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Past talks in the Seminar on Applications in Mathematics series can be found at:
http://www.math.virginia.edu/Institute/SAM.htm |
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Friday, November 21, 2003:
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Thursday, November 20, 2003:
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Tuesday, November 18, 2003:
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Tuesday, November 11, 2003:
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Topology Seminar, Thursday, November 6, 2003:
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Thursday, November 6, 2003:
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Geometry Seminar, Tuesday, November 4, 2003:
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Friday, October 31, 2003:
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Thursday, October 30, 2003:
The Dunkl-Cherednik algebras (aka the `rational' Cherednik algebras) is an interesting family of associative algebras related to a finite reflection group G. They appear as a deformation (in fact, the universal deformation) of the ring of G-equivariant differential operators when the usual partial derivatives are replaced by the Dunkl differential-reflection operators. On the other hand, they can also be viewed as a natural (`rational') degeneration of the double-affine Hecke algebras introduced by I. Cherednik. The Dunkl-Cherednik algebras have a rich representation theory which strikingly resembles the classical representation theory of semisimple complex Lie algebras. The purpose of this talk is to give a survey of (part of) this theory with a view towards applications in geometry and mathematical physics. (The talk is based on results of Ch. Dunkl, E. Opdam, R. Rouquier, and my joint work with P. Etingof and V. Ginzburg.) |
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Friday, October 10, 2003:
Zelevinsky showed that any equioriented type A quiver locus is isomorphic to an open dense subset of a Schubert variety of a partial flag variety. We show how this implies that the associated quiver polynomial is the ratio of evaluated double Schubert polynomials. Because the denominator is a simple product of linear forms and Schubert polynomials admit a nice combinatorial description, the ratio formula is a powerful tool for understanding quiver polynomials. For example, we used it to prove a conjecture of Buch and Fulton, who posited an explicit combinatorial formula for the quiver polynomial as a sum of products of Schur polynomials. This is joint with Allen Knutson and Ezra Miller. |
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Thursday, October 9, 2003:
Quiver polynomials via Grobner deformation Abstract: Thom studied cohomology classes given by the locus of points where a given map between manifolds has rank at most r. Rephrasing and generalizing, Buch and Fulton considered collections of vector bundles and maps between them, organized by a quiver (directed graph). Instead of a single integer r, the kinds of degeneracies are described by (equivalence classes of) representations of the quiver. There exist universal formulae of a combinatorial flavor for the cohomology classes of degeneracy loci, given by evaluating a universal polynomial (that depends only on the quiver) at the Chern roots of the bundles, the classical example being the Giambelli-Thom-Porteous formula. We indicate a method for computing these universal polynomials that involves the Grobner deformation of the orbit closure of a quiver representation. Our method leads naturally to beautiful combinatorial formulae, which give geometric explanations for the formulae and which apply to the computation of classical intersection numbers and genus zero 3-point Gromov-Witten invariants for the flag variety. The talk will be elementary with a nice running example. This is joint work with Allen Knutson and Ezra Miller. |
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Wednesday, September 24, 2003:
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Wednesday, September 17, 2003
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Tuesday, September 16, 2003:
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Monday, September 15, 2003:
Semisimple representation theory of quantum groups leads to polynomial invariants of links (including Jones, HOMFLY, and Kauffman polynomials). A quite different collections of algebraic tools is needed to go one dimension up and produce quantum invariants of link cobordisms and surfaces in the four-space. We will describe what appears to be the first non-trivial example of such invariant, and review the algebraic structures in its foundation: nonsemisimple representations, homotopy and derived categories of complexes, Frobenius rings and functors. |
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Monday, September 15, 2003:
We explain the construction and properties of a bigraded homology theory of links whose Euler characteristic is the Jones polynomial. |
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May 20-24, 2003 |
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Wednesday, April 30, 2003: |
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Friday, April 25, 2003: |
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Wednesday, April 23, 2003:
This is an experimental physicist's(*) overview of instances of
the use of group representation theory in "old" physics, in particular molecular spectroscopy, and in the novel field of quantum computing, via the hidden subgroup problem.
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Wednesday, April 9, 2003: |
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Wednesday, April 2, 2003: |
| [Combined notes for Ward's Intro to Coding Theory lecture series] |
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Sunday, March 16 and Monday March 17, 2003: Informal discussions of ongoing projects in coding theory and
a colloquium titled, "Some recent results in permutation decoding," by Jennifer Key.
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December 6-8, 2002 |
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