Let X/k be a smooth variety over a finite field. Motivated by work of Corlette-Simpson over the complex numbers, we formulate a conjecture that certain rank 2 local systems on X come from families of abelian varieties. After an introduction to l/p-adic companions, we explain how the existence of a complete set of p-adic companions can be used to approach the conjecture. We also prove Lefschetz theorems for families of abelian varieties over F_q, analogous to work of Simpson over C. This is joint work with A. Pál.

We discuss analog of several classical results about mean values of multiplicative functions over F_q[X] explaining some features that are not present in the number field setting. In the first part of the talk, which is based on the joint work with C. Pohoata and K. Soundararajan, we describe spectrum of multiplicative functions over F_q[x]. In the second part of the talk (based on the joint work with A. Mangerel and J. Teravainen), we will focus on the "corrected" function field analog of the Erdos Discrepancy Problem.

Given a random n x n matrix over the finite field of p elements for a fixed prime p, we will compute the probability that its Jordan normal form contains a specified 0-Jordan block. As n goes to infinity, we will see that our answer converges to the Cohen-Lenstra distribution, which is conjectured to compute the probability that the class group of a random imaginary quadratic field has a specified p-part (when p is odd). We will see why this happens by making connections between our statistics of random matrices and a heuristic distribution of finite abelian groups given by Cohen and Lenstra.

Much of the talk is from a joint work with Yifeng Huang and Zhan Jiang. We will not assume any background from the audience beyond basic graduate algebra classes.