Carlos Simpson

Title: Structure at infinity of character varieties and Higgs bundle moduli spaces

Abstract:
The character variety or Betti moduli space of representations of the fundamental group is not very easy to understand
algebraically. If we consider a normal crossings compactification, which has to be non-canonical because of the action of
the mapping class group, then we can form the dual boundary complex, a simplicial complex characterizing the intersections of
boundary divisor components. The link at infinity maps to the realization of this complex. On the other hand, the Higgs bundle
moduli space has a nice compactification depending on the complex structure of the Riemann surface, and it admits the
Hitchin map to an affine space. This yields a map from the link at infinity to the sphere at infinity of the Hitchin base.
We conjecture that the dual boundary complex on the Betti side is a sphere and that these two maps are the same.
I'll present some evidence for this conjecture, relate it to the P=W conjecture, and discuss how the geometric picture appears
to be related to the subject of Talk 1, touching on Kontsevich-Soibelman wall-crossing along the way.