Starting in the winter quarter (Monday Jan 4, to  be precise), I will be giving a graduate course on random matrices, with lecture notes to be posted on this blog.  The topics I have in mind are somewhat fluid, but my initial plan is to cover a large fraction of the following:

  • Central limit theorem, random walks, concentration of measure
  • The semicircular and Marcenko-Pastur laws for bulk distribution
  • A little bit on the connections with free probability
  • The spectral distribution of GUE and gaussian random matrices; theory of determinantal processes
  • A little bit on the connections with orthogonal polynomials and Riemann-Hilbert problems
  • Singularity probability and the least singular value; connections with the Littlewood-Offord problem
  • The circular law
  • Universality for eigenvalue spacing; Erdos-Schlein-Yau delocalisation of eigenvectors and applications

If time permits, I may also cover

  • The Tracy-Widom law
  • Connections with Dyson Brownian motion and the Ornstein-Uhlenbeck process; the Erdos-Schlein-Yau approach to eigenvalue spacing universality
  • Conjectural connections with zeroes of the Riemann zeta function

Depending on how the course progresses, I may also continue it into the spring quarter (or else have a spring graduate course on a different topic – one potential topic I have in mind is dynamics on nilmanifolds and applications to combinatorics).