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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.

Bio:

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. He works on algebraic geometry and related fields.

 Video：http://archive.ymsc.tsinghua.edu.cn/pacm_lecture?html=Geometry_of_the_P_W_conjecture_and_beyond.html