Tuesdays 10:30-12:00 in room 7218.
Organisers: Justin Roberts, Nitu Kitchloo
As usual for our Tuesdays seminar, this is a ``learning seminar'' and
most participants will be expected to give talks. In the first meeting
on Tuesday Sept 27, we will give a brief introduction and then survey
the contents of the various lectures, so that participants can pick
their favorite topic.
The goal is to read the following paper:
The Yang-Mills equations on Riemann surfaces,
by Atiyah and Bott. There are lots of ideas and techniques in this
paper, but it is very elegantly and carefully written and contains
plenty of exposition. I have often felt that reading and digesting
this paper would be a great way to get a really solid grounding in
advanced geometry and topology. This is the main reason we have chosen
it for the subject of our seminar.
The paper concerns certain "moduli spaces" M(G,S) associated to a Lie
group G (typically SU(N)) and a two-dimensional surface S. (A moduli
space is just a space parametrising equivalence classes of some kind
of object.) The ubiquity and importance of these particular spaces
come from the fact that they can be described in lots of different
ways:
1. as spaces of flat G-connections on a principal bundle on S
2. as spaces of group homomorphisms pi_1(S) -> G, up to conjugacy
3. if S is given a complex structure (making it a Riemann surface), as
spaces of holomorphic vector bundles on S
4. as the symplectic quotients of the spaces of all
connections by the action of the gauge group
5. as spaces of minima of the "Yang-Mills energy", a function defined
on the space of all connections
Consequently, the geometry and topology of these spaces touches on
areas such as algebraic geometry, representation theory, gauge theory,
physics, etc...
The main goal of the paper is to compute the cohomology of the moduli
spaces, and the main tools are Morse theory and equivariant
cohomology, though there is also a method making striking use of the
Weil conjectures from number theory!
How much of the paper we get through remains to be seen. At least if we get
stuck we can always ask Raoul for help!
Provisional outline of lecture topics:
Sept 27: Justin Roberts Introductory meeting
Oct 4: Mike Gurvich Equivariant cohomology
Oct 11: Michael Hansen Morse theory
Oct 18: Dave Clark Connections and their curvature
Oct 25: Maia Averett The topology of gauge groups
Nov 1: Ben Cooper The Yang-Mills functional
Nov 8: Sean Raleigh Flat connections on surfaces
Nov 15: Nitu Kitchloo Holomorphic bundles
Nov 22: Maia Averett The cohomology of the moduli space of bundles
Nov 29: Justin Roberts Hermitian-Einstein-Yang-Mills theory
justin@math.ucsd.edu
