Title: The Higher Cycle Operations on Graphs
Speaker: Eric Katz (Ohio State University)
Abstract: The cycle pairing on leafless graphs takes a pair of cycles to their oriented intersection. While purely combinatorial, it arose in Picard-Lefschetz theory, a branch of both algebraic geometry and analysis, as a way of studying monodromy of families of algebraic curves, variations of Hodge structures, and asymptotics of period integrals. The cycle pairing, once properly packaged, determines a graph up to two moves by the graph Torelli theorem of Caporaso and Viviani, which makes use of a classical theorem of Whitney. In this talk, we introduce the higher cycle operations, a mildly non-Abelian extension which reflects not just the edges in a cycle but their relative ordering. We relate this cycle pairing to asymptotics of iterated integrals and variations of Hodge structures on the fundamental group. We discuss the recent proof of the higher graph Torelli theorem with Raymond Cheng which is an analogue of the unipotent Torelli theorem of Hain and Pulte.
Seminar URL: https://u.osu.edu/probability/spring-2017/