Geometry and topology of wild character varieties

Wild character varieties parametrize monodromy
representations of flat meromorphic connections on compact Riemann
surfaces. They are classical objects with remarkable geometric and
topological properties. In the past twenty years new insights from
algebraic geometry lead to precise conjectures on the topological
structure and complexity of character varieties.  I will recall some
of these conjectures and will sketch a strategy for approaching
them. In particular I will describe recent joint works with Chuang,
Diaconescu, Donagi, and Nawata in which we use dualities in geometry
and physics to extract cohomological invariants of wild character
varieties from enumerative Calabi-Yau geometry and refined
Chern-Simons invariants of torus knots.