### UW Algebra Seminar

Abstracts

**Speaker:** Amnon Yekutieli, Ben Gurion University

**Title:** Perverse sheaves and dualizing complexes over
noncommutative ringed schemes

**Date:** October 8, 2002

**Abstract:**
I will discuss an attempt at Grothendieck Duality on
noncommutative spaces. Since in the case of affine noncommutative
spaces (i.e. rings) Grothendieck Duality is pretty well understood,
and on the other hand we don't even know what is a noncommutative
space in general, we consider an intermediate case: a noncommutative
space Y that's an affine fibration over a commutative scheme X. That's
a fancy way to say that Y=(X,A) where A is a sheaf of quasi coherent
noncommutative rings on X. We call (X,A) a quasi coherent ringed scheme.
As usual in such circumstances, we encounter the problem of gluing. On
each affine open set U in X we have a rigid dualizing complex for A|_U
from the ring construction, and these are compatible on intersections.
But how to glue these complexes globally? One should note that Cousin
complexes, the solution devised by Grothendieck for gluing dualizing
complexes, will not work in the noncommutative world due to well known
obstructions.
Instead we choose to use perverse sheaves. This is a gluing method
invented by Bernstein-Beilinson-Deligne-Gabber in the context of
geometry of singular spaces. We discovered that the Auslander
condition of dualizing complexes over noncommutative rings (a very
algebraic property) is exactly what is needed to define perverse
modules over a noncommutative ring. And furthermore using a few nice
features of the theory, we can also extend the definition from rings
to noncommutative ringed schemes. Finally it turns out that rigid
dualizing complexes are themselves perverse sheaves of bimodules
(namely on the product X^2), so we can glue the local pieces together.
I will explain what are dualizing complexes and what they are good for
(concentrating on the noncommutative side). Then I'll discuss perverse
sheaves, the Auslander condition and how they interact. I'll finish by
sketching our construction.
The work is joint with J.J. Zhang (Seattle).

**Speaker:** Isabella Novik, University of Washington

**Title: **
Algebraic shifting

**Date:** October 15, 2002

**Abstract:**
Algebraic shifting introduced by Gil Kalai is an algebraic operation
that given a simplicial complex $\Gamma$ produces a shifted
complex $\Delta(\Gamma)$. This new complex has a simpler
combinatorial structure, yet it shares with $\Gamma$
several combinatorial, topological, and algebraic properties
such as face numbers, (topological) Betti numbers, extremal
(algebraic graded) Betti numbers, etc.
In the talk I will survey existing results and
will present several new ones on algebraic shifting and its
connections to commutative algebra and algebraic geometry.
This is a joint work with Eric Babson and Rekha Thomas.

**Speaker:** Karen Smith, University of Michigan

**Title:** Uniform Approximation of Valuation ideals

**Date:** October 17, 2002

**Abstract:**
Let E be a divisor on a variety X contracted to a point
on the affine plane under some proper birational map.
The E determines a valuation of the function field k(x, y)
and the corresponding valuation ideals I_n in k[x, y] consisting of
functions whose values are at least n form a decreasing collection of
primary ideals. An important question is: how close are these ideals
to being powers of some fixed ideal? If E is the exceptional divisor of
of a blowup at a point, then the ideals I_n are powers of the
corresponding maximal ideal, but in general, this is a difficult problem.
In the talk, we will discuss the background needed to appreciate this
problem and describe recent work towards a solution. This is
joint work with Lawrence Ein and Rob Lazarsfeld.

**Speaker:** David Berenstein, Institute for Advanced Study

**Title:** Singularities and their resolutions: a perspective from open strings

**Date:** October 22, 2002

**Abstract:**
I will address how string theory gives a natural theory to
describe
resolutions of algebro-geometric singularities by noncommutative algebras.
I will describe what requirements the noncommutative algebras need to satisfy.
Then, I will
give various examples of such resolutions (mostly Calabi-Yau singularities in
dimension 3), and some techniques that allow one
to show that they satisfy all of the
required conditions. I will also describe why the homological Ext-functors
are the natural data that the string theory provides, and why the natural
invariant for the resolution is
the bounded derived category of finitely generated modules over the algebra.

**Speaker:**

**Title:**

**Date:** October 29, 2002

**Abstract:**

**Speaker:** Doug Lind, University of Washington

**Title:** Commutative Algebra and Dynamical Systems

**Date:** November 5, 2002

**Abstract:**
The study of d commuting automorphisms of a compact
abelian group corresponds, via Pontryagin duality, to the
study of modules over the ring R of Laurent polynomials in d
commuting variables with integer coefficients. Natural
dynamical notions such as ergodicity, mixing, and
expansiveness translate into algebraic properties of the
corresponding R-module, giving rise to some off-beat
questions, and answers, in commutative algebra. Such
algebraic Z^d-actions are the only class of Z^d-actions
studied so far with a reasonably complete dynamical theory.
I'll describe, using several concrete examples,
this correspondence. Next I'll give a dynamics/algebra
"dictionary", where the prime ideals associated to an
R-module play a fundamental role. Finally, I'll describe
recent joint work with Einsiedler, Miles, and Ward that
describes the subdynamics of algebraic Z^d-actions in terms
of the complex "amoeba" of an ideal in R (the logarithmic
image of its complex variety), and show why we think that
p-adic versions of the amoeba are necessary for a complete
picture.

**Speaker:** Stefan Kebekus, Universitat Bayreuth

**Title:**
Families of Rational Curves and a Characterization of the
Projective Space

**Date:** November 19, 2002

**Abstract:**
In the beginning 80s of the last century, S. Mori showed in his
groundbreaking works that many interesting complex-projective
Manifolds
are covered by rational curves. As many of the geometrical
properties of
those spaces are reflected in the geometry of the rational curves
that
they contain, the study of families of rational curves has become a
standard tool of classification theory and Fano-geometry. In this,
the
rational curves that are of minimal degree are of particular
interest
because they are in many ways similar to projective lines. In the
talk we
will make this statement precise, and show how to use rational
curves to
prove a long-standing conjecture that characterizes the projective
space
as the space that contains the most rational curves.

**Speaker:** David Ben-Zvi, University of Chicago

**Title:** D-modules and Solitons

**Date:** December 3, 2002

**Abstract:**
We describe joint work with T. Nevins on the geometry of D-modules on
singular varieties and on smooth curves. Thinking of D-modules on a
variety Y as bundles on Y with infinitesimal parallel transport, we
show that they do not change when Y develops cusp singularities.
Thinking of D-modules on a smooth curve X as bundles (or in special
cases, configurations of points) on a noncommutative version of the
cotangent bundle to X, we use them to find a geometric bridge between
soliton equations (such as the KP and KdV hierarchies) and many-body
integrable Hamiltonian systems.

**Speaker:** Adrian Iovita, University of Washington

**Title:** Hidden Structures

**Date:** December 10, 2002

**Abstract:**
Let Z be an algebraic variety over a finite extension of Q_p
and let D^i denote its i-th de Rham cohomology group. As first
envisioned by Grothendieck D^i has frequently enough "hidden structure"
to characterize the arithmetic properties of Z. Coleman and I have been
able to make this structure "visible", i.e. explicit in some cases.
I will explain how deformation theory can be used
for this purpose.

*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).
*

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