Geometry of the P=W conjecture and beyond

Time:Fri., 8:30-9:30 am Oct. 20, 2023

Venue:Tencent Meeting 221-820-248

Speaker:Junliang Shen 沈俊亮 Yale University


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.


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.

DATEOctober 20, 2023
Related News
    • 0

      Reflections on the Hodge Conjecture from an Arithmetic Geometer

      Speaker James D. Lewis is Professor at University of Alberta. His current research interests are in regulators of (higher) algebraic cycles into Hodge cohomology theories (such as Beilinson's absolute Hodge cohomology). With his colleague Matt Kerr, together initially with Stefan Mueller-Stach, they provided over the course of 10 years an explicit description of the Beilinson-Bloch regulators i...

    • 1

      Dowling-Wilson conjecture and equivariant compactification of the vector group

      Abstract:In the first part of the talk, I will give a survey of the proof of the Dowling-Wilson conjecture using the Schubert variety of a hyperplane arrangement. The Schubert variety of a hyperplane arrangement is an equivariant compactification of the vector group with finitely many orbits. In the second part of the talk, we will discuss a recent work of Colin Crowley characterizing Schubert...