Topic: Quantum lattice systems at intermediate temperatures February 7

Presenter: Daniel Ueltschi, Princeton University

Topic: Mather's theory on Lagrangian systems and connecting orbits February 8

Presenter: Jeff Xia, Northwestern University

Topic: Estimates of Green's functions and the kicked rotor problem February 8

Presenter: J. Bourgain, Institute for Advanced Study

Topic: Reeb chords and symplectic geometry February 8

Presenter: Klaus Mohnke, Stanford University

Abstract: Using Gromov's construction of holomorphic disks with boundary on a Lagrangian submanifold I will prove that for each Legendrian knot in the standard contact three sphere and any corresponding contact one form there exists a Reeb orbit starting and ending on the knot. I will also discuss other situations in which one can derive the existence of such Reeb chords.

Topic: p-adic L-functions of elliptic curves at supersingular primes February 8

Presenter: Robert Pollack, Harvard University

Abstract: The p-adic L-function of an elliptic curve at an ordinary prime has finitely many zeroes all encoded in a single polynomial. This polynomial has great conjectural arithmetic importance via the Main Conjecture. The supersingular case is very different as the L-series is known to have infinitely zeroes. This talk will attempt to shed some light on the arithmetic nature of this situation.

Topic: Radon Transform, inversion formulas, and a glimpse of singularity February 9

Presenter: Hadi Jorati, Princeton University

Abstract: The so called Radon transform, introduced early this century by Funk and Radon, has applications in analysis and tomography. After going around the subject for a little while, we will begin looking for generalizations and specially wonder what happens if we introduce a singular kernel to the problem and when the singularity goes on a manifold. If time permits I will sketch a proof by Christ, Nagel, Stein and Wainger about L^p bddness of singular Radon transform. It should be accessible to a wide range of audience.

Topic: Ricci curvature, minimal volumes, and Seiberg-Witten theory February 9

Presenter: Claude LeBrun, SUNY Stony Brook

Topic: A problem on differential forms coming from economics February 12

Presenter: Louis Nirenberg, Courant Institute

Topic: Taking the rough with the smooth - controlling short time behaviour February 12

Presenter: Terry Lyons, University of Oxford

Abstract: Consider a continuously evolving system with state $y$ subject to some external stimulus (or control) and modelled by the equation modelled by $dy^i(t)=f^{i,j}(y(t))dx^j(t)$. Consider the functional relating control $x$ and response $y$. In many application settings the control is far from smooth (wind on a bridge) and it becomes interesting theoretically, as well as for numerical analysis to ask how one should approximate to $x$ if one wishes to efficiently capture enough information to compute $y$ accurately. It turns out that there are interesting answers that suggest that the correct approach is via iterated integrals (or in fancy language: generalised loops and bi-algebras) and one produces an algebraic transform of the path $x$ somewhatanalogous to a fully non-commutative version of Fourier series. At least in theoretical examples, this approach can be very computationally effective. We discuss the faithfulness of this algebraic representation of paths.

Topic: Jet schemes of locally complete intersection canonical singularities February 13

Presenter: M. Mustata, University of California at Berkeley

Topic: What's (Genuinely) New in Constrained Optimization February 13

Presenter: Margarent Wright, Bell Laboratories, Lucent Technologies, Murray Hill, New Jersey

Abstract: Methods for finding the best value of a function are relatively easy to motivate and explain when there are no restrictions on the variables. Once constraints are added, however, the situation is much less clearcut. Solution techniques and convergence analysis can become so nonintuitive and so complicated that it is difficult to determine the connections, if any, between an apparently new approach and previous suggestions. We shall examine some of the most popular ideas today for treating constrained optimization problems, with special attention to novelty and related properties. This will be a fairly general talk (for non-experts in optimization).

Topic: Capacity of Quantum Channels February 14

Presenter: Mary Beth Ruskai, University of Massachusetts, Lowell

Topic: The Principle of Functoriality and Classical Groups February 14

Presenter: J. Arthur, University of Toronto and the Institute for Advanced Study

Abstract: The principle of functoriality is a far reaching, but quite precise, conjecture of Langlands that relates fundamental arithmetic information with equally fundamental analytic information. The arithmetic information arises from the solutions of algebraic equation. It includes data that classify algebraic number fields, and more general algebraic varieties. The analytic information arises from spectra of differential equations and group representations. It includes data that classify irreducible representations of reductive groups. We shall review the conjecture in elementary terms. We shall then describe an important special case that applies to representations of classical matrix groups. The ultimate goal would be to understand the representations of these groups in terms of representations of general linear groups.

Topic: Random Perturbations of 2-D Periodic Hamiltonian Flows February 15

Presenter: Leonid Koralov, Princeton University

Abstract: We consider the motion of a particle in a periodic two dimensional flow perturbed by small (molecular) diffusion. The flow is generated by a divergence free zero mean vector field. The long time behavior corresponds to the behavior of the homogenized process - that is diffusion with the constant diffusion matrix (effective diffusivity). We obtain the asymptotics of the effective diffusivity when the molecular diffusion tends to zero. In thecase of cellular flows the effective diffusivity has the order of the square root of molecular diffusion.

