1) Geometry of moduli spaces of connections on curves.

The aim of this talk is to describe some basic situations where moduli
spaces of meromorphic connections on vector bundles on complex
algebraic curves arise. Subsequently it will be shown how these
examples are linked together and may be generalised.
The themes I hope to cover include:

a) Quantum differential equations and nonlinear braid group actions in
the theory of semisimple Frobenius manifolds

b) Drinfeld-Jimbo quantum groups, Poisson Lie groups and the quantum
Weyl group actions

c) Hitchin integrable systems, their meromorphic and genus zero
versions (related to work of Moser and
Adams-Harnad-Hurtubise-Previato) and their deformations: isomonodromy
systems

If time permits I will describe some of the philosophy (and theorems)
of nonabelian Hodge theory underlying this circle of ideas, and the
(new) hyperkahler manifolds that arise.


Main references:
a) Stokes Matrices, Poisson Lie Groups and Frobenius Manifolds
Invent. Math. 146, (2001) 479-506
b) Symplectic Manifolds and Isomonodromic Deformations
Adv. in Math. 163, (2001) 137-205
c) Wild non-abelian Hodge theory on curves (with O. Biquard)
Compos. Math. 140 (2004), no. 1, 179-204


2) Geometric braid group actions

I will explain how the well-known symplectic actions of braid and
mapping class groups on character varieties (spaces of representations
of the fundamental group of a Riemann surface) have a natural
generalisation when one considers moduli spaces of wild/irregular
connections on curves. The simplest perspective is to generalize the
notion of `algebraic curve with marked points'(which appears in the
usual, tame, setting) to the notion of an `irregular curve'. The "wild
mapping class group" then appears as the fundamental group of the
moduli space
of admissible deformations of an irregular curve, generalising the
usual moduli space of curves. These actions arise by integrating a
natural nonlinear connection, called
the (irregular) isomonodromy connection or the nonabelian
Gauss-Manin connection, on a family of wild character varieties fibred
over a space of admissible deformations of irregular curves.
Mathematically it seems there is just one class of completely natural
nonlinear flat connections, so it is not surprising that they appear
in many different places.
The simplest example (from 2001) explains the geometry
underlying the "quantum Weyl group" action defined by generators and
relations by Lusztig, Soibelman, and Kirillov-Reshetikhin.
One possible way to describe these connections is via wall-crossing
formulae reminiscent of those of Kontsevich-Soibelman.

Main references:
a) G-bundles, Isomonodromy and Quantum Weyl Groups
Int. Math. Res. Not. 22, (2002) 1129-1166  (arXiv:0108152)
b) Geometry and braiding of Stokes data; Fission and wild character varieties
Annals of Math., to appear (arXiv:1111.6228)



3) Simply-laced isomonodromy systems

In this talk some of the simplest examples of (irregular) isomonodromy
systems will be considered in detail (as systems of integrable
nonlinear differential equations), generalising the isomonodromy
system of Jimbo-Miwa-Mori-Sato (1979) that they studied in relation to
the quantum nonlinear Schrodinger equation. In particular a link with
Nakajima quiver varieties will be described. This leads to a precise
relation between moduli spaces of connections on curves and many
Kac-Moody root systems/Weyl groups. In particular it explains the
Okamoto (affine Weyl group)  symmetries of the fourth, fifth and sixth
Painleve equations, and puts these symmetries into the larger context
of Weyl groups for not-necessarily-affine Kac-Moody root systems. On
one hand this explains why there are such symmetries, via the
Fourier-Laplace transform, and on the other hand it shows where such
exotic root systems occur in nature. The appearance of such root
systems and Weyl groups, beyond the affine case, seems to distinguish
this theory from earlier work on soliton equations. Such (nonlinear)
isomonodromy systems often have "quantum" (or "linear") analogues. For
example  Schlesinger system quantizes to the KZ connection
and similarly the JMMS system quantizes to the the FMTV connection, of
Felder-Markov-Tarasov-Varchenko (2000). This work
on simply-laced isomonodromy systems suggests there are more
quantum connections generalizing the KZ and FMTV connections.

Main references:
a) Irregular connections and Kac-Moody root systems
June 2008 (arXiv:0806.1050)
b) Simply-laced isomonodromy systems
Publ. Math. IHES 116, No. 1 (2012) 1-68 (arXiv:1107.0874)