Student Seminar@Stat MSU, Spring 2014

Details

Time: 4:30pm-6:00pm Wednesday.

Location: C405 Wells Hall (except as noted)

Food served at C405 Wells Hall.

Overview

The theory of random matrices is a rich topic in mathematics. Beside being interesting in its own right, random matrices play fundamental roles in various areas such as statistics, mathematical physics, combinatorics, theoretical computer science, number theory, and numerical analysis, to mention a few. A famous example here is the study of physicist E. Wigner, who used the spectrum of random matrices as a model in nuclear physics, and consequently discovered the fundamental semicircle law which describes the limiting distribution of the eigenvalues of a random hermitian matrix.

Special random matrices models where the entries are iid complex or real Gaussian random variables (GUE, GOE or Wishart) have been studied in detail. However, much less was known about general models, as the above-mentioned study relies very heavily on properties of the Gaussian distribution. In the last ten years or so we have witnessed considerable progress on several fundamental problems concerning general models, such as the Circular law conjecture or the Wigner-Dyson-Mehta conjecture. More importantly, these new results are proved using novel approaches which seem to be applicable to many other problems.

Our seminar will focus on some fundamental results and methods for the random matrices to get general ideas about it and get inspired for our own research.

Schedule

Main Reference

• • T. Tao, "Topics in Random Matrix Theory"

Related Materials

• • G. W. Anderson, A. Guionnet, O. Zeitouni, "An Introduction to Random Matrices"

  • Z. Bai and J. W. Silverstein, "Spectral Analysis of Large Dimensional Random Matrices"

  • R. Vershynin, "Introduction to The non-Asymptotic Analysis of Random Matrices"

• A. Guionnet, "Free Probability and Random Matrices"

• A. Edelman, B. D. Sutton, Y. Wang, "Random Matrix Theory, Numerical Computation and Applications"

  • M. Rudelson, "Lecture Notes on Non-Asymptotic Theory of Random Matrices"

  • C. Bordenave and D. Chafai, "The Circular Law"

  • S. Kargin and E. Yudovina, "Random Matrix Theory"

Useful Links

• 2013 AMS Short Course on Random Matrices

• Princeton University ORFE Student Seminar

• UCLA MATH 254A : Topics in Random Matrix Theory