Description

Random matrix theory is the stirring of linear algebra with probability. Through mathematical serendipity it connects to subjects such as analysis, combinatorics, representation theory, statistical mechanics, etc. The study of random matrices began in statistics with Wishart in the 1930s, and then became popular within physics by the work of Wigner in the 1950s. Nowadays it finds significant applications in statistics and computer science, for instance, in the principal component analysis of big datasets.

The project will be an introduction to methods and phenomena in random matrix theory. Topics may include the Wigner semi-circle law, method of moments, determinantal point processes, the Tracy-Widom law and the fun problem of the longest increasing subsequence in a random permutation. Students may also pursue other topics depending on their interest.

There will be opportunities for computer simulations. You can do some neat computer experiments with random matrix data. Try the following for yourself. Take a 100 x 100 matrix A whose entries are independent Normal random variables. Consider the matrix B = A + (transpose of A). Matrix B is symmetric and has 100 real eigenvalues. Plot their histogram. What do you see?

Prerequisites

2H Probability and Analysis in Many Variables are essential.

Complex Analysis and any level III probability course will be useful but not required.

Resources

Students will be guided to lecture notes for certain topics. The following textbooks are a useful introduction and provide background.

• G. Akermann, J. Baik and P. D. Francesco. Oxford handbook of random matrix theory.
• G. W. Anderson, A. Guionnet and O. Zeitouni, An introduction to random matrices.
• D. J. Daley and D. Vere-Jones. An introduction to the theory of point processes.
• P. J. Forrester, Log gases and random matrices.
• D. Romik, The surprising mathematics of longest increasing subsequences.

Get in touch by email if you have questions.