## Algebraic Geometry Seminar: Jim Bryan (UBC)

- Date: 01/10/2011

Jim Bryan (UBC)

University of British Columbia

What is the probability that two randomly chosen matrices with entries in a finite field commute? : On the motivic class of the commuting variety and related problems.

In 1960, Feit and Fine were interested in the question posed by the title and to answer it, they found a beautiful formula for the number of pairs of commuting n by n matrices with entries in the field F_q. Their method amounted to finding a stratification of the variety of commuting pairs of matrices into strata each of which is isomorphic to an affine space (of various dimensions). Consequently, their computation can be interpreted as giving a formula for the motivic class of the commuting variety, that is, its class in the Grothendieck group of varieties. We give a simple, new proof of their formula and we generalize it to various other settings. This is joint work with Andrew Morrison.

3:00pm-5:00pm, WMAX 110