UIUC Dept. of Mathematics Seminar Calendar

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                        30

Wednesday, November 5, 2008

Algebra, Geometry and Combinatorics Seminar
3:00 pm in 345 Altgeld Hall, Wednesday, November 5, 2008

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Dave Anderson (University of Michigan)
Positivity in the equivariant K-theory of flag varieties
Abstract: The torus-equivariant K-theory of a (generalized) flag variety G/P is an algebra over a Laurent polynomial ring. This algebra has two natural bases: the basis of structure sheaves of Schubert varieties, and its dual basis under the Euler-Poincare pairing. The structure constants for multiplication in either basis are Laurent polynomials. Based on a wealth of evidence, Griffeth--Ram and Graham--Kumar conjectured that the coefficients of these polynomials are positive (with respect to a certain choice of generators for the polynomial ring). These conjectures generalize theorems of Graham and Brion, on equivariant cohomology and non-equivariant K-theory, respectively. In joint work with Stephen Griffeth and Ezra Miller, we prove these positivity conjectures. I will explain our methods, which combine earlier work of Brion with new equivariant transversality techniques.

Mathematics Colloquium: Coble Memorial Lectures
4:00 pm in 245 Altgeld Hall, Wednesday, November 5, 2008

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Peter Ozsváth (Columbia University)
Heegaard Floer homology and knots II
Abstract: Heegaard Floer homology is an invariant for low-dimensional manifolds defined using Heegaard diagrams and holomorphic disks, constructed in joint work with Zoltán Szabó. These constructions can be modified to give an invariant for knots in the three-sphere, knot Floer homology, which has the structure of a bigraded Abelian group whose graded Euler characteristic is the Alexander polynomial. Unlike the Alexander polynomial, however, knot Floer homology contains precise geometric information about the knot: it encodes the knot genus, and also it can be used to determine whether or not the knot is fibered.

In the first two lectures, I will describe the main properties of the invariants, some topological applications, and a sketch of their construction. I will also describe a purely combinatorial formulation for the case of knots in the three-sphere, using grid diagrams. This formulation was discovered in joint work with Ciprian Manolescu and Sucharit Sarkar, and then further refined in joint work with Manolescu, Szabó, and Dylan Thurston.

Please join us at 3:30 p.m. in 331 Altgeld for cookies and coffee before the lecture.