UW Algebra Seminar
Abstracts
Speaker:
Gordon Heier, University of Michigan and Bochum University
Title:
An effective uniform bound for the Shafarevich Conjecture over
function fields
Date:
October 4
Abstract:
The Shafarevich Conjecture over function fields (now Theorem of
Parshin-Arakelov) asserts the finiteness of isomorphism classes of
families
of compact hyperbolic curves over a compact curve with prescribed
degeneracy
locus. After giving some motivation from the related case of
effective
finiteness theorems for maps between compact hyperbolic manifolds,
we will
discuss how to obtain an effective solution to the Shafarevich
Conjecture.
On Wednesday October 5 Heier will speak in the DG/PDE seminar
(Algebraic methods in the theory of finite type domains)
and that talk might
be of interest to algebraic geometers.
Speaker:
Wee Liang Gan, MIT
Title:
Harish-Chandra homomorphisms and symplectic reflection algebras for
wreath-products
Date:
October 11
Abstract:
I will speak on a current project with Etingof, Ginzburg and
Oblomkov
in which we constructed analogues of Harish-Chandra homomorphism
for invariant differential operators on the space of
representations of
certain quivers. This gives a quantum Hamiltonian reduction
description
of the spherical subalgebras of symplectic reflection algebras for
wreath-product groups. As an application, we constructed shift
functors
for the symplectic reflection algebras.
Speaker:
Conan Leung, Chinese University of Hong Kong
Title:
Gromov-Witten invariants of K3 surfaces
Date:
October 18 --- 2:30 in LOW 216
Abstract:
This is a joint seminar with DG/PDE. It will be at 2:30 in
LOW 216.
Speaker:
Konstanze Rietsch, King's College, London, and University of Waterloo
Title:
Mirror families for flag varieties G/P and the Peterson variety
Date:
October 25
Abstract:
We give a Lie-theoretic construction of a conjectural mirror family
in the sense of Givental to a general flag variety G/P, and show
that this mirror family recovers the Peterson variety presentation
for the quantum cohomology rings qH*(G/P)_{(q)}.
Speaker:
Max Lieblich, Princeton University
Title:
A geometric approach to period-index problems
Date:
November 1
Abstract:
The Brauer group is a fundamental invariant with deep roots
in both algebra and geometry. It simultaneously captures information
about the arithmetic of fields and the transcendental cohomology of
algebraic varieties (even over finite base fields!). A particular
algebraic problem -- first described by Brauer -- is to relate the two
numbers one can attach to any Brauer class: the period and the index.
I
will describe this classical problem in detail in the case of function
fields and discuss a way of approaching it in this case which makes
use
of modern geometric methods (algebraic stacks, rationally connected
varieties, moduli spaces of vector bundles, etc.). These techniques
have
yielded some positive results to date and seem to be potentially
useful
for the future.
Speaker:
Alexei Oblomokov, IAS
Title:
Generalized double affine Hecke algebras of higher rank.
Date:
November 8
Abstract:
We define generalized double affine Hecke algebras (GDAHA) of
higher rank, attached to a non-Dynkin star-like graph D. This
generalizes GDAHA of rank 1 defined earlier by Etingof, Oblomkov
and Rains. If the graph is extended D_4, then GDAHA is the
algebra defined by Sahi, which is a generalization of the
Cherednik algebra of type BC_n. We prove the formal PBW theorem
for GDAHA, and parametrize its irreducible representations in the
case when D is affine (i.e. extended D_4, E_6, E_7, E_8)
and q=1. We formulate a series of conjectures regarding
algebraic properties of GDAHA. We expect that, similarly to how
GDAHA of rank 1 provide quantizations of del Pezzo surfaces, GDAHA
of higher rank provide quantizations of deformations of Hilbert
schemes of these surfaces. The proofs are based on the study of
the rational version of GDAHA and differential equations of
Knizhnik-Zamolodchikov type. The talk is based on the joint paper
with Etingof and Gan.
Speaker:
Jason Bell, Simon Fraser University
Title:
Critical density vs. density in projective varieties
Date:
November 15
Abstract:
An infinite subset of a projective variety X is critically
dense if it has finite intersection with every proper closed
subvariety of X. We consider subsets of X of the form
{s^n(x) | n\in Z} where s \in Aut(X). Rogalski has conjectured that
for a
variety over a field of characteristic 0, such a set is
critically dense
if and only if it is Zariski dense. We look at recent progress on
this
conjecture, including its solution in the case that X is a
projective
surface. We also look at what can be said when X is a projective
variety
over a field of positive characteristic.
Speaker:
Title:
Date:
November 22
Abstract:
Speaker:
Julia Pevtsova, University of Washington
Title:
Representations and cohomology of finite group schemes I
Date:
November 29
Abstract:
In his seminal papers of 1976, Quillen introduced the study of
cohomology
and geometry arising from it as a tool to obtain intrinsic
information
about
finite groups. The most celebrated result is "Quillen stratification
theorem" which allows to stratify the variety defined by the
cohomology
ring
of a finite group $G$ in terms of the varieties associated to
elementary
abelian subgroups of $G$. A numerical corollary of Quillen's theorem
is
that
the Krull dimension of $H^*(G,k)$ for a finite group $G$ equals the
``elementary abelian rank" of $G$.
Quillen's geometric methods were further applied to the study of
$G$-modules
leading to the development of rank and support varieties and the
connection
between the two.
A parallel theory was subsequently developed by Friedlander and
Parshall
in
the context of p-restricted Lie algebras where the geometry turned
out to
be
surprisingly different and rather more complicated.
Nonetheless, it is possible to construct a uniform general theory
for all
finite group schemes - of which finite groups and restricted Lie
algebras
are particular examples - bringing together cohomology and
representation
theory in one coherent geometric picture. The theory is partially
based
on
the "Detection of cohomology elements modulo nilpotents for finite
group
schemes" theorem of Suslin which, in turn, is built upon the
original
result
of Quillen for finite groups.
Speaker:
Julia Pevtsova, University of Washington
Title:
Representations and cohomology of finite group schemes II
Date:
December 6
Abstract:
Having tied cohomology and representation theory together in the
first
lecture, I will mention some applications to the structure of the
stable
module category of a finite group scheme and then use the developed
geometry
to part from cohomology and go into more representation theoretic
direction.
I will introduce new numerical and geometric invariants associated
to a
finite dimensional representation which can be viewed as
generalizations
of ``support varieties", and state some open questions.
Parts of the talk will be based on a joint work with Eric
Friedlander and
Andrei Suslin.
Speaker:
Karl Schwede, University of Washington
Title:
On F-injective and Du Bois singularities
Date:
December 13
Abstract:
For the last 25 years people have been aware of relationships
between singularities defined by the action of Frobenius and
singularities
related to the minimal model program. With the definition of tight
closure for ideals in characteristic p in the late 1980s, a
dictionary
between the two classes of singularities began to be seriously
established. By the mid 1990's, a relationship was proved between
F-regular, F-rational and F-pure singularities with log terminal,
rational, and log canonical singularities respectively. I will talk
about
research further extending this dictionary to F-injective and Du
Bois
singularities.
To request disability accommodations, contact the Office of the ADA
Coordinator, ten days in advance of the event or as soon as possible:
543-6450 (voice); 543-6452 (TDD); 685-3885 (FAX); access@u.washington.edu (E-mail).
Back to Algebra Seminar