I wil discuss connections between geometry and arithmetic, with implications for both sides.
Character varieties are spaces that parametrize representations of
fundamental groups of Riemann surfaces with punctures, with prescribed
local monodromies around the punctures. Via Simpson's correspondence,
they are diffeomorphic to moduli spaces of semistable Higgs bundles. A
conjecture of Hausel, Letellier and Rodriguez-Villegas gives an
explicit formula for the Betti numbers of these spaces, which hints on
a connection with Hilbert schemes. In another development Gorsky,
Oblomkov, Rasmussen and Shende conjecture a connection between Hilbert
schemes and homological invariants of torus knots and links. The
purpose of this talk is to show that certain cell decompositions of
character varieties produce an explicit connection between the two
conjectures, allows us to calculate the cohomologies in some examples,
and implies the so-called curious hard Lefschetz property.
The moduli space of holomorphic differentials (with prescribed zeros and poles) on nonsingular curves is not compact since the curve may degenerate. I will discuss a compactification of these strata in the moduli space of Deligne-Mumford stable pointed curves, which includes the space of canonical divisors as an open subset. The theory leads to geometric/combinatorial constraints on the closures of the strata of holomorphic differentials and as
a consequence, one can determine the cohomology classes of the strata. This is joint work with Rahul Pandharipande.
According to a classical theorem of Elie Cartan, compact Lie groups have trivial second homotopy. I shall explain the analogue of this result
in algebraic geometry (in all characteristics). I shall also show how to compute low-degree homotopy groups of homogeneous spaces of algebraic groups in
terms of invariants of the group and the stabilizer. This is joint work with Cyril Demarche.