2024-02-05T09:00:00+01:00
End: 2024-02-08T11:00:00+01:00
Location: No location set

by Dr Giorgio Cipolloni

Monday 5 Feb 2024, 09:00 → Thursday 8 Feb 2024, 11:00 Europe/Rome

Description

Lecturer:
Dr. Giorgio Cipolloni (Princeton University)
gc4233 AT princeton DOT edu

Course description
In 1955 E. Wigner had the revolutionary vision that energy level statistics of heavy nuclei are universal, and suggested Random (Hermitian) Matrices as a simple toy model to prove this phenomenon in a mathematical rigorous way. This is known as Wigner-Dyson-Gaudin-Mehta (WDM) conjecture. Since then, it has been conjectured that random matrix statistics are actually much more universal, appearing in number theory (Montgomery conjecture for Riemann zeta function), dynamical systems (Bohigas-Giannoni-Schmidt conjecture), and neural networks.

In this course we will briefly present these conjectures and give a short overview of interesting open questions in random matrix theory. Then we will walk through a new short proof of the WDM conjecture that uses recently developed techniques relying on

1) Dynamical proof of local laws (i.e. law of large numbers for resolvents).

2) Optimal relaxation of the Dyson Brownian motion.

This course is designed as an introduction to the fundamentals of random matrix theory, which can be useful for diverse questions in mathematical physics, machine learning, and number theory.

Schedule
All lectures will be held at the Main Lecture Hall in Viale Crispi 7 building, according to the following schedule:

Mon Feb 5: 9-11
Tue Feb 6: 9-11
Wed Feb 8: 15:15 - 17:15
Thur Feb 9: 9-11