UW Algebra Seminar
Balazs Szegedy, Microsoft
Congruence subgroups and product decompositions in arithmetic groups
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.
Sunil Chebolu, University of Washington
Subcategories of modules and complexes.
It is well known that the derived category of a commutative ring is
the stable homotopy category of spectra. So one can use various tools and
homotopy theory to understand these derived categories. This approach,
in the 1980's, proved to be very fruitful. For instance, several concepts
problems in ring
theory have been studied by formulating them in a homotopy invariant
passing to its
derived category. I will illustrate this strategy by a theorem of Mark
wide subcategories of modules using thick subcategories of complexes.
Luis Garcia, Virginia Tech
Algebraic Geometry of Bayesian Networks
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.
we study the algebraic varieties defined by the
conditional independence statements of Bayesian
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.
Yuri Berest, Cornell University
A_infinity-modules and Calogero-Moser spaces
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
(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.
Elena Mantovan, UC Berkeley
Moduli spaces of p-divisible groups with relation to the local Langlands
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.
Sverre Smalo, University of Trondheim
Co- and Contravariantly finite subcategories
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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