UW Algebra Seminar

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.

Date: October 29, 2002


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.
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