Abstract: One knows there is a close analogy between the ring Z of integral numbers and algebraic curves X defined over a finite field. Langlands has conjectured that either over Z or over X some kinds of harmonic analysis objects (automorphic representations of GL(r) ) and some kind of algebraic objects (l-adic representations of rank r of the Galois group) should correspond to each other. This is proved over algebraic curves X, generalizing Drinfeld's proof for the case of rank r=2. One combines Grothendieck's l-adic cohomology and fixed points formula, the geometry of Drinfeld shtukas and the Arthur-Selberg trace formula.

Topic: The Infinite Numbers of Georg Cantor October 27

Presenter: John Conway, Princeton University

Topic: Non-bipartite graphs and their smallest eigenvalue October 28

Presenter: Benny Sudakov, Princeton University

Abstract: A folklore result mentioned often in the early papers of Erd\H{o}s, asserts that any loopless graph contains a bipartite subgraph with at least half of its edges. This has been extended in several papers. The maximum number of edges in a bipartite subgraph of a $d$-regular graph $G$ can be bounded in terms of the smallest eigenvalue of (the adjacency matrix of) $G$. Indeed, if $G$ has $n$ vertices and smallest eigenvalue $\lambda_n$, then it contains no bipartite subgraph with more than $n(d- \lambda_n)/4$ edges. The smallest eigenvalue of $G$ is closely related to some other properties of its bipartite subgraphs. In this talk we obtain another result based on this relation.

Let $G=(V,E)$ be a graph on $n$ vertices with diameter $D$, maximum degree d and eigenvalues $\lambda_1 \geq \lambda_2 \geq \ldots \geq \lambda_n$. If $G$ is non-bipartite then $$\lambda_n \geq -d + 1/((D+1)n).$$

This statement is optimal, up to a constant factor and improves some previous results. We also show how this result implies tight bounds on mixing rate of the random walk on d-regular graphs. (Joint work with Noga Alon.)

Topic: Mass aggregation in a sticky gas October 28

Presenter: Jaroslaw Piasecki, University of Warsaw

Topic: A homology associated to a contact form on a space of October 28

dual Legendrian curves: definition and hypotheses

Presenter: Abbas Bahri, Rutgers University

Presenter: Jean Mawhin, Université de Louvain Rm 705

Time: 4 p.m.

Topic: Estimates for multilinear forms associated to the wave equations.

Presenter: Damiano Foschi, Princeton University

Time: 5 p.m.

Location: Rutgers Hill Ctr., Rm.705

Topic: Failure of Brown Representability in Derived Categories October 28

Presenter: Daniel Christensen, Institute for Advanced Study

Topic: Elliptic Integrals and Intersection Theory October 29

Presenter: Matthew Kerr, Princeton University

Topic: A degree for the contact mapping class group October 29

Presenter: Charles Epstein, Uninversity of Pennsylvania

Topic: Numerical Determination of Chaotic Transfers; or, Orbit November 2

Design for Cheap Spaceflight

Presenter: Edward Belbruno, PACM & IOD

Topic: The Surreal Numbers November 3

Presenter: John Conway, Princeton University

Topic: Lie groups, maximal functions and pointwise ergodic theory November 8

Presenter: Amos Nevo, Princeton University & Technion Haifa

Topic: Multigrid methods and applications November 8

Presenter: Jinchao Xu, Penn State University

Abstract: The speaker will first give a brief description on the state of the art on multigrid methods for solving partial differential equations and then present some recent results and applications.

Topic: Implementation and Application of Machine-Checked Proofs November 9

Presenter: Andrew Appel, Princeton University

Topic: Flag manifolds, quantum cohomology, K-theory and Toda lattices November 9

Note: Special time & place

Presenter: A. Givental, UC Berkley

Topic: Ample divisors on holomorphic symplectic fourfolds November 9

Presenter: Brendan Hassett, University of Chicago

Abstract: Let S be a polarized K3 surface. The Picard group Pic(S) may be regarded as an integral quadratic form with respect to the intersection pairing. There is a dictionary between the geometry of S and the arithmetic properties of this form. For example, there are criteria for the existence of smooth rational curves in terms of the integers represented by the form. This yields a simple arithmetic description of the ample cone of S. Our goal is to extend this dictionary to certain higher dimensional analogs to K3 surfaces, known as holomorphic symplectic manifolds. These include punctual Hilbert schemes (i.e., desingularized symmetric products) of K3 surfaces. We give a conjectural framework generalizing the picture for K3 surfaces and provide evidence for our conjectures.

Topic: Implementation and Application of Machine-checked Proofs November 9

Presenter: Andrew Appel, Princeton Univiversity Department of Computer Science

Topic: A review of the chaotic hypothesis November 10

Presenter: Giovanni Gallovotti, University of Rome

Topic: On a theorem of Jordan November 10

Presenter: J-P Serre, College de France

Topic: TBA November 10

Presenter: John Conway, Princeton University

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: TBA November 11

Presenter: Kim Froyshev, Harvard University

Topic: Counting Orbits of Branched Covers of an Elliptic Curve November 12

Presenter: David Goldberg, Princeton University

Topic: Introduction to Symplectic Geometry November 12

Presenter: Yakov Eliashberg, Princeton University

Topic: Newton diagrams and Ho 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 in November 16

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

Title: TBA November 18

Presenter: Ronnie Lee, Yale University

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: Lie Groups and Quantum Circuits November 23

Presenter: Robert Solovay, UC Berkeley

Topic: Introduction to Symplectic Geometry 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: Introduction to Symplectic Geometry November 30

Presenter: Yakov Eliashberg, Princeton University

Topic: Constructive criteria for locatization in random operators December 1

Presenter: Michael Aizenman, Princeton University

Title: TBA December 2

Presenter: Feng Luo, Rutgers University

Topic: Introduction to Symplectic Geometry 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: Introduction to Symplectic Geometry December 9

Presenter: Yakov Eliashberg, Princeton University