Akshat Mudgal

On commuting pairs of matrices

Given an integer d>1 and a finite, non-empty set X of d x d matrices with integer entries, one is interested in counting the number of matrices A, B in X such that AB = BA. Recently, Browning–Sawin–Wang investigated this question in the case when d>=3 and X is the set of all integer matrices with entries in {-N,...,N}. Inspired by their work, I analysed the d=2 case of this problem but having replaced {-N,...,N} with an arbitrary, finite set of integers. The latter connects nicely to topics in incidence geometry, sum-product phenomenon and growth in groups.

In this talk, I will give a survey of the above results and the methods involved, and mention some more progress on this type of question when d=3. Part of this involves joint work with Jonathan Chapman.