Park City Mathematics Institute Research Course-Random Matrices Project Abstract

Research Course-Random Matrices

Authors: Brian Hopkins, Brian Hsu, Lynde Nanson, Melvin Peralta, Daniel Persia, Hilda Prado

 

This teacher group participated in the Undergraduate Summer course "Introduction to Random Matrices" led by Mihai Stoiciu of Williams College. The group attended the 1pm lecture and then spent an hour debriefing and working on problem sets, with Mathematica (available free to PCMI participants) being a primary tool. Initial classes (held before the teachers arrived, unfortunately) reviewed linear algebra, probability theory, and measure theory. Initial random matrix examples included entries selected randomly from the normal distribution, which demonstrated the Circle Law. The most time was spent on the Gaussian Beta Ensembles from physics, including symmetric, Hermitian, and symplectic matrices, leading to the Semicircle Law and exploring local distributions of eigenvalues (which are different near the boundary than inside the interval; some of this work is so new that it is under review for publication). Considering the largest eigenvalues introduced the Tracy--Widom distribution, which dates from 2002 and has recently found wide applications. Later topics included the unitary matrices of Circular Beta Ensembles whose eigenvalues are distributed on the unit circle.

 

Course resources are available at http://sites.williams.edu/mstoiciu/pcmi/. Among the resources, we particularly recommend the short article on the Tracy--Widom distribution from Quanta.

 

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