October 28: Junliang Shen

Title: Hitchin systems, hyper-Kaehler geometry, and the P=W conjecture

Abstract:

Lagrangian fibrations play a crucial role in the study of hyper-Kaehler geometry and integrable systems. The P=W conjecture by de Cataldo, Hausel, and Migliorini suggests a surprising connection between the topology of Lagrangian fibrations and Hodge theory. In this talk, we will first discuss a compact version of this phenomenon, based on joint work with Andrew Harder, Zhiyuan Li, and Qizheng Yin. Then we will focus on interactions between compact and noncompact hyper-Kaehler geometry. Such connections lead to new progress on the P=W conjecture for Hitchin systems and character varieties. This is joint work with Mark de Cataldo and Davesh Maulik.