UW Algebra Seminar
Abstracts

Speaker: Karl Schwede, University of Washington
Title: On F-injective and Du Bois singularities
Date: January 10
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.
Speaker: Gene Abrams, University of Colorado
Title: Algebraic precursors to graph C*-algebras: the Leavitt path algebras
Date: January 17
Abstract: Most of the rings one encounters as basic examples have the ``Invariant Basis Number" property: for every pair of positive integers m and n, if the free left R-modules R^m and R^n are isomorphic, then m=n. There are, however, many important classes of rings which do not have this property. At first glance such rings might seem pathological but they arise quite naturally in a number of contexts (e.g. as endomorphism rings of infinite dimensional vector spaces), and possess a significant (perhaps surprising) amount of structure. We describe a class of such rings, the (now-classical) Leavitt algebras, and then describe their recently developed generalizations, the Leavitt path algebras. One of the nice aspects of this subject is that pictorial representations (using graphs) of the algebras are readily available. In addition, there are strong connections between these algebraic structures and a class of C*-algebras, a connection which is currently the subject of great interest to both algebraists and analysts.
Speaker: Aravind Asok, University of Washington
Title: Equivariant Vector Bundles and Representations
Date: January 24
Abstract: Q: What do the following problems have in common? 1) If A and B are two n x n Hermitian matrices, what can be said about the eigenvalues of A+B in terms of the eigenvalues of A and B? 2) Can one give a uniform construction of the finite dimensional representations of a compact group? A: Both questions can be formulated as problems can be related to the geometry of algebraic varieties X with action of a group G and can be phrased in terms of equivariant vector bundles on X. I'll define what an equivariant vector bundle is, and briefly describe why their study is important. I'll then discuss work describing in ``linear algebraic" terms the category of equivariant vector bundles on certain varieties with group action. Along the way, I'll discuss what this has to do with the two problems above.
Speaker:
Title:
Date: January 31
Abstract:
Speaker: Amer Iqbal, University of Washington
Title: Quivers and Geometry
Date: February 7
Abstract: Quivers are natural objects that can be used to encode information about supersymmetric gauge theories. String theory has been used to "engineer" these supersymmetric gauge theories using geometry of Calabi-Yau 3-folds. How the quiver arises from the geometry is an interesting question. We will construct quivers for various non-compact Calabi-Yau 3-folds (e.g., total space of O(-3) over P^2) using exceptional collection of sheaves. On the mirror side we will show that these quivers are given by 3-cycles and their intersection numbers.
Speaker: Amer Iqbal, University of Washington
Title: Quivers and Geometry
Date: February 14
Abstract: Quivers are natural objects that can be used to encode information about supersymmetric gauge theories. String theory has been used to "engineer" these supersymmetric gauge theories using geometry of Calabi-Yau 3-folds. How the quiver arises from the geometry is an interesting question. We will construct quivers for various non-compact Calabi-Yau 3-folds (e.g., total space of O(-3) over P^2) using exceptional collection of sheaves. On the mirror side we will show that these quivers are given by 3-cycles and their intersection numbers.
Speaker: Tatyana Chmutova
Title: Rational Cherednik algebras of dihedral type
Date: February 21
Abstract: For every finite reflection group one can define the corresponding family of Cherednik algebras. The representation theory of these algebras is somewhat similar to the representation theory of simple Lie algebras. In particular one can define the category O of modules over a Cherednik algebra. The main problem about this category is to find the characters of the irreducible modules in O and their multiplicities in standard modules. This problem is solved only for a few special cases. In this talk I will describe the category O for the Cherednik algebra corresponding to the dihedral group.
Speaker: Adam Nyman, University of Montana
Title: Arithmetic noncommutative projective lines
Date: February 28
Abstract: There are several reasonable approaches to constructing noncommutative analogues of projective lines over a field L. We begin by presenting an approach due to M. Van den Bergh. In this approach, the usual homogeneous coordinate ring of the projective line is replaced by the n.c. symmetric algebra of an L-L bimodule. Let K be a subfield of L. An arithmetic n.c. projective line over K is the projectivization of the n.c. symmetric algebra of a K-central L-L bimodule. If L=K(t), such a space is the generic fiber of a n.c. Hirzebruch surface. In case L is finite over K, we show that the classification of arithmetic n.c. projective lines over K is dictated by the arithmetic of the extension L/K. Our main tool for carrying out the classification is an invariant analogous to the Hilbert polynomial of a graded ring.
Speaker: Rajesh Kulkarni, Michigan State University
Title: Maximal orders on algebraic surfaces
Date: March 7
Abstract: There is an essential dichotomy in the study of noncommutative projective surfaces. Namely, the corresponding graded algebras either have trivial center (the base field) or they are orders in central simple algebras over the function fields of (commutative) surfaces. The local study of such orders was carried out by Artin, Van den Bergh and others in the 80's. In this talk, we will discuss the global geometry of maximal orders on surfaces. Namely, we will explain work on classification of such orders. Time permitting, we will discuss some explicit constructions of orders.
Speaker:
Title:
Date: March 14
Abstract:

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