L-functions
One of the behaviours Michael explores deals with the zeros of the Riemann zeta function; at which complex values does this function equal zero. To explore this, Michael is using random matrix theory. “There are ways to interpret matrices as functions. What we’re comparing,” explains Michael. “Is not the L-function to the matrix, but the L-function to the characteristic polynomial of the matrix. That random matrices should have anything to do with number theory remains somewhat of a mystery and suggests a deep theory waiting to be uncovered.”
Random matrix theory gives Michael some intuitions about L-functions, which he then explores numerically. “Globally they share properties, but if you look more closely there’s a difference. We managed – just from the inspiration provided by random matrix theory – to make guesses as to what happens on the number theory side by adding an extra layer of arithmetic,” Michael notes. “Statistically, L-functions exhibit the same properties as the characteristic polynomials of certain kinds of large matrices, specifically unitary matrices.”
University of Waterloo Mathematics, Annual Report 2006