Topic: Atoms in neutron star magnetic fields: High B asymptotics at fixed Z November 17

Presenter: Jakob Yngvason, University of Vienna

Topic: Anderson localization for quasi-periodic Schroedinger equations November 17

Presenter: M. Goldstein, IAS, University of Toronto

Abstract: The purpose of this talk to explain recent progress in the

subject achieved in the works by J.Bourgain, M.Goldstein, W.Schlag.

Topic: Geometry, Logic & Physics November 17

Presenter: John Conway, Princeton University

Topic: Minimal surfaces in normed spaces, asymptotic volume of Finsler tori November 18

and Besicovitch-type inequalities

Presenter: D. Burago, Pennsylvania State University

Topic: Non-equivalent separated nets and Jacobians of Lipschits homeomorphisms November 18

Presenter: D. Burago, Pennsylvania State University

Topic: Global Strichartz estimates and nonlinear existence results for non-trapping November 18

perturbations of the D'Alembertian

Presenter: Chris Sogge, John Hopkins University

Location: Rutgers, Hill Center Room 705, (tea is available in the 7th floor lounge of Hill Center 3:30 - 4:30 p.m.)

Title: Surgery Formula for spectral invariants of 3-manifolds November 18

Presenter: Ronnie Lee, Yale University

Abstract: From the work of Atiyah, Patodi and Singer, there are a number of interesting topological invariants of

manifolds obtained by applying geometric operators such as the Dirac or self-dual operators coupled with some natural

bundles and studying the associated eta or spectral flow invariants. Recently there have been a renewed interest on this

subject when the underlying manifold can be decomposed into two pieces as in the surgery situation. In my talk, I will give

a survey of my joint work with Sylvain Cappell and Edward Miller on studying a decomposition formula for spectral flows

in this situation as well as the work of Ulrich Bunke and Wojciechowski on a corresponding formula for eta invariants. As

an application, I will explain how these formulae can be applied to study the Casson-type invariants of 3-manifold in the

setting of Seiberg-Witten gauge theory as developed by Weiman Chan and Carey-Marcolli-Wang.

Topic: Complete reducibility for groups and for Tits buildings November 19

Presenter: J.P. Serre, College de France

Topic: Lefschetz formulas and zeta functions attached to lattices in November 19

semisimple Lie groups

Presenter: Anton Deitmar, Princeton/Exeter

Topic: Exact relations for effective tensors of composites: November 19

Towards a complete solution

Presenter: Yury Grabovsky, Temple University

Abstract: Composite materials are media that look homogeneous but in fact have complex structure (microstructure) when viewed under a microscope. These materials are finding their way into our everyday lives in objects such as skis, golf clubs, automobiles, aircraft, computers, construction components of buildings and bridges, sensors and actuators many many more. It is an important and a formidable task to predict the properties (called effective properties) of such media theoretically. The most serious obstruction in our way is the strong dependence of the effective properties of composite materials on the microstructure. So it comes as a nice surprise to come across exact formulae relating an effective tensor of a composite to the tensors of its constituents regardless of the microstructure. Such formulae have been discovered before and were rightfully regarded as rare jewels in the subject. In my talk I will describe the general theory of such formulae that we call exact relations. The new machinery allows one to harvest all exact relation in a context of virtually any coupled linear physical problem including conductivity, elasticity, piezo-electricity, and many others. The application of representation theory of rotation groups SO(2) and SO(3) makes obtaining actual exact relations feasible in rather high dimensional settings. The current work has given rise to a set of new questions in group representations, that is currently

being transformed into a beautiful theory by one of my collaborators.

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: TBA November 30

Presenter: Horng-Tzer Yau, Courant Institute

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: TBA December 2

Presenter: Feng Luo, Rutgers University

Topic: TBA 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.

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