Abstracts

(The names of the speakers are linked to the schedule of their respective talks.)

Speaker: Michel Brion Title: Hodge polynomials of some complete intersections      Abstract: PDF

Speaker: Alessio Corti Title: Quantum cohomology of toric stacks  Abstract: I describe joint work in progress with Coates, Iritani and Tseng. I give a ring presentation with generators and relations which, in good cases, is the equivariant quantum orbifold cohomology of a toric stack.

Speaker: Alicia Dickenstein Title: Classifying smooth lattice polytopes via toric fibrations  Abstract: In a recent paper Batyrev and Nill suggested that there should be a bound, N(d), such that every lattice polytope of degree d and dimension at least N(d) decomposes as a Cayley sum. We give a sharp answer to this question for smooth Q-normal lattice polytopes. We show that any smooth Q-normal lattice polytope P of dimension n and degree d is a Cayley sum of strictly combinatorially equivalent polytopes if n is greater than or equal to 2d+1. These polytopes correspond to particularly nice toric fibrations, namely toric projective bundles. The proof  relies on the study of the nef value morphism associated to the corresponding toric embedding.  Joint work with Sandra di Rocco and Ragni Piene.

Speaker: Allen Knutson Title: Frobenius splitting and juggling patterns Abstract:  While the intersection of two reduced subschemes is typically not reduced, if the two subschemes are compatibly split w.r.t. the same Frobenius splitting, then the intersection is compatibly split, which implies that it is reduced. I'll recall the foundations of this theory, and give two criteria for a hypersurface of degree n in n variables to be compatibly split, one involving counting F_p-points. Then I'll describe an algorithm that, under special circumstances, finds all the subvarieties that are compatibly split w.r.t. a given splitting. (This was recently proved to be a finite collection by Schwede and by Kumar-Mehta.) In the Grassmannian the subvarieties correspond 1:1 to juggling patterns. This part is joint with Thomas Lam and David Speyer.

Speaker:  Milen Yakimov Title: Poisson structures on flag varieties The geometry of the standard Poisson structure on a full flag variety is closely related to the intersection of Schubert cells with respect to a pair of opposite Borel subgroups. Similarly, on a partial flag variety the standard Poisson structure sees the Lusztig stratification. Such Poisson structures have special properties for cominuscule flag varieties and even more for Grassmannians. This can be used to give another proof of a recent cyclicity result of Knutson, Lam and Speyer for the Lusztig stratification of Grassmannians. Furthermore, we will prove that the quantizations of the restrictions of these Poisson structures to Schubert cells are precisely the quantum nilpotent algebras U^w_q of De Concini, Kac, and Procesi. There are certain quantum analogs of the results of Ramanathan and Kempf that intersections of opposite Schubert varieties are reduced, and Schubert varieties are linearly defined. They are due to Gorelik and Joseph. Based on those, we will work out a very concrete description of the spectra of U^w_q

Speaker:Pablo Parrilo Title: An invitation to convex geometry Abstract:  Convex algebraic geometry is an emerging field at the interface of convex optimization and real algebraic geometry. In this talk we describe the motivation, central questions, and associated computational techniques, building on the notions of sums of squares and semidefinite programming. In particular, we will focus on semialgebraic sets defined as convex hull of algebraic varieties, discussing a natural generalization of the theta body of a graph to general polynomial ideals.

Speaker: Ravi Vakil Title: The ring of invariants of n points on the projective line Abstract:  The GIT quotient of a small number of points on the projective line has long been known to have beautiful geometry. For example, the case of six points is intimately connected to the outer automorphism of S_6. We extend this picture to an arbitrary number of points, completely describing the equations of the moduli space. (In some sense there is only one equation.) The case of eight points is particularly entertaining. This is joint work with Ben Howard, John Millson, and Andrew Snowden.

Speaker:  Seth Sullivant Title: Finiteness theorems in algebraic statistics Abstract:  I will describe a range of new finiteness results for statistical models, in particular, results which say that, up to symmetry, many models in random variables with state space "tending to infinity" have finite implicit descriptions. The focus will be on applications of these ideas to Markov bases (i.e. generating sets of toric ideals), but the techniques apply to many other statistical models. The results follow from developing a suitable notion of a finite Groebner basis in polynomial rings in infinitely many variables under the action of a monoid, which leads to a theory that may be of independent interest. This is joint work with Chris Hillar.