Research Seminar in Mathematics (MTH G450 - 30757)
Organized by Professor Jonathan Weitsman
Guest Speaker: Melissa Liu
Columbia University
Title: Moduli spaces of flat bundles over a nonorientable surface
Date: Tuesday, November 18, 2008
Time: 1:00 p.m.
Location: 509 Lake Hall
Pretalk I (1:00 - 2:00): Yang-Mills connections on orientable and nonorientable surfaces I
Break (2:00 - 2:30)
Pretalk II (2:30 - 3:00): Yang-Mills connections on orientable and nonorientable surfaces II
Abstracts of the pretalks:
In "The Yang-Mills equations over Riemann surfaces", Atiyah and Bott studied Yang-Mills functional over a Riemann surface from the poing of view of Morse theory. Nan-Kuo Ho and I generalized their study to all closed, compact, connected, possibly nonorientable surfaces. I will review the work of Atiyah and Bott and describe my joint work with Ho.
Department tea (3:30 - 4:00)
Research Talk (4:00 - 5:00): Moduli spaces of flat bundles over a nonorientable surface.
Abtract:
Let G be a compact Lie group, and let S a connected, closed,
orientable or nonorientable surface. The moduli space of
flat G-bundles over S can be identified with
Hom(\pi_1(S), G)/G. When S is orientable, the G-equivariant
Poincare series of the representation variety Hom(\pi_1(S),G)
can be computed by the Atiyah-Bott recursion relations derived
from the Morse stratification of the Yang-Mills functional.
I will describe computations of the G-equivariant Poincare series of
Hom(\pi_1(S), G) for a nonorientable surface S
when G=U(2), SU(2), U(3), SU(3). Unlike the orientable case,
the Morse stratification of the Yang-Mills functional is not
perfect, and the real kirwan map is not surjective. This is a joint
work with Nan-Kuo Ho.
Discussion at 5:00 p.m. followed by dinner with the speaker.
Previous Talks:
Fall 2008:9/16, 9/23, 10/07,10/28
Started: 23 September 2008 Last modified: 10 November 2008
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