Anosov representations, Hodge theory, and Lyapunov exponents

Abstract

Discrete subgroups of semisimple Lie groups arise in a variety of contexts, sometimes "in nature" as monodromy groups of families of algebraic manifolds, and other times in relation to geometric structures and associated dynamical systems. I will discuss a class of such discrete subgroups that arise from certain variations of Hodge structure and lead to Anosov representations, thus relating algebraic and dynamical situations. Among many consequences of these relations, I will explain Torelli theorems for certain families of Calabi-Yau manifolds (including the mirror quintic), uniformization results for domains of discontinuity of the associated discrete groups, and also a proof of a conjecture of Eskin, Kontsevich, Moller, and Zorich on Lyapunov exponents. The discrete groups of interest live inside the real linear symplectic group, and the domains of discontinuity are inside Lagrangian Grassmanians and other associated flag manifolds. The necessary context and background will be explained.

Speaker

Simion Filip receive his PhD in June 2016 from the University of Chicago under the supervision of Alex Eskin. He is interested in the connections between dynamical systems and algebraic geometry, in particular between Teichmüller dynamics and Hodge theory. His recent interests also involve K3 surfaces and their special geometric properties. Simion has been appointed as a Clay Research Fellow for a term of five years beginning 1 July 2016.