Course title:               Math 21-499: Introduction to Undergraduate Research

Instructor:                 Steven J. Miller (sjm1@williams.edu)

Random Matrix Theory: The goal would be to look at the distribution of eigenvalues of certain structured ensembles of matrices. These projects require linear algeba, combinatorics and basic probability. I am also very interested in trying to find a random matrix analogue of Rankin-Selberg convolution (i.e., continuing a bridge of connections between number theory  and random matrix theory;  this theme is described in greater detail in the L-function section below).

• Papers
  • Odlyzko: Distribution of spacings between zeros of the zeta function: One of the classics in the field of the phenomenal agreement between the zeros of the Riemann zeta function and the eigenvalues of matrices.
  • Mezzadri: How to generate random matrices from the classical compact groups: if we choose to numerically explore eigenvalues of the classical compact groups, this is a great place to begin.
  • Lehman: Survey of the Semi-Circle Law (former student of mine, written at a research class at Princeton).
  • Zakharevich: Paper on Generalization of the Semi-Circle Law (former student of mine at AIM).
  • Toeplitz Ensemble Papers:
    • Hammond - Miller: Eigenvalue spacing distribution for real symmetric Toeplitz matrices (former student of mine at AIM and OSU).
    • Miller - Massey - Sinsheimer: Eigenvalue spacing distribution for palindromic Toeplitz matrices (former students at OSU and Brown).
    • Jackson - Pham - Miller: Eigenvalues of highly palindromic Toeplitz matrices (former SMALL students).
    • Kologlu - Kopp - Miller: Eigenvalues of symmetric period m-circulant matrices (former SMALL students)
  • Distribution of eigenvalues of weighted, structured matrix ensembles
  • Limiting Spectral Measures for Random Matrix Ensembles with a Polynomial Link Function
• Talks
  • Introduction to Random Matrix Theory (7 minutes): https://youtu.be/PlpxOWxaNfM
  • Part I (Classical RMT, Intro L-fns, Dirichlet): http://youtu.be/2PuUbk6gUMM (from a two part number theory talk, beginning is appropriate)