UW Algebra Seminar

Speaker: Balazs Szegedy, Microsoft
Title: Congruence subgroups and product decompositions in arithmetic groups
Date: April 6
Abstract: An arithmetic group, roughly speaking, consists of the integral points of a matrix group which is defined by polynomial equations. The most familiar example is SL(n,Z). The theory of arithmetic groups is an exciting meeting point of number theory, group theory, geometry, and combinatorics. We give a short introduction to the subject and present some recent results.

Speaker: Sunil Chebolu, University of Washington
Title: Subcategories of modules and complexes.
Date: April 13
Abstract: It is well known that the derived category of a commutative ring is formally very similar to the stable homotopy category of spectra. So one can use various tools and ideas from stable homotopy theory to understand these derived categories. This approach, initiated by Hopkins in the 1980's, proved to be very fruitful. For instance, several concepts and problems in ring theory have been studied by formulating them in a homotopy invariant fashion by passing to its derived category. I will illustrate this strategy by a theorem of Mark Hovey which classifies wide subcategories of modules using thick subcategories of complexes.

Speaker: Luis Garcia, Virginia Tech
Title: Algebraic Geometry of Bayesian Networks
Date: April 20
Abstract: We develop the necessary theory in algebraic geometry to place Bayesian networks into the realm of algebraic statistics. This new field uses computational commutative algebra for problems in experimental design and statistical modelling. In particular, we study the algebraic varieties defined by the conditional independence statements of Bayesian networks. Moreover, Bayesian networks with hidden variables are related to the geometry of higher secant varieties. Finally, a complete algebraic classification is given for Bayesian networks on at most three random variables and one hidden variable. The relevance of these results for model selection is discussed.

Date: April 27

Date: May 4

Speaker: Yuri Berest, Cornell University
Title: A_infinity-modules and Calogero-Moser spaces
Date: May 11
Abstract: We reexamine the well-known correspondence between the space of isomorphism classes of projective ideals of certain non-commutative algebras (e.g., the 1st Weyl algebra) and associated quiver varieties (e.g., Calogero-Moser spaces). We will give a new construction of this correspondence based on the notion of an A_infinity-model (A_infinity-envelope) of a projective module adapting the standard approach of rational homotopy theory. Though perhaps less geometric than other methods, our construction is much simpler and seems more natural from the point of view of homological deformation theory.

Speaker: Elena Mantovan, UC Berkeley
Title: Moduli spaces of p-divisible groups with relation to the local Langlands correspondences.
Date: May 18
Abstract: Rapoport and Zink constructed certain moduli spaces of p-divisible groups as local analogues of Shimura varieties. Just as Shimura varieties are conjecturally related to the Langlands correspondences over number fields, the Rapoport-Zink spaces are conjecturally related to the Langlands correspondences over local fields. I will discuss a conjecture explaining how the geometry of the Rapoport-Zink spaces reflects Langlands functoriality.

Speaker: Sverre Smalo, University of Trondheim
Title: Co- and Contravariantly finite subcategories
Date: May 25
Abstract: The notions of co- and contravariantly finite subcategories of categories of modules over varying rings will be introduced. Examples of constructions preserving these notions and some of the properties inherited by these subcategories will be given.

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