Tuesday, February 2, 2021
02/02/2021 - 9:00am
In this talk I will introduce my thesis work in progress to prove a Gross-Zagier formula for CM cycles over Shimura curves. The formula connects the global height pairing of special cycles in Kuga varieties over Shimura curves with the derivatives of the L-functions associated to weight-2k modular forms. As a key original ingredient of the proof, I will introduce some harmonic analysis on local systems over graphs, including an explicit construction of Green's function, which we apply to compute some local intersection numbers.
02/02/2021 - 4:00pm
Abstract: A train track map is a self-map of a graph with particularly nice properties. Train track maps and their cousins, relative train track maps, were developed by Bestvina and Handel in 1992 to prove the Scott conjecture: the fixed subgroup of an automorphism of a finite-rank free group has rank bounded by the rank of the free group. Since then, relative train track maps, particularly in their modern incarnation as CTs, have become perhaps the main tool in studying outer automorphisms of free groups. We will meet (relative) train track maps and describe a generalization of them to graphs of groups. As an application, we will see an index inequality implying a version of the Scott conjecture for automorphisms of free products.