UW Algebra Seminar
Abstracts

Speaker: Monty McGovern, University of Washington
Title: Annihilators and associated varieties of Harish-Chandra modules
Date: January 21, 2003

Abstract: Simple Harish-Chandra modules over complex semisimple Lie algebras are typically realized as subquotients of much larger modules which tend to obscure their basic properties. In this talk I will show how to compute three basic invariants attached to a Harish-Chandra module over a classical group via combinatorial algorithms. The results will be incomplete but suggestive.

Speaker: Christopher Hacon, University of Utah
Title: Characterization of Abelian Varieties
Date: January 28, 2003

Abstract: In this talk we will illustrate the following results conjectured by Kollar: Theorem 1: Let X be a smooth complex projective variety with P_2(X)=1. Then X maps surjectively to its Albanese variety (which is a complex projective torus of dimension h^0(\Omega ^1_X)). In particular dim(X)+1 > h^0(\Omega ^1_X). Theorem 2: Let X be a smooth complex projective variety with P_2(X)=1 and dim(X)=h^0(\Omega ^1_X). Then X is birationally equivalent to a complex torus. (Both theorems are joint work with A. J. Chen)

Speaker: Sandor Kovacs, University of Washington
Title: Birational classification of algebraic varieties
Date: February 4, 2003

Abstract: In this talk I will review the current state of birational classification of varieties. One of the central notions discussed is 'Kodaira dimension'. I will also talk about the Iitaka fibration and how it allows us to concentrate on varieties whose Kodaira dimension is either negative, 0, or equal to their (usual) dimension. Finally, time permitting, I will discuss a few classes with negative Kodaira dimension. An alternative title of the talk is '1954, 1970, 1990'. The audience is invited to solve the implicit riddle in this title.

Speaker: Sandor Kovacs, University of Washington
Title: Dubbies or no Dubbies?
Date: February 11, 2003

Abstract: This is a report on recent joint work with Stefan Kebekus. The main topic is the quest to find an answer to the question, 'How far are minimal degree rational curves from being a line?' I will also mention how this question relates to the classification question discussed in the previous talk.

Speaker: Srikanth Iyengar, MSRI
Title: The Gorenstein dimension of modules over local homomorphisms
Date: February 18, 2003

Abstract: I will present some aspects of a theory of Gorenstein dimension for modules (over commutative local rings) that are finite via some local homomorphism, being developed by Sather-Wagstaff and myself. Our motive for setting up this machinery has been to prove certain results concerning the Frobenius endomorphism of commutative local rings. I hope to discuss these as well.

Speaker: Kristin Lauter, Microsoft
Title: Complex multiplication methods for generating curves over finite fields
Date: February 25, 2003

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Speaker: Balazs Szendroi, Utrecht University
Title: Families of derived equivalences in mirror symmetry
Date: March 4, 2003

Abstract: Kontsevich' Homological Mirror Symmetry Conjecture predicts the existence of interesting equivalences of derived categories of coherent sheaves in families of varieties with trivial canonical bundle. The talk explores some interesting examples of such families, and some related conjectures.

Speaker: Michael Van Opstall, University of Washington
Title: Moduli of Algebraic Varieties
Date: March 11, 2003

Abstract: After completing the program of finding good representatives of birational equivalence classes of algebraic varieties, we collect related varieties together into families. Moduli spaces are algebraic varieties whose points parameterize varieties with certain properties, and whose subvarieties correspond to families of varieties. The study of families of varieties is then equivalent to the study of the geometry of their moduli spaces. For algebraic curves, the moduli theory is well developed and many interesting geometric results are known. Some results have been obtained for surfaces, but they are primarily discouraging (the moduli spaces are disconnected, some of their connected components are reducible, and some of their irreducible components are non-reduced). I will outline the theory for curves and surfaces and use deformation theory and results for curves to prove new results for surfaces.

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Date: March 18, 2003

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