### UW Algebra Seminar

Abstracts

**Speaker:** Paul Smith, University of Washington

**Title:** The McKay Correspondence, I

**Date:** July 9, 2002

**Abstract:**
Yet one more survey talk, similar to dozens of others, regarding
the connection between the regular polyhedra, the finite subgroups
of SL(2,C), the Dynkin diagrams of types A,D,E,
the Kleinian singularities and rational double points, their
resolutions, the intersection matrix/graph of the components of the
exceptional fiber, and lots more.

**Speaker:** Paul Smith, University of Washington

**Title:** The McKay Correspondence, II

**Date:** July 16, 2002

**Abstract:**
The continuation.

**Speaker:** Gene Abrams, University of Colorado at Colorado
Springs

**Title:** Surprising isomorphisms between matrix rings

**Date:** July 23, 2002

**Abstract:**
We give some examples of rings $R$ having the property that
different sized matrix rings $M_n(R)$ and $M_m(R)$ are isomorphic
but the free left $R$-modules ${}_RR^n$ and ${}_RR^m$ are not. We
then
show that two seemingly disparate examples
of such rings are in fact isomorphic. The first example utilizes a
direct
limit, while the second involves an intersection process.

**Speaker:** Francesc Perera, Queen's University, Belfast

**Title:** Non-stable K-Theory for two classes of rings.

**Date:** July 30, 2002

**Abstract:**
We shall discuss some recent results concerning the
K-theoretical behaviour of the class of exchange rings and that of
QB-rings, that show some surprising similarities. Definitions and
examples will also be given, and some open problems presented.

**Speaker:** Ken Goodearl, Univ. of California at Santa Barbara

**Title:** Algebraic tori acting on quantized coordinate rings

**Date:** August 6, 2002

**Abstract:**
Under the heading of ``quantum groups'', there are many algebras
viewed
as ``quantized'' versions of coordinate rings of affine algebraic
groups or varieties. These algebras support very natural actions of
algebraic tori, actions which provide some framework for
understanding
these algebras. In particular, in ``generic'' situations (when
suitable
parameters are not roots of unity), the prime spectrum of the
algebra
has a finite stratification in which each stratum is homeomorphic
to a
classical scheme, namely the prime spectrum of a Laurent polynomial
ring over a field. I will introduce the general framework arising
from
rational actions of tori on noncommutative algebras, discuss some
of
the ideas behind the framework, and illustrate the picture as it
applies to some quantized coordinate rings.

**Speaker:** Birge Huisgen-Zimmermann, Univ. of California at Santa Barbara

**Title:** Finite dimensional representation theory by way of
Grassmannians

**Date:** August 13, 2002

**Abstract:**
I will review the classical variety of representations of
a fixed dimension d over a finite dimensional algebra, the pertinent
GL(d)-action which this variety carries, and the problem of
understanding
the closures of the orbits under this action. (The points in the closure
of an orbit GL(d).X are called degenerations of X.) I will then
introduce and explore Grassmannian alternates to these traditional
varieties, which encode representation-theoretic information in a far
more accessible geometric form. The main result presented along this line
addresses the structure of degenerations.

**Speaker:** Edward S. Letzter, Temple University

**Title:**
Topological aspects of noncommutative affine spaces

**Date:** August 16, 2002

**Abstract:
**
In the first part of this talk we describe an elementary
generalization of the Zariski topology, applicable to the set of
isomorphism classes of simple modules over a ring R. This topology is
noetherian when R is noetherian, and this topology can distinguish
(e.g.) between the Weyl algebra and a field. In the second part of this
talk we discuss continuity, and related properties, for morphisms of
noncommutative spectra.

**Speaker:** Iain Gordon, Univ. of Glasgow

**Title:** Symplectic reflection algebras (Part I)

**Date:** August 20, 2002

**Abstract:**
Symplectic reflection algebras were introduced in 2000 by Etingof
and Ginzburg, providing a uniform setting for earlier work of many
authors on deformations of Kleinian singularities and more generally
certain quiver varieties, and Dunkl operators for Weyl groups.
In the first lecture, we will discuss the basic properties of symplectic
reflection algebras (including the defintion!), focusing on links
to ring theory, quiver varieties and certain singular algebraic
varieties.
In the second lecture, we will apply some finite dimensional
representation theory for symplectic reflection algebras to say
something about relatively recent (i.e.the last 10 years) problems
in the invariant theory of Weyl groups.

**Speaker:** Iain Gordon, Univ. of Glasgow

**Title:** Symplectic reflection algebras (Part II)

**Date:** August 27, 2002

**Abstract:**
Symplectic reflection algebras were introduced in 2000 by Etingof
and Ginzburg, providing a uniform setting for earlier work of many
authors on deformations of Kleinian singularities and more generally
certain quiver varieties, and Dunkl operators for Weyl groups.
In the first lecture, we will discuss the basic properties of symplectic
reflection algebras (including the defintion!), focusing on links
to ring theory, quiver varieties and certain singular algebraic
varieties.
In the second lecture, we will apply some finite dimensional
representation theory for symplectic reflection algebras to say
something about relatively recent (i.e.the last 10 years) problems
in the invariant theory of Weyl groups.

*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).
*

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