UW Algebra Seminar

Speaker: Dan Nakano, University of Georgia
Title: Support varieties for Weyl modules
Date: July 2, 2004

Abstract: Support varieties were developed around 25 years ago in pioneering work of Alperin and Carlson. The varieties allow one to provide a bridge between the representation and cohomology theory for restricted Lie algebras. In my talk I will present recent progress on connecting support varieties for Weyl modules with the classical theory of nilpotent orbits.

Speaker: Koen De Naeghel, Limburgs Universitair Centrum, Belgium
Title: Hilbert schemes of points on quantum projective planes
Date: August 11, 2004

Abstract: In algebraic geometry subschemes of dimension zero and degree n on P^2 are parameterized by the Hilbert scheme of points Hilb_n(P^2). Set-theoretically such a subscheme corresponds to n points in the plane. We replace P^2 by noncommutative deformations called quantum projective planes P^2_q. By definition this is a noncommutative projective scheme which has as coordinate ring a Koszul three dimensional Artin-Schelter regular algebra A. The Hilbert scheme of points Hilb_n(P^2_q) for such a noncommutative plane was recently constructed by Nevins and Stafford. Its objects are graded, rank one, torsion-free A-modules up to shift of grading. In general there appear, in stark contrast to the commutative case, reflexive objects which form an open subset of this Hilbert scheme. We give the possible Hilbert series and minimal resolutions of the (reflexive) objects of Hilb_n(P^2_q).

Speaker: Koen De Naeghel, Limburgs Universitair Centrum, Belgium
Title: On the incidence between strata of the Hilbert scheme of points on the projective plane
Date: August 18, 2004

Abstract: The Hilbert scheme of points on P^2 has a natural stratification given by the Hilbert series of the corresponding ideal sheaves. This stratification is related to the properties of linear systems on P^2. Unfortunately the precise inclusion relation between the closures of the strata is unknown. Under a technical condition this problem was recently solved by Guerimand in the special case where the Hilbert series of the strata are ``as close as possible'', i.e. when there is no intermediate Hilbert series which is numerically possible. We give a new proof of Guerimand's result based on deformation theory. In our approach the technical condition is not necessary and furthermore we are able to treat other incidence problems as well. Our new proof was found while investigating the corresponding noncommutative problem, that is, for Hilbert schemes of points on generic quantum projective planes. Alhough the research for this noncommutative problem is still in progress, we compare its solution with the commutative case.

Speaker: Pramathanath Sastry, Department of Mathematics, University of Toronto
Title: Residues via the Deligne-Verdier approach
Date: September 16, 2004

Abstract: The Deligne Verdier approach to duality, while elegant, has the drawback that explict formulae for traces (or residues) are hard to deduce from it, even for projective space. Recent work by the speaker on non-flat base change for Cohen-Macaulay maps makes it possible to get around this problem, and I plan to speak a little on the strategy involved in getting an explicit hold of residues (or equivalently--traces).
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