Cours spécialisé
Symplectic geometry and gauge theory on Riemann surfaces
Philip Boalch
Email : Philip.Boalch à ens point fr
Présentation
The aim of the course is to introduce and study some fundamental moduli spaces involving connections on bundles on complex curves. Such moduli spaces pervade mathematics, from work on the Langlands program to string theory.
Contenu
-
Symplectic and quasi-Hamiltonian geometry,
- Jacobian varieties and their nonabelian analogue; the Narasimhan--Seshadri
theorem
- Higgs bundles and flat connections; nonabelian
Hodge theory on curves
- Riemann-Hilbert correspondence, character varieties and the
Stokes phenomenon
- Geometric braid group actions and
isomonodromy
Prérequis
We will often use the language of differential geometry so some
such background would be very useful, as would some expertise on
Riemann surfaces.
Bibliographie
-
A. Alekseev, A. Malkin and E. Meinrenken, Lie group valued
moment maps, Journal of differential geometry 48, (3) 1998,
pp.445-495, arXiv:dg-ga/9707021
- M. Audin, Lectures on
gauge theory and integrable systems, in: Gauge Theory and
Symplectic Geometry, (ed.s Hurtubise and Lalonde), Kluwer NATO ASI
Series C: Maths & Phys. 488, 1995
- N.J. Hitchin, Gauge
theory on Riemann surfaces, in: Lectures on Riemann surfaces
(Trieste, 1987), 99-118, World Sci. Publishing, Teaneck, NJ,
1989
- C. T. Simpson, The Hodge filtration on nonabelian
cohomology, in: Algebraic geometry---Santa Cruz 1995,
Proc. Sympos. Pure Math.62, pp. 217--281, AMS 1997,
arXiv:alg-geom/9604005