Topic: A review of the chaotic hypothesis November 10

Presenter: Giovanni Gallovotti, University of Rome

Topic: On a theorem of Jordan (moved at the last minute to a larger room) November 10

Presenter: J-P Serre, College de France (McDonnell A01)

Topic: The Surreal Numbers November 10

Presenter: John Conway, Princeton University

Topic: Packing odd cycles in graphs of large connectivity November 11

Presenter: Dieter Rautenbach, CNRS (Paris)

Abstract: Thomassen proved that a $2^{3^{9k}}$-connected graph either contains $k$ vertex-disjoint odd cycles or a set of at most $2k-2$ vertices hitting all odd cycles. In this talk, we show that the above statement is still valid for $2000k$-connected graphs, which is essentially best possible. Furthermore, we will show how the method might also be used to prove the existence of odd subdivisions in highly connected graphs. This is joint work with Bruce Reed.

Title: Correlation kernels arising in asymptotic representation theory November 11

Speaker: Grigori Olshanskii, IITP (Moscow) and University of Pennsylvania

Abstract: Representation theory of "big" groups (like the infinite symmetric group) provides new interesting examples of random point processes. The aim of the talk is to discuss these examples, their connection to random matrix theory and computation of the correlation functions. This is a recent joint work with Alexei Borodin (U. Penn).

Title: Morse theory of harmonic forms and Near-Minimal Singular Foliations November 11

Presenter: Gabriel Katz, Harvard University

Topic: Mathematics of soap bubbles and crystal growth November 11

Presenter: Jean Taylor, Rutgers University

Abstract: Geometric Measure Theory has been developed to provide a mathematical framework in which to solve problems such as finding surfaces of least area with a given boundary. (Think about what ``boundary'' might mean in the context of soap films on wire frames!) Although many proofs of theorems in the subject are $\epsilon$-$\delta$ extravaganzas, the net result can be mathematical models for quite everyday things such soap bubbles.

Recently, variational methods have been successfully applied to model crystal growth problems. Approximate flows are created for any given time step, by doing a sequence of minimizations. One then shows that there is a limit to the approximate flows as the time step goes to zero. This idea has been applied to model (theoretically and computationally) motion by mean curvature, grain growth in polycrystalline materials, and dendritic crystal growth such as occurs in snowflakes. Even single crystal turbine blades can have dendritic patterns of concentration in their microstructure.

Topic: Complete reducibility for groups and for Tits buildings November 12

Presenter: J.P. Serre, College de France

Topic: Counting Branched Covers of an Elliptic Curve November 12

Presenter: David J. Goldberg, Princeton University

Topic: Introduction to Symplectic Field Theory November 12

Presenter: Yakov Eliashberg, Princeton University

Topic: Property (T) and Kazhdan constants for generic hyperbolic groups November 12

Presenter: Andrej Zuk, ENS Lyon

Topic: Newton diagrams and the decay of oscillatory integrals November 15

Presenter: Jacob Sturm, Rutgers University

Topic: Morphological Instabilities of Surfaces and Growing Films November 15

Presenter: David Srolovitz, PMI/MAE, Princeton University

Abstract: While surface tension is capable of stabilizing flat surfaces against shape perturbations in unstressed solids, the presence of a stress can destabilize the surface. I will begin by discussing thermodynamic and kinetic issues associated with the stability of surfaces of stressed solids. I will show that non-hydrostatic stresses will destabilize flat surfaces of isotropic solids at wavenumbers that depend on the stress, surface tension and elastic modulus. I will then show numerical evidence that these instabilities produce cusp like features that can be thought of in terms of cracks. Next, I will consider the stability of the surface of a growing film and demonstrate that multilayer films can be stabilized against this form of morphological instability.

Topic: Arithmeticity of discrete subgroups containing lattices November 16

in horospherical subgroups

Presenter: Hee Oh, Princeton University

Abstract: Let G be a connected simple Lie group with finite center. A unipotent subgroup of G is called "horospherical" if it is the unipotent radical of a proper parabolic subgroup. Margulis conjectured that if the (real) rank of G is at least 2, any discrete subgroup of G containing lattices in a pair of opposite horospherical subgroups is an arithmetic subgroup of G. This conjecture has been settled in many cases, in particular, the cases when G is split over real (but not locally isomorphic to SL_3(R)). I will talk about some ideas in the proof as well as discuss open cases of the conjecture.

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

Finsler tori 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 November 18

non-trapping 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 symmetrizedrandom 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: Universality in 2D Ising Model December 2

Presenter: Haru Pinson, IAS

Title: TBA December 2

Presenter: Feng Luo, Rutgers University

Topic: Introduction to Symplectic Field Theory December 2

Presenter: Yakov Eliashberg, Princeton 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.