Nonabelian varieties and wild monodromy
Every smooth compact complex algebraic curve X has an abelian variety attached
to it: its Jacobian. Topologically this can be viewed as the space of
representations of the fundamental group of X into the one dimensional unitary
group U(1), the circle. This suggests the non-abelian generalisation: replace
U(1) by U(n) for n>1, and leads to the Mumford-Narasimhan-Seshadri moduli spaces
of stable vector bundles, related to Andre Weil's project of constructing
nonabelian theta functions. More recently Hitchin and others looked at a
complexified set-up, replacing U(n) by GL_n(C). This leads to a rich class of
non-compact non-abelian moduli spaces. I'll explain how this story extends to
the case where X is allowed to be non-compact, focusing mainly on the
topological and symplectic descriptions of the moduli spaces, and the notion of
{\em wild monodromy representations}, generalising the familiar fundamental
group representations above.