UW Algebra Seminar

Speaker: Zachary Treisman, University of Washington
Title: Algebraic Hyperbolicity
Date: April 8, 2003

Abstract: Consider the following question. Given a smooth complex projective variety X, is there a nonconstant holomorphic map C --> X? If not, X is said to be Kobayashi hyperbolic. I will survey some of the recent work on the algebro-geometric meaning of this property and of related notions of hyperbolicity for quasi-projective varieties. For example, hyperbolicity is strongly related to being of general type.

Speaker: Ravi Vakil, Stanford University
Title: A geometric Littlewood-Richardson rule
Date: April 15, 2003

Abstract: I will describe an explicit geometric Littlewood-Richardson rule, interpreted as deforming the intersection of two Schubert varieties so that they break into Schubert varieties. There are no restrictions on the base field, and all multiplicities arising are 1; this is important for applications. This rule should be seen as a generalization of Pieri's rule to arbitrary Schubert classes, by way of explicit homotopies. It has a straightforward bijection to other Littlewood-Richardson rules, such as tableaux, and Knutson and Tao's puzzles. This gives the first geometric proof and interpretation of the Littlewood-Richardson rule. It has a host of geometric consequences, which I may describe, time permitting. The rule also has an interpretation in K-theory, suggested by Buch, which gives an extension of puzzles to K-theory. The rule suggests a natural approach to the open question of finding a Littlewood-Richardson rule for the flag variety, leading to a conjecture, shown to be true up to dimension 5. Finally, the rule suggests approaches to similar open problems, such as Littlewood-Richardson rules for the symplectic Grassmannian and two-flag varieties.

Speaker: Herb Clemens, University of Utah
Title: Mimicking a proof of the classical theorem of Abel for zero cycles on surfaces of geometric genus zero
Date: April 22, 2003


Speaker: Mircea Mustata, Clay Institute/Harvard University
Title: Contact loci in arc spaces
Date: April 29, 2003

Abstract: The traditional way of looking at singularities (in Mori theory, for example) is based on certain invariants associated to divisors. I will present a different approach which is based on the geometry of the space of arcs of a given variety. One can get in this way a very explicit correspondence between divisors and certain sets of arcs on the variety, which can be considered as a version of Nash's correspondence. This is based on joint work with L. Ein and R. Lazarsfeld.

Speaker: Eric Sharpe, University of Illinois
Title: BRST=Ext
Date: May 6, 2003

Abstract: We shall review some recent work relevant to understanding the role derived categories play in physics. In particular, we shall answer one of the most basic questions one can ask -- how to see explicitly that Ext groups count massless states in open strings? We shall strive to answer physics questions, using mathematical technology.

Speaker: Ragnar-Olaf Buchweitz, University of Toronto
Title: How large is the Centre of a Derived Category?
Date: May 13, 2003

Abstract: Mainly motivated by the homological mirror symmetry conjecture, invariants of triangulated or derived categories have attracted much attention in recent years. One of the invariants worthwhile studying is the graded centre of the derived category of a ring or space, as it is naturally a graded commutative ring and target of a universal homomorphism from the Hochschild cohomology ring, encoding the theory of characteristic classes. In this talk, we present some basic examples and report on what little is known so far in general.

Speaker: Greg Smith, Barnard College (Columbia University), New York
Title: Orbifold Chow Rings of Simplicial Toric Varieties
Date: May 20, 2003

Abstract: We give a combinatorial description of the orbifold Chow ring of a simplicial toric variety. More precisely, we work with a class of smooth Deligne-Mumford stacks for which the coarse moduli space is a simplicial toric variety. If the underlying toric variety admits a crepant resolution, then there is an explicit flat deformation connecting the orbifold Chow ring and the Chow ring of the resolution. This is joint work with Lev Borisov and Linda Chen.

Speaker: Daniel Rogalski, University of Washington
Title: Projectively Simple Rings
Date: May 27, 2003

Abstract: We report on recent joint work with James Zhang and Zinovy Reichstein concerning graded rings which have as few two-sided ideals as possible. The study of these naturally leads to an interesting purely geometric question: which algebraic varieties have automorphisms which fix no subvarieties? We will discuss the case of abelian varieties in particular.

Date: June 3, 2003


Speaker: Wim Veys (Katholieke Universiteit Leuven)
Title: Stringy invariants for algebraic varieties with general singularities.
Date: June 10, 2003

Abstract: Batyrev associated interesting invariants, the stringy Euler number and stringy E--function, to algebraic varieties with at worst log terminal singularities (these are in some sense the "mild" singularities in algebraic geometry). They are defined in terms of a good resolution of the given variety. He used them for example to formulate a topological mirror symmetry test for pairs of certain Calabi--Yau varieties. It is a natural question whether one can extend these invariants beyond the log terminal case. Now at first sight this log terminality condition, and certainly the weaker condition that all log discrepancies are different from zero in a good resolution, seems essential for the definition. Nevertheless we associate generalized stringy invariants to 'almost all' algebraic varieties, more precisely to those without strictly log canonical singularities. Note that this includes cases were some log discrepancies can be zero. For surfaces our constructions are unconditionally; in higher dimensions we assume the Minimal Model Program.
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