Deligne's approach to the Riemann-Hilbert problem taught us that
representations of the fundamental group (say of a punctured complex
algebraic curve) correspond to regular singular connections on
algebraic vector bundles, i.e. to those connections whose solutions
satisfy a polynomial growth condition.
Thus one obtains a vast family of generalisations of spaces of
fundamental group representations by considering more general,
irregular, connections.
The aim of this course is to describe (without proof) the notion of
generalised monodromy data, Stokes data, classifying such objects and
some of the geometry of their moduli spaces. For example one obtains
new complete hyperkahler manifolds in this way, and a geometric
understanding of the so-called quantum Weyl group.
1) Review of nonabelian Hodge theory on curves and its irregular extension
2) Stokes data
3) Symplectic geometry of moduli spaces of Stokes data
4) Braiding of Stokes data (irregular mapping class groups)