# Jerusalem Mathematics Colloquium

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Thursday, 3rd January 2002, 4:00 pm

Mathematics Building, Lecture Hall 2

##

Jonathan Weitsman

(University of California at Santa Cruz)

The geometry of Hamiltonian Loop Group Spaces

** Abstract: **
We study Hamiltonian actions of the loop group
*LG*, where *G* is a compact Lie group, on symplectic
Banach manifolds. In the case where the moment map
for the action is proper, we prove an infinite-dimensional
analog of Kirwan's surjectivity theorem, which shows
that the equivariant cohomology ring of the manifold
gives a set of generators for the cohomology ring of
the reduced space. Examples where this theorem apply
arise from Yang-Mills theory in two dimensions as well
as from coadjoint orbits of the loop group.

The method of proof involves the study of an appropriate
generalization of Morse-Kirwan theory to infinite
dimensions, and the construction of equivariantly
perfect Morse-Kirwan functions on Hamiltonian *LG*-spaces.
In the case of the smallest coadjoint orbit of the loop
group, our function coincides with the energy function
whose perfection plays a key role in the classical
proof of Bott periodicity. In the case of Hamiltonian
*LG*-spaces arising from Yang-Mills theory, the function
we study is closely related to the Yang-Mills functional
whose Morse-theoretic properties were first studied by
Atiyah and Bott.

(Joint work with R. Bott and S. Tolman)

Coffee, Cookies at the faculty lounge at 3:30.

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