Friday, January 11, 2008 at 4:00 PM in HA 335
Todor Milanov, Stanford University
Picard-Lefschetz periods and integrable hierarchies
If X is a projective manifold with sufficiently many rational curves then there is an algebraic formalism that allows us to reconstruct the higher genus Gromov-Witten (shortly GW) invariants of X in terms of the genus-0 ones and the higher genus GW invariants of a point. It is still an open problem to determine whether the Gromov-Witten invariants of such manifolds X are governed by integrable hierarchies. So far it is known that the GW theory of a point is governed by the KdV hierarchy and the GW theory of the projective line by a certain extension of the 1-Toda lattice hierarchy.
On the other hand the algebraic formalism in GW theory makes sense in singularity theory, i.e., the study of isolated critical points of holomorphic functions. So we may define the analogue of GW invariants and ask the same question, is it true that they are governed by integrable hierarchies. In a joint work with A. Givental we showed that the answer is positive in the case of singularities of type A, D, or E. In this talk I would like to give an introduction to this subject and to advertise an approach to integrable systems based on vertex operators, Hirota quadratic (also known as bi-linear) equations, and Picard-Lefschetz periods. No previous knowledge of Gromov-Witten or singularity theories will be assumed.Friday, January 25, 2008 at 4:00 PM in HA 335
Seth Sullivant, Harvard University
Algebraic Statistics
Algebraic statistics advocates algebraic geometry as a useful language for discussing statistical and probabilistic problems. The starting point is the observation that many statistical models are described by algebraic constraints or parametrizations. I will try to illustrate this connection with some examples including Gaussian conditional independence models, log-linear models, and phylogenetic models.
Departmental tea at 3:30 pm in HA 243.
Dr. Sullivant is a candidate for an assistant professorship in the Mathematics Department.Tuesday, February 19, 2008 at 4:00 PM in HA 335
Jiping Zhang, Peking University, China
Arithmetical theory of finite groups and applications
Arithmetical properties of various invariants of finite groups have been studied intensively, especially in recent years on the sets of character degrees, lengths of conjugacy classes or element orders, etc. We survey some recent progress and applications (including to Cayley graphs).Friday, February 29, 2008 at 4:00 PM in HA 335
David Speyer, MIT
Sortable elements - beyond finite type
In a finite Coxeter group W, Nathan Reading introduced "sortable elements" in order to relate two of the objects enumerated by W-Catalan numbers - the variables in the W-cluster algebra and the noncrossing partitions for W. Research started by Nathan Reading, and completed by he and I, gives very precise and simple connections between sortable elements, cluster algebras, non-crossing partitions and semi-invariants of quiver representations. More recently, we have found analogues of our results that hold for all Coxeter groups, not only the finite ones. I will explain this work, and describe some of the intriguing new phenomena which appear when we leave the finite case. I will not assume any knowledge of clusters, non-crossing partitions or quivers and I will try to assume only a vague familiarity with Coxeter groups.Monday, March 10, 2008 at 4:00 PM in HA 330
Jiang Zeng, Université Claude Bernard Lyon-I, France
Addition theorems via continued fractions
We show connections between a special type of addition formulas, a theorem of Stieltjes and Rogers. We use different techniques to derive the desirable addition formulas. We apply our approach to derive special addition theorems for Bessel functions and confluent hypergeometric functions. We also derive several additions theorems for basic hypergeometric functions. Applications to the evaluation of Hankel determinants are also given. This talk is based on a joint work with Mourad Ismail.
Friday, March 14, 2008 at 4:00 PM in HA 335
Patricia Hersh, Indiana University
Regular cell complexes in total positivity
We give a new criterion for determining whether a finite CW complex is regular. This involves both combinatorial conditions on the closure poset and also topological conditions on the codimension one cell incidences. We will also discuss how this applies to a conjecture of Fomin and Shapiro that certain stratified totally non-negative spaces with the Bruhat intervals as their closure posets are regular CW complexes.Monday, March 24, 2008 at 4:00 PM in HA 330
Alberto De Sole, Harvard University
Poisson vertex algebras in the theory of Hamiltonian equations
We discuss some algebraic structures relevant to the theory of Hamiltonian equations and their integrals of motion. In particular, we discuss the relation between the notions of Hamiltonian operator and of Poisson vertex algebra. We also describe how to use pairs of compatible Poisson vertex algebra structures to construct infinite hierarchies of integrable Hamiltonian equations.Friday, March 28, 2008 at 4:00 PM in HA 335
Cristiano Husu, University of Connecticut
The Jacobi identity for vertex operators, and standard A_1^{(1)} and A_2^{(2)}-modules
The Jacobi identity for relative twisted vertex operators is, roughly speaking, the Jacobi identity for vertex operator algebras generalized by means of the correction factors that preserve the structure of the identity in the general case of relative twisted operators. The application of the identity to the A_{1}^{(1)} and A_{2}^{(2)} weight lattices shows how the correction factors form the generalized commutator and anti-commutator relations for the Z-operator construction of standard modules. In the A_{1}^{(1)}-case, multi-operator extensions of the Jacobi identity also describe the relationship between Z-operators and the generating function identities for the annihilating ideals of standard modules.Friday, April 18, 2008 at 4:00 PM in HA 335
Dimitar Grantcharov, San Jose State University
Categories of weight modules of Lie algebras
In the early 20th century H. Weyl classified all finite-dimensional representations of the classical Lie algebras in terms of the so-called character formula. Following works of G. Benkart, D. Britten, S. Fernando, V. Futorny, F. Lemire, A. Joseph and others, in 2000, O. Mathieu achieved a major breakthrough in the representation theory by obtaining an infinite dimensional analog of Weyl's result for the so called weight modules. In this talk we will discuss the recent developments of Mathieu's ideas and methods. More precisely, results related to the structure of the indecomposable weight modules will be presented. These results are a part of an ongoing joint project with V. Serganova.Friday, April 25, 2008 at 4:00 PM in HA 335
Tom Lada, NC State University
L-infinity modules vs OCHA's (Open-Closed Homotopy Algebras)
We will review these two related versions of the action of an L-infinity algebra on a vector space. OCHAs were inspired by open-closed string field theory. Several examples will be exhibited.
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Seminar Organizers: Bojko Bakalov and Kailash C. Misra