Algebra, Geometry and Combinatorics Day (AlGeCom) is a one day, informal meeting of mathematicians from the University of Illinois, Purdue University and nearby universities, with interests in algebra, geometry and combinatorics (widely interpreted).Further details will be posted here as they become available. Or you may contact the University of Illinois organizers Hal Schenck and Alexander Yong, or the Purdue organizers Uli Walther and Saugata Basu .Next event : Spring, 2011Date: March 26 (Sat), 2011Location: Department of Mathematics, Purdue University.The talks will held at MATH 175.Speakers and schedule:Cofee and snacks 8h30 - 9h30
Javid Validashti (Kansas) 9h30 - 10h30 Title: .A numerical condition for equisingularityAbstract: . Multiplicity-based criteria for integral dependence play a significant role in equisingularity theory, where one would like to use numerical invariants to distinguish between members of a given a family of singularities. These criteria are based on the Hilbert-Samuel or Buchsbaum-Rim multiplicity and their variations, which rely on some kind of finiteness conditions. To explore this problem, we introduce a few notions of multiplicity without any finiteness assumption and we show that these invariants can be used in detecting integral dependence of modules and characterizing equisingularity conditions numerically. Parts of this talk are based on joint works with Bernd Ulrich and Steven L. Kleiman. Joe Lipman (Purdue) 11h - 12hTitle: Residues, Duality, and the Fundamental Class of a scheme-mapAbstract: .The duality theory of coherent sheaves on algebraic varieties goes back to Roch's half of the Riemann-Roch theorem for Riemann surfaces (1870s). In the 1950s, it grew into Serre duality on normal projective varieties; and shortly thereafter, into Grothendieck duality for arbitrary varieties and more generally, maps of noetherian schemes. This theory has found many applications in geometry and commutative algebra. We will sketch the theory in the reasonably accessible context of a variety V over a perfect field k, emphasizing the role of differential forms, as expressed locally via residues and globally via the fundamental class of V/k. (These notions will be explained.) As time permits, we will indicate some connections with Hochschild homology, and generalizations to maps of noetherian (formal) schemes. Even 50 years after the inception of Grothendieck's theory, some of these generalizations remain to be worked out. Evgeny Mukhin (IUPUI) 14h - 15hTitle: Title: Schubert Calculus and Representation Theory. Abstract: Schubert Caculus computes index of intersection of Schubert varieties in Dave Anderson (Washington) 16h - 17hTitle: Okounkov bodies, toric degenerations, and polytopesAbstract: Given a projective variety X of dimension d, a "flag" of subvarieties Y_i, and a big divisor D, Okounkov showed how to construct a convex body in R^d, and in the last few years, this construction has been developed further in work of Kaveh-Khovanskii and Lazarsfeld-Mustata. In general, the Okounkov body is quite hard to understand, but when X is a toric variety, it is just the polytope associated to D via the standard yoga of toric geometry. I'll describe a more general situation where the Okounkov body is still a polytope, and show that in this case X admits a flat degeneration to the corresponding toric variety. As an application, I'll describe some toric degenerations of flag varieties and Schubert varieties, and explain how the Okounkov bodies arising generalize the Gelfand-Tsetlin polytopes. FoodThere
are many restaurants within walking distance to the campus
(including Indian, Chinese, Irish, Middle-eastern, Thai, Japanese,
Korean, Vietnamese, Mexican etc.). There are also several
coffee-houses in and around campus as well as across the river in the
town of Lafayette. See here for dining options in the Lafayette-West Lafayette area.We will go for dinner on Sat evening at 6.30 to the Nine Irish Brothers. Accommodation Parking Parking is free on Saturdays on campus. The most convenient parking garage is on N. University street adjacent to the Math building. |