Topic: Quantum Classical Crossover in the Spin 1/2 XXZ Chain April 7

Presenter: Barry McCoy, SUNY Stony Brook

Topic: Three-manifold invariants and the Theta-Divisor April 7

Presenter: Zoltan Szabo, Princeton University

Abstract: In this talk I will discuss an invariant for three-manifolds, which we found recently with Peter Ozsvath. The invariant is defined by using a Heegaard decomposition of the three-manifold along a Riemannian surface and studying how the Theta-Divisor of behaves when the surface is degenerated along some curves that are naturally associated to the Heegaard decomposition. We prove that this invariant is independent of the Heegaard decomposition, and so it gives a topological invariant. We also relate this invariant with more classical invariants: Alexander polynomial, Turaev torsion and Casson invariant. The close relationship between our invariant and the Seiberg-Witten invariant for 3-manifolds will also be discussed.

Topic: Asymptotics of set partitions and the failure and indirect success April 8

of analytic methods

Presenter: Andrew Odlyzko, AT&T Labs

Abstract: The asymptotic behavior of the number of set partitions of an n-element set into blocks of distinct sizes is determined. This behavior is more complicated than is typical for set partition problems. Although there is a simple generating function, the usual analytic methods for estimating coefficients fail in the direct approach, and elementary approaches combined with some analytic methods are used to obtain most of the results.

Topic: On the distribution of the iterates of a class of dynamical systems April 8

associated with sigma-delta quantization and the related exponential sums

Presenter: Sinan Gunturk, Princeton University

Topic: Trace of infinite wedge, tori coverings, and random partitions April 8

Speaker: Andrei Okounkov, University of Chicago

Abstract: I shall discuss a formula which describes: 1) characters of the algebra of differential operators on the circle, 2) ramified coverings of a torus, 3) shape and all order fluctuations of a random partition. Talk will be based on joint work with S. Bloch and work in progress with A. Eskin. Talk will be independent from the talk on 04/09, but related to it.

Topic: From Representation Theory to Homotopy Groups April 8

Presenter: Don Davis, Lehigh University

Topic: Models of 3D Euler vorticity equations April 8

Presenter: Diego Cordoba, Princeton University & IAS

Topic: Rogers-Ramanujan identities: a convergence of q series, partitions, April 8

algebra, function theory and physics

Presenter: Barry McCoy, SUNY

Topic: Dual Pairs and Random matrix Theory April 9 IAS

Presenter: Martin R. Zirnbauer, Princeton University

Abstract: The traditional tool for computing the correlations of random matrix eigenvalues are orthogonal polynomilas. In this talk an alternative method, rooted in the theory of dual pairs and symplectic geometry, is presented. By tensoring the oscillator representation of the metaplectic group with the spinor representation of the spin group, a representation p of an orthosymplectic Lie supergroup OSp is obtained. Its character is the square root of a superdeterminant, and serves as a generating function for the eigenvalue correlations of unitary matrices. We then focus on the circular random matrix ensembles defined over the classical compact Lie groups. Their correlation functions are computed by considering any one of the dual pairs O(N) X OSp(2n|2n), U(N) X GL(n|n), or Sp(N) X OSp(2n|2n) inside the representation p of OSp. On integrating voer the first factor with Haar measure, and using Howe's exposition of classical invariant theory, the correlation functions become characters of an irreducible highest-weight representation of the second factor. In the large-N limit, the latter can be evaluated by a supersymmetric generalization of the Duistermaat-Heckman theorem.

Topic: Random matrices and random permutations April 9

Presenter: Andrei Okunkov, University of Chicago

Abstract: I shall discuss recent results about the connection between GUE and the Plancherel measure on the set of representations of the symmetric group, including the proof of a version of a conjecture due to Baik, Deift, and Johansson. Talk will be independent from the talk on 04/08, but related to it.

Topic: Nonlinear biharmonic equations with negative exponents April 9

Presenter: Xu Xingwang, Princeton University

Topic: Evolving real hypersurfaces on the trace of Levi form April 9

Presenter: Gerhard Huisken, Princeton University

Topic: Weak solutions of the Euler equations. Part II April 9

Presenter: Alexander I. Shnirelman, Princeton University

Topic: TBA April 13

Presenter: Dietmar Salamon, ETH University, Zurich Switzerland

Topic: An analogue for GL(n) of Serre's conjecture April 13

Presenter: Avner Ash, Ohio State University

Topic: Sharp Lieb-Thirring inequalities in high dimension and stability of matter April 14

Presenter: Ari Laptev, Royal Institute of Technology, Stockholm

Topic: Gauss-Bonnet integral and conformal compactification in dimension four April 14

Presenter: Paul Yang, Princeton University

Abstract: In this talk I will motivate the discussion by reviewing the Cohn-Vossen inequality for complete surfaces of finite total curvature and the results of Huber and Finn relating the Gauss-Bonnet integral to an isoperimetric constant for the surface. I then introduce a fourth order invariant as a variant of the four dimensional Chern-Gauss-Bonnet integrand. Analysis of the this equation allows us to extend the resultsof Cohn-Vossen/Huber/Finn to four dimensions.

Topic: Kazhdan-Lusztig Polynomials for 321-hexagon-avoiding permutations April 15

Presenter: Sara Billey, MIT

Abstract: In 1990, Deodhar proposed a combinatorial framework for determining the Kazhdan-Lusztig polynomials $P_{x,w}$ in the case where $W$ is any Coxeter group. We explicitly describe the combinatorics in the case where $W=A_n$ (the symmetric group on $n+1$ letters) and the permutation $w$ is 321-hexagon-avoiding. Our formula can be expressed in terms of a simple statistic on all subexpressions of of any fixed reduced word for $w$. As a consequence of our results on Kazhdan-Lusztig polynomials, we show that the Poincar\'e polynomial of the intersection cohomology of the Schubert variety corresponding to $w$ is $(1+q)^{l(w)}$ if and only if $w$ is 21-hexagon-avoiding. We also give a sufficient condition for the Schubert variety $X_w$ to have a small resolution. The results extend easily to those Weyl groups whose Coxeter graphs have no branch points ($B_n$, $F_4$, $G_2$). This work is joint with Gregory Warrington.

Topic: Adaptive structured mesh refinement for possible singularities April 16

in 3D fluid and plasma flows

Presenter: Christiane Marliani, NYU Courant Institute

Topic: Representations of covering groups of SL(n) April 20

Presenter: Jeffrey D. Adams, University of Maryland

Topic: TBA April 27

Presenter: Jeffrey Achter, University of Massachusetts

Topic: On the rank of elliptic curves April 29

Presenter: Joseph Silverman, Brown University

Topic: The what and the why of quantum mechanics April 30

Presenter: Sheldon Goldstein, Rutgers University