MATH 705, Seminar in Probability (39948R), Spring 2013.
Class meetings: W 1-3 pm, in KAP 245.


 

Information on this and linked pages changes frequently.

Organizer: Sergey Lototsky.
Office: KAP 248D.
Phone: (213) 740-2389.

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

  • [TT] Terence Tao ``Topics in random matrix theory'' Graduate Studies in Mathematics, vol. 132, AMS, 2012
  • [AGZ] Greg W. Anderson, Alice Guionnet, and Ofer Zeitouni ``An Introduction to Random Matrices'' Cambridge Unversity Press, 2009.
    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

    January 16

    January 23

    January 30

    February 6

    February 13

    February 20

    February 28

    March 6

    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.