Georgia Institute of Technology College of Sciences

First talk at 4:00 by by Ananth Shankar (MIT http://math.mit.edu/~ananths/)

Exceptional splitting of abelian surfaces over global function fields.

Let A denote a non-constant ordinary abelian surface over a global function field (of characteristic p > 2) with good reduction everywhere. Suppose that $A$ does not have real multiplication by any real quadratic field with discriminant a multiple of $p$. Then we prove that there are infinitely many places modulo which $A$ is isogenous to the product of two elliptic curves. If time permits, I will also talk about applications of our results to the p-adic monodromy of such abelian surfaces. This is joint work with Davesh Maulik and Yunqing Tang.

Second talk at 5:15 Jordan Ellenberg (University of Wisconsin http://www.math.wisc.edu/~ellenber/)

What is the tropical Ceresa class and what should it be?

This is a highly preliminary talk about joint work with Daniel Corey and Wanlin Li.  The Ceresa cycle is an algebraic cycle canonically attached to a curve C, which appears in an intriguing variety of contexts; its height can sometimes be interpreted as a special value, the vanishing of its cycle class is related to the Galois action on the nilpotent fundamental group, it vanishes on hyperelliptic curves, etc.  In practice it is not easy to compute, and we do not in fact know an explicit non-hyperelliptic curve whose Ceresa class vanishes.  We will discuss a definition of the Ceresa class for a tropical curve, explain how to compute it in certain simple cases, and describe progress towards understanding whether it is possible for the Ceresa class of a non-hyperelliptic tropical curve to vanish.  (The answer is:  "sort of”.)  The tropical Ceresa class sits at the interface of algebraic geometry, graph theory (because a tropical curve is more or less a metric graph), and topology: for we can also frame the tropical Ceresa class as an entity governed by the mapping class group, and in particular by the question of when a product of commuting Dehn twists can commute with a hyperelliptic involution in the quotient of the mapping class group by the Johnson kernel.