### UW Algebra Seminar

Abstracts

**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

**Abstract:**

**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.

**Speaker:**

**Title: **

**Date:** June 3, 2003

**Abstract:**

**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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543-6450 (voice); 543-6452 (TDD); 685-3885 (FAX); access@u.washington.edu (E-mail).
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