# Counting matrices over finite fields and the topology of the moduli space

Let g be a positive integer. What is the number of all 2g-tuples of invertible matrices (A_1,…,A_g, B_1,…,B_g), with entries in a finite field F_q, satisfying

[A_1,B_1][A_2,B_2]…[A_g,B_g]=Id?

The first half of the talk is about discussing why this question is interesting. We shall see that this counting problem is intimately related to the problem of describing topological invariants of certain moduli spaces which arise naturally in geometry and physics. The second half of the talk concerns computing the above number using combinatorial representation theory. There will be a break in between the two halves to allow people to flee.

Based on an ongoing project with David Baraglia.

### About Pure mathematics seminars

We present regular seminars on a range of pure mathematics interests. Students, staff and visitors to UQ are welcome to attend, and to suggest speakers and topics.

Seminars are usually held on Tuesdays from 3pm to 3.50pm.

Talks comprise 45 minutes of speaking time plus five minutes for questions and discussion.

#### Information for speakers

Researchers in all pure mathematics fields attend our seminars, so please aim your presentation at a general mathematical audience.

#### Contact us

To volunteer to talk or to suggest a speaker, email Ole Warnaar or Travis Scrimshaw.