Abstract

Given a compact Riemann surface C, nonabelian Hodge theory relates topological and algebro-geometric objects associated to C. Specifically, complex representations of the fundamental group are in correspondence with algebraic vector bundles on C, equipped with an extra structure called a Higgs field.

This gives a transcendental matching between two very different moduli spaces for C: the character variety (parametrizing representations of the fundamental group of C) and the so-called Hitchin moduli space of C (parametrizing vector bundles with Higgs field). In 2010, de Cataldo, Hausel, and Migliorini proposed the P=W conjecture, which gives a precise link between the topology of the Hitchin space and the Hodge theory of the character variety, imposing surprising constraints on each side.

I will introduce the conjecture, review its recent proofs, and discuss how the geometry hidden behind the P=W phenomenon is connected to other branches of mathematics.

Speaker 

Junliang Shen finished his Ph.D at ETH Zurich under Rahul Pandharipande in 2018. He was a Moore Instructor at MIT from 2018-2021, before moving to Yale as an Assistant Professor.

His research area is algebraic geometry, particularly the study of moduli spaces. He is interested in using tools from algebraic geometry to solve questions and conjectures rooted in topology, geometry, and mathematical physics. He was awarded the SwissMAP Innovator Prize in 2018.