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.
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