Topic: Matrix coefficients of unitary representations of semisimple Lie groups November 22

Presenter: Hee Oh, Princeton University

Abstract: Let G be a semisimple algebraic group over a local field k (not of characteristic 2) and K a good maximal compact subgroup of G. A unitary representation of G is called "class one" if it has a non-trivial K-invariant vector. This talk concerns the matrix coefficients of a class one unitary representation of G with respect to K-invariant vectors. We present a class of pointwise bounds for all matrix coefficients (with respect to K-invariant vectors) of the class one part of its unitary dual (of course, except the trivial representation). These are sharper than those previously obtained by Howe and Colwing. In particular, these pointwise bounds are optimal for G=SL_n(k) (n>2) or Sp_{2n}(k) (n>1). I will try to explain the construction of these bounds for SL_n(k) case.

Topic: Scherk's First Surface, Twist-Grain-Boundaries and All That November 22

Presenter: Randall Kamien, University of Pennsylvania

Abstract: Large twist-angle grain boundaries in layered structures are often described by Scherk's first surface whereas small twist-angle grain boundaries are usually described in terms of an array of screw dislocations. I will discuss this and other minimal surfaces and will show that there is no essential distinction between minimal surface and topological defect descriptions and that, in particular, their comparative energetics depends crucially on the core structure of their screw-dislocation topological defects.

Topic: Introduction to Symplectic Field Theory November 23

Presenter: Yakov Eliashberg, Princeton University

Topic: A differential analogue of Kummer theory on semi-abelian varieties November 23

Presenter: D. Bertrand, Universite de Paris 6

Abstract: Kummer theory on a semi-abelian variety studies the fields of definition of the division points of given rational points. These extensions are controlled by the unipotent radical of the image $G$ of the Galois representations attached to certain one-motives $M$. In the 80's, $R_u(G)$ was described in detail by K. Ribet, who showed that it can be abelian only under a strong condition of a geometric nature on $M$. Linear differential equations $M$ yield monodromy representations, or more generally, differential Galois groups $G$, whose unipotent radicals can be computed with precisely the same cohomological tools as Kummer theory. We shall thus describe necessary and sufficient conditions for $R_u(G)$ to be `a big as possible' (a typical illustration is here given by polylogarithms), and show that when $M$ decomposes into three irreducible factors, $R_u(G)$ can be abelian only under a strong duallity condition, similar to Ribet's. The (much simpler) proof of the latter fact is based on a study of signs in Lagrange's bilinear concomitants.

Topic: Zeros of partition functions at first-order phase transitions November 23

Presenter: Roman Kotecky, Charles University, Prague

Topic: Symmetrized random permutations and random matrices November 29

Presenter: Jinho Baik, Princeton University/IAS

Abstract : Recently, asymptotic statistics of the longest increasing subsequence of a random permutation (Ulam's problem) turns out to have a connection with random GUE matrix. After quick review, we consider symmetrized random permutations to obtain connections with GOE, GSE random matrices, and discuss related topics such as random involutions and certain asymmetric random walk. Analytic issue of the problem is a double scaling limit of certain Toeplitz/Hankel determinants. (Joint work with Eric Rains.)

Topic: Introduction to Symplectic Field Theory November 30

Presenter: Yakov Eliashberg, Princeton University

Topic: Congruences of Neron model for tori and the Artin conductor November 30

Presenter: J. Yu, Princeton University

Topic: Constructive criteria for locatization in random operators December 1

Presenter: Michael Aizenman, Princeton University

Topic: Geometry of the group of contact transformations December 1

Presenter: Yakov Eliashberg, Princeton University

Topic: Universality in 2D Ising Model December 2

Presenter: Haru Pinson, IAS

Topic: Introduction to Symplectic Field Theory December 2

Presenter: Yakov Eliashberg, Princeton University

Title: Homeomorphisms and simple loops on surfaces December 2

Presenter: Feng Luo, Rutgers University

Topic: Higher Regularity for Globally Area Minimizing Foliations December 3

Presenter: Penny Smith, Lehigh University

Topic: Derivation of the Euler equation from Quantum mechanics December 7

Presenter: Horng-Tzer Yau, Courant Institute

Topic: The eigenvalue problem and the saturation conjecture December 7

Presenter: William Fulton, Unviersity of Michigan

Topic: Smooth dynamics and new theoretical ideas in December 8

nonequilibrium statistical mechancis

Presenter: David Ruelle, I.H.E.S.

Topic: Universality in 2D Ising Model December 9

Presenter: Haru Pinson, IAS

Topic: Introduction to Symplectic Field Theory December 9

Presenter: Yakov Eliashberg, Princeton University

Title: Planar Polygons and Special Lagrangians in Calabi-Yau Manifolds December 9

Presenter: Ciprian S. Borcea, Rider University

Abstract: The possible configurations, up to orintation-preserving isometry, for a planar $n$-gon with prescribed length for each of its edges, make-up a compact space, which is, in general, a smooth, orientable manifold of dimension $(n-3)$.Its topological type varies according a chamber structure for admissible edge-length-vectors, and can be investigated by means of Morse theory, geometric invariant theory, symplectic and toric geometry.

In adequate coordinates, the defining equations are algebraic, and yield families of complex projective varieties whose real points are the above configuration spaces. In particular, a construction used by Darboux for quadrilaterals, leads, in arbitrary dimension, to Calabi-Yau varieties. The singularities of the latter are away from the real locus, and resolutions to Calabi-Yau manifolds will contain identifiable types of special Lagrangians.

A conjecture of Strominger, Yau, and Zaslow suggests that myrror symmetry for pairs of Calabi-Yau manifolds corresponds geometrically to a duality of fibrations in special Lagrangian tori, and indeed, we do find special Lagrangian tori at appropriate points in our family.

A different, yet related complexification, and thus other examples of special Lagrangian tori on Calabi-Yau manifolds, can be obtained from the non-Euclidean scenario. We investigate in more detail the families of K3 surfaces and Calabi-Yau threefolds associated to configuration spaces of pentagons and hexagons.

Topic: TBA December 14

Presenter: Karen Smith, University of Michigan

Title: TBA December 16

Presenter: Paul Feehan, Ohio State University

Saturday, December 18, 1999

Fine Hall

11:00 Coffee, Common Room

11:30 Victor Guillemin "The Strong Szego Limit Theorem"

1:30 Sergiu Klainerman "PDE as a unified subject"

2:30 Chris Sogge "Global estimates for the wave equation"

3:30 Alice Chang "Linking the Zeta functional determinant to problems in PDE"

All are welcome! To help the organizers, please e-mail conf99@math.princeton.edu before December 9, 1999 if you are

planning to attend. Week of November 15 - 19, 1999