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Friday, November 4, 2005

**Abstract:** The topology of symplectic quotients is a topic of great interest in many different contexts, including combinatorics, algebraic geometry, representation theory, and gauge theory. We first give an overview and motivation for this subject, and then discuss a surjectivity result which expresses the K-theory of a symplectic quotient M//G in terms of the equivariant K-theory of the original manifold M. This result is the natural K-theoretic analogue of the Kirwan surjectivity theorem for rational cohomology. Along the way, we prove a K-theoretic version of a key lemma of Atiyah and Bott, which states that the equivariant K-theory Euler class of a G-bundle is not a zero divisor, provided that an S^1 subgroup fixes precisely the zero seciton. This lemma is a key result in equivariant symplectic geometry, and (time permitting) we discuss some further applications of this lemma. This is joint work, and work in progress, with Gregory D. Landweber.