Symplectic manifolds and isomonodromic deformations
Given any connection on an algebraic vector bundle on a smooth complex curve,
there are various pieces of topological data one can define, and then use to
classify such connections. I'll compare various approaches and show how the
notion of ``wild monodromy'' leads to a simple presentation of the resulting
moduli space (the wild character variety). Many varieties well-known in
representation theory occur as examples. Then I'll review the TQFT-type approach
to their symplectic/Poisson structures, using the framework of quasi-Hamiltonian
geometry (complementing the original analytic approach from 1999). Finally I'll
discuss the notion of ``wild Riemann surface'' that involves recognising that
some of the boundary conditions behave exactly like the moduli of the curve, and
how varying this structure leads to the ``wild mapping class group'' and its
action by algebraic Poisson automorphisms on the wild character variety,
generalising the usual (tame) case. The final step in this project (the twisted
case) is joint work with D. Yamakawa (arXiv:1512.08091).