ÃÛÌÒ´«Ã½

Degrees & Accolades:

Ph.D. (Dalhousie University)

Research Profile:

Random matrices are matrices with random entries. The eigenvalue distribution of random matrices are widely used in statistics, physics, mathematical finance, wireless communications, as well as many areas of mathematics.

Free independence is a new kind of independence that is especially useful for studying random matrices. Free probability, while running in parallel with classical probability, constantly surprises us with new twists on the old theory.

As this is a very new subject there are many simple questions that are just waiting to be investigated. I am currently recruiting graduate students.

Research Areas:

Problem 1: If one has a polynomial, p, with real roots, then the derivative, p′ also has real roots, and these roots interlace those of p. We get a similar phenomenon with principal minors of matrices. Namely let A be a self-adjoint n × n matrix and B be the (n − 1) × (n − 1) minor obtained by deleting the last row and the last column. Then the eigenvalues of B interlace those of A. When the entries of A are suitably random there is a connection between these two phenomena, but there is much more to do here using the concept of free independence.

Problem 2: Every permutation has a genus, which is a nonnegative integer describing the genus of a surface on which one can draw the cycles of the permutation in a non-crossing way. One can lower the genus by punching a hole. This has only been explored at small genera. More information here would have a big impact on free probability.

Problem 3: The two questions above have interpretations involving analytic functions of several variables. Most the current work involves only formal power series. Fonding domains where there was convergence would be very beneficial.