UW Algebra Seminar
Abstracts
Speaker:
Sverre Smalo, University of Trondheim
Title:
The finitistic dimension conjectures.
Date:
September 30
Abstract:
Let A be an algebra, and M a left A-module. The projective
dimension of M is an important numerical invariant. If one takes Nagata's
example of a commutative noetherian algebra of infinite Krull-dimension,
then it is not hard to produce modules of arbitrary projective
dimensions. One can also obtain this by just using finitely generated
modules. However, if one insists that the algebra A is of finite
k-dimension, then no example of this sort is known. This question was raised
in a paper by Bass in 1960. In fact, in that paper Bass raised two question
about modules of finite projective dimension over a finite dimensional algebra
A.
Question 1. Is the supremum of the projective dimension of all modules
of finite projective dimension equal to the supremum of finite projective
dimension of finitely generated modules of finite projective dimension?
Question 2. Is the supremeum of the projective dimension of the finitely
generated modules of finite projective dimension finite?
The answer to the first of these question is NO; the first example of
an algebra where these two numbers differ was given by Birge
Huisgen-Zimmermann. I will in this lecture give another and somewhat simpler
example.
I will also give some results and classes of algebras where the second
question is answered positively. The second question is still open in
general.
Speaker:
Sverre Smalo, University of Trondheim
Title:
Quivers, algebras, representations, and almost split sequences
Date:
October 7
Abstract:
In representation theory of finite dimensional assosiative algebras, the
Gabriel quiver is an important tool. I will explain the
relation between representations of quivers and modules, then
introduce almost split sequences, and the Auslander Reiten-quiver.
In the end I will discuss some infinite quivers and give a
criterion when the category of finite dimensional representations
of these infinite quivers have almost split sequences.
Speaker:
Joseph Gubeladze, San Francisco State University
Title:
Toric varieties with huge Grothendieck group
Date:
October 14
Abstract:
In each dimension n>2 there are many projective simplicial toric varieties whose Grothendieck groups of vector bundles are at least as big as the ground field. In particular, the conjecture that the Grothendieck groups of locally trivial sheaves and coherent sheaves on such varieties are rationally isomorphic fails badly.
Speaker:
Title:
Date:
October 21
Abstract:
Speaker:
Amnon Yekutieli, Ben Gurion University
Title:
On Deformation Quantization in Algebraic Geometry
Date:
October 28
Abstract:
We study deformation quantization of Poisson algebraic varieties.
Using the universal deformation formulas of Kontsevich, and an
algebro-geometric approach to the bundle of formal coordinate
systems over a smooth variety X, we prove existence of
deformation quantization of the sheaf of functions O_X
(assuming the vanishing of certain cohomologies). Under slightly
stronger assumptions we can classify all such deformations.
Speaker:
Diane Maclagan
Title:
Toric Hilbert schemes and representations of the McKay Quiver
Date:
November 4
Abstract:
The McKay quiver M_G is associated to a finite group G
contained in SL(n,C). When G is abelian, moduli of representations of M_G
are (variants of) the toric Hilbert scheme associated to G. I will
introduce these objects, and explain our polyhedral description of them.
This is joint work with Alastair Craw and Rekha Thomas.
Speaker:
Paul Hacking
Title:
Compact Moduli of Hyperplane Arrangements
Date:
November 18
Abstract:
The minimal model program suggests a compactification of the moduli space
of hyperplane arrangements which is a moduli space of `generalised
arrangements'. Here, a generalised arrangement consists of a (possibly
reducible) variety X which is a degeneration of projective space together
with a collection D_1,..,D_n of codimension 1 subvarieties obtained as
limits of hyperplanes. For example, in the one dimensional case, the
generalised arrangements are stable curves of genus 0 with n marked
points. We prove that the compactification coincides with a compactification
defined by Kapranov using a quotient construction, and deduce a fairly
explicit description of the generalised arrangements. Our work exploits
connections with matroids, polyhedral subdivisions, toric geometry and
logarithmic geometry. We illustrate our results by describing the case of 6
lines in the plane.
Speaker: Vitaly Vologosky
Title: The extended Prym and Torelli maps, I
Date:
December 2
Abstract:
This talk is about the extended Torelli map.
Classically, for every nonsingular curve C one can define
a principally polarized abelean variety: Jacobian JC.
Torelli's famous theorem states that the curve is uniquely
determined by its Jacobian. There is a good generalization
of the Jacobian in the case of nodal curves.
This provides the extended
Torelli morphism on corresponding moduli spaces.
I will discuss definitions of Jacobians, the
related periodic lattice polytope decompositions
and generalization of the Torelli's theorem.
Speaker: Vitaly Vologosky
Title: The extended Prym and Torelli maps, II
Date:
December 9
Abstract:
This talk is about the extended Prym map.
As well as Jacobians, Prym varieties form
an important class of abelian varieties.
The same way as the construction of the Jacobian
produces the Torelli map, the construction of the
classical Prym produces the classical Prym map. Prym
varieties, the natural extension of the classical Prym map
and properties of this map
are the topic of this second talk.
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