Department of

Mathematics

Seminar Calendar
for events the day of Friday, November 4, 2005.

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Questions regarding events or the calendar should be directed to Tori Corkery.
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Friday, November 4, 2005

3:00 pm in 141 Altgeld Hall,Friday, November 4, 2005

The K-theory of symplectic quotients

Megumi Harada   [email] (University of Toronto)

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.

4:00 pm in 341 Altgeld Hall,Friday, November 4, 2005

Interpreting Random Graphs in Pseudofinite Fields

Ozlem Beyarslan (University of Illinois at Chicago)

Abstract: A {\bf pseudofinite field} is an infinite field satisfying all first-order properties in the language {0,1,+,x} of rings which hold in {\bf all} finite fields, like having an extension of degree $n$ for every natural number $n>0$. Pseudofinite fields exist and they can be realized for example as ultraproducts of finite fields. An {\bf n-ary random graph} is a set X with a symmetric and irreflexive n-ary relation R (i.e., $R\subseteq X^n$,m such that R is $\Sym(n)$-closed and no two entries of an element of R are equal) such that for any two finite and disjoint subsets A and B of X^{n-1}, there is an $x\in X$ such that R(a,x) and $\neg R(b,x)$ for all $a\in A$ and $b\in B$. In 1974 J.L. Duret interpreted a random binary graph in a pseudofinite field. This has some important model theoretic consequences. We will show that we can interpret a random n-ary graph in pseudofinite fields.