UC Berkeley

Commutative Algebra and Algebraic Geometry Seminar

April 22, 2008

939 Evans Hall


12:17AM: Finite group representations and local subgroups

Marty Isaacs

There appear to be a number of unexplained (and unproved) connections between the representations of a finite group G and the representations of certain subgroups of G. Given a prime p, the normalizer in G of a nontrivial subgroup of p-power order is called a "p-local" subgroup of G. It is the representations of these p-local subgroups that seem to influence the representation theory of G. This talk will discuss some of the outstanding conjectures in this area, with an emphasis on the McKay conjecture and some of its variations.

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3:45PM: Theta dualities on moduli spaces of sheaves on K3 surfaces

Alina Marian

When studying spaces of sheaves in algebraic geometry, there are natural divisors to consider in products of two, in a sense complementary, moduli spaces of sheaves on curves or surfaces. Such divisors, which can be viewed as analogues of the classical theta divisors in the Jacobian of a curve, induce maps between spaces of sections of corresponding line bundles on the two moduli spaces. Remarkably, these maps are isomorphisms in the case of curves, and have been conjectured to be isomorphisms for some moduli spaces of sheaves on surfaces, a phenomenon known as strange (or theta) duality. After explaining the problem in general, I will focus on the case of moduli of sheaves on a K3 surface. When smooth, these moduli spaces have an irreducible holomorphic symplectic structure, and the geometry is especially beautiful. I will explain the main features of this setup. I will then give concrete examples of the duality, providing evidence for the general statement in this setting. If time permits, I will also allude to the case of sheaves on abelian surfaces. The talk will partly be based on work in progress with Dragos Oprea, but will also be an introduction to the basic facts in this field.

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5:00PM: Polynomial Bounds for Rings of Invariants (after Derksen)

Cynthia Vinzant

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