Title: Counting representations of arithmetic groups and points of schemes

Abstract:

We will discuss the following question: How many irreducible representations of a given dimension n do groups like SLd(Z) have?
We will see how this question is related to the number of Z/nZ-points of certain schemes. 
Those are related to singularities of moduli spaces, pushforward of smooth measures, commutators of random elements in finite groups, 
jet schemes and more. 

As a result of those connections, we will show that the number of such representations is bounded by a polynomial in n
whose degree is universally bounded for high rank arithmetic groups (by 40). 

This is a joint project with Nir Avni.