Class meetings: W 1-3 pm, in KAP 245.

Office Hours: MWF 10:30-11:30am.

Walk-ins and appointments at other time are welcome.

** The objective this semester: **
To discuss random matrices and related topics. The main references are the
books

as well as research and survey papers on the subject.

The participating students are expected to take part in the discussions, make several presentations, and take turn writing summaries of each meeting.

** What we did **

March 13: Presentation by Haining about unimodal permutations

March 27: General discussion of the
Haar measure notes.
The conclusion: to construct the Haar measure on a compact topological group,
construct a suitable
bounded linear functional on the space of continuous functions on the group.
Then the Haar measure is whatever represents this functional according to the
Riesz-Markov-Kakutani representation theorem.

April 3: First, a summary by Li of the Riesz-Markov-Kakutani representation theorem for the dual of the space of continuous functions on a locally compact Hausdorff space. Then John's summary (following the Haar measure notes) of the construction of the functional that is represented by the Haar measure.

April 10: Presentation by Gene on Chapter 2 of the [AGZ] book. The main question: understanding the log-Sobolev inequality.

April 17: Two separate discussions (for technical reasons) of Chapter 3 of the [AGZ] book on Hermite polynomials and their applications in the study of the fine asymptotic properties of eigenvalues of Gaussian matrices. Here is a summary by Li. The main question: how many of the results will continue to hold if the normal distribution is replaced with something else (e.g. uniform) and the Hermite polynomials are replaces with the corresponding orthogonal polynomials (Legendre in the uniform case)?

April 24: A discussion of Chapter 4 of [AGZ], especially the part about eigenvalues of a matrix made of Brownian motions. Here is a summary by Radoslav.

May 1: A discussion of Chapter 5 of [AGZ] about free probability. John provided a handout and led the discussion.