Houston Kirk Colloquium: Nonabelian Hodge theory for Riemann surfaces and the P=W conjecture

Abstract: Often in algebraic geometry, there are nontrivial identifications between algebraic structures and purely topological structures.   Given a compact Riemann surface $C$, a very interesting example of this phenomenon comes from nonabelian Hodge theory (developed in the 1980's) which relates topological and algebro-geometric structures on $C$. Namely, complex representations of the fundamental group are in correspondence with algebraic vector bundles, equipped with an extra structure called a Higgs field. This gives a transcendental matching between two very different moduli spaces associated with the Riemann surface: the character variety (parameterizing representations of the fundamental group) and the Hitchin moduli space (parameterizing Higgs bundles).   In this talk, I want to introduce this circle of ideas and talk about recent progress and questions in the area, specifically the so-called P=W conjecture about the cohomology of these moduli spaces.

Host: Carl Lian

Reception to follow at Cupples I, Room 200 (Lounge) from 2:00pm to 3:00pm