UW Algebra Seminar
Abstracts

Speaker: Sunil Chebolu, University of Washington
Title: Wide subcategories over noetherian regular rings.
Date: October 5
Abstract: A wide subcategory of modules is an abelian subcategory that is closed under extensions. When R is a regular coherent ring, wide subcategories of finitely presented R-modules have been classified by Mark Hovey. (The lattice of these subcategories is isomorphic to the lattice of certain specialisation closed subsets of Spec(R).) I will use this isomorphism to prove a Krull-Schmidt theorem for wide subcategories. This gives us two interesting corollaries: The first one is an application to K-theory and the second can be regarded as a structure theorem for finitely generated modules over a noetherian regular ring.
Speaker: James Zhang, University of Washington
Title: Homological transcendence degree
Date: October 12
Abstract: Let D be a division algebra over a base field k. The homological transcendence degree of D, denoted by Htr(D), is defined to be the injective dimension of the enveloping algebra of D (i.e., the tensor product over k of D with its opposite). We show that Htr has several useful properties which the classical transcendence degree has. We extend some results of Resco, Rosenberg, Schofield and Stafford, and compute Htr for several classes of division algebras. The main tool for the computation is Van den Bergh's rigid dualizing complex. This is joint work with Amnon Yekutieli.
Speaker: Murray Elder, University of St. Andrews
Title: A context-free and a counter language of geodesics for Baumslag-Solitar groups
Date: October 19
Abstract: In this talk I will describe a language of geodesic normal forms for the Baumslag-Solitar group BS(1,2) with its standard generating set, and show that the language is accepted by a one-counter automaton. It follows that the language is both context-free and "counter". I will discuss these formal language classes and prove some interesting results about how they are interrelated. The work uses a lot of geometric and combinatorial arguments, and the talk should accessible to anyone who knows the definition of a group.
Speaker: Peter Trapa, University of Utah
Title: Shimura correspondences for split real groups
Date: October 26
Abstract: About 30 years ago, Shimura showed that one could learn something about automorphic forms for SL(2,R) by studying the nonalgebraic metaplectic double cover Mp(2,R). Stripped of any arithmetic content, the relevant representation theoretic statement amounts to the observations that the metaplectic complementary series of Mp(2,R) is "half" the length of the spherical complementary series of SL(2,R). I'll explain the precise meaning of this statement and show how it generalizes to other split real groups. This is joint work with Adams, Barbasch, Paul, and Vogan.
Speaker: Matthew Kerr, University of Chicago
Title: 0-cycles and higher Abel-Jacobi Maps
Date: November 2
Abstract: One of the central results in 19th-century algebraic geometry is Abel's theorem, which in part shows how certain transcendental integrals behave as (complete) invariants of rational equivalence classes of 0-cycles (or sums of points) on a curve. In the last half-century there has been substantial effort to understand what happens in higher (co)dimension (e.g., points on a surface), with fruits such as the introduction of Chow groups (of cycles modulo rational equivalence), Griffiths's Abel-Jacobi map and Mumford's theorem, along with relations to web geometry and iterated integrals. More recently, however, there have been the conjectures of Bloch and Beilinson and the work they have inspired -- various attempts to construct (a) the motivic filtration they predict on Chow groups, and (b) invariants to detect cycles in the graded pieces. Two parallel theories have sprung up: an arithmetic approach represented by Raskind's work on l-adic higher Abel-Jacobi mappings; and a Hodge-theoretic one introduced in work of M. Saito, Griffiths, Green, and Lewis. We will attempt a very brief review of selected parts of this history, and discuss some of our own recent results on invariants (and new 0-cycles rationally inequivalent to 0) that arise out of the Hodge-theoretic approach, especially for 0-cycles on products of curves.
Speaker: Alex Wolfe, University of Michigan
Title: Volumes of Line Bundles on Algebraic Varieties
Date: November 9
Abstract: We describe an invariant for line bundles L on an algebraic variety X, the volume, that that is closely related to the Riemann-Roch problem. It descends to a continuous function on real homology classes of divisors on X, and a natural question is to compute what the function is and see what it tells us about X. We will give some answers to such questions for projective bundles.
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Date: November 16
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Speaker: Paul Hacking, Yale and MSRI
Title: Conic bundles and noncommutative geometry
Date: November 23
Abstract: A conic bundle is a 3-fold fibred over a surface with general fibre a smooth rational curve. These varieties occupy an important position in the classification of 3-folds as described by Mori's minimal model program (MMP). We show how a version of the MMP for noncommutative surfaces allows the construction of particularly simple birational models of conic bundles, verifying a conjecture of Corti. Along the way we give a explicit description of the noncommutative singularities which arise. These singularities have many nice properties, for example, they are simultaneously Clifford algebras and quotients of Azumaya algebras. This is joint work with Daniel Chan and Colin Ingalls.
Speaker: Jim Carrell, University of British Columbia
Title: Vector Fields and Equivariant Cohomology
Date: November 30
Abstract: An old result of myself and David Lieberman that says that the cohomology algebra of a smooth complex projective variety admitting holomorphic vector field V with zeroes is the associated graded of a certain filtered ring defined on the zeroes of V. Often, this ring is the cohomology ring of the zero set itself. Until recently, there hasn't been an honest geometric proof. I plan to review this result and discuss how equivariant cohomology fills in the gap.
Speaker: Joost Slingerland, Microsoft
Title: Quantum group symmetry breaking and confinement in planar physics
Date: December 7
Abstract: I give a short introduction to the way in which quantum groups (quasitriangular Hopf algebras) may appear in the description of symmetries in planar physics, followed by a discussion of the breaking of quantum group symmetries. The quantum group symmetry breaking formalism which I will describe generalises the usual description of the breaking of symmetries described by groups. The quantum group is broken down to a certain Hopf subalgebra, which may subsequently be projected onto a Hopf quotient. I apply the general theory to gauge theories in which the gauge group is broken down to a finite group. These theories enjoy a quantum group symmetry which includes the gauge symmetry. The various ways of breaking this symmetry lead to phases of the theory which have different spectra of confined and non-confined particles.

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