# Some things I know and some I don’t know about moduli spaces of Higgs bundles

# Some things I know and some I don’t know about moduli spaces of Higgs bundles

**In-Person Talk **

I report on two joint works: with my current student Siqing Zhang, and with Davesh Maulik (MIT), Junliang Shen (Yale) and Siqing Zhang. The Dolbeault moduli space of Higgs bundles over a complex algebraic curve is one of the ingredients in the Nonabelian Hodge Theory of the curve. Much is known and much is not known about this theory. From my current point of view, I consider some of the structures on the cohomology ring of these moduli spaces. I will start by introducing the P=W conjecture in Nonabelian Hodge Theory, mostly as motivation for the two joint works.The first work provides a cohomological shadow of a (strictly speaking non-existing) Nonabelian Hodge Theory for curves over fields of positive characteristic, and it unearths a new pattern for moduli of Higgs bundles in positive characteristic, which we call p-multiplicativity. The second work applies the first over a finite field to provide indirect evidence for the P=W conjecture over the complex numbers.