Topic: Variational problems from quantum field theory: February 15

The Chern-Simons-Higgs model

Presenter: Juergen Jost, Max Planck Institute, Leipzig

Topic: Holomorphic disks and applications for 3-manifolds February 15

Presenter: Zoltan Szabo, Princeton University

Topic: TBA February 19

Presenter: Paul Barford, Wisconsin University

Topic: Branched covers of the projective line and the Chow ring of the moduli February 20

space of curves

Presenter: R. Vakil, MIT

Topic: Higher-period ordered phases on the Bethe lattice February 21

Presenter: James Freericks, Georgetown University

Topic: On the smooth ergodic theory of some examples of parabolic flows February 21

Presenter: Giovanni Forni, Princeton University

Abstract: We will present recent results on the behaviour of ergodic averages of smooth functions for two examples of `parabolic' conservative flows: generic area-preserving flows (with saddle-like singularities) on higher genus surfaces and horocycle flows on (compact) surfaces of constant negative curvature. We prove that the deviation of ergodic averages from the leading behaviour determined by the ergodic theorem exhibits a power-law decay controlled by invariant distributions. In the case of flows on higher genus surfaces this result was part of a series of conjectures by M.Kontsevich and A.Zorich. The proofs are based on the analysis of the hyperbolicity properties of the appropriate `renormalization' dynamics, related to the Teichmuller flow on the moduli space in the case of flows on higher genus surfaces and to the geodesic flow in the case of horocycle flows.

Topic: Mather's theory on Lagrangian systems and connecting orbits February 22

Presenter: Jeff Xia, Northwestern University

Topic: Group negative curvature for atoroidal 3-manifolds with essential laminations February 22

Presenter: David Gabai, Caltech

Topic: A characterization of isoparamteric hypersurfaces of Clifford type February 23

Presenter: Gary Jensen, Washington University in St. Louis

Topic: L^p and dispersive estimates for the wave equation with the inverse-square February 26

potential

Presenter: Shadi Tahvildar-Zadeh, Rutgers University

Topic: Stochastic Optimization Problems in Finance February 26

Presenter: Ronnie Sircar, ORFE, Princeton University

Topic: TBA February 28

Presenter: Jinho Baik, Princeton University

Topic: TBA March 2

Presenter: Gui Changfeng, University of Connecticut and UBC

Topic: TBA March 5

Presenter: Walter Strauss, Brown University

Topic: Absorbing Boundary Conditions for Acoustics March 5

Presenter: Jan Hesthaven, Brown University

Abstract: The numerical solution of wave-dominated problems in domains of infinite extend often require careful attention to the design and application of artificial absorbing boundary conditions to enable an accurate, efficient and robust solution of the infinite problem using a smaller finite computational domain. Although this problem is almost as old as computational modeling itself and approximate solutions numerous, it remains one of the central, yet essentially open, issues in the accurate solution of a multitude of problems in, e.g., electromagnetics, gas-dynamics, aero-acoustics, and non-linear optics. Solutions to such problems become ever more important as the development of computational methods and resources enables the high-order accurate solution of very large problems over very long periods of time where even very low levels of reflections from the artificial boundary can prohibit the expected fidelity of the solution. The 1994 introduction of the Perfectly Matched Layer (PML) methods, consisting of a sponge layer capable of absorbing all incoming waves, regardless of their frequency and angel of incidence, seemed at first to essentially eliminate this critical issue for problems of electromagnetics and, shortly thereafter, for related problems in acoustics and linear elasticity. However, subsequent analysis of these scheme has exposed many problems and many open questions to address. In this talk we shall focus the attention on the construction and analysis of PML methods for problems in acoustics. We shall begin by showing that the original approach by which the PML equations are obtained, utilizing a non-physical splitting of the equations, leads to loss of strong wellposedness of the partial differential equations and, subsequently, the possibility of exponential instability of the semi-discrete form under low-order perturbations. As we shall discuss briefly, this is a general result for splitfield formulations of PML methods as illustrated by examples from electromagnetics, acoustics, and elasticity. We continue by discussing PML schemes for the special case of ambient acoustics before addressing the more general, and much more complex, question of PML schemes for general convective aero-acoustics. Rather than using physical arguments, we present a general mathematical procedure that enables the derivation of a strongly wellposed PML scheme for the case of a constant mean flow. Computational experiments show its superior performance but also exposes a very curious problem with this, and all other PML methods, when subjected to a special excitations. We shall conclude by explaining this issue and propose a solution. This work has been done in collaboration with Saul Abarbanel (Tel Aviv University) and David Gottlieb (Brown University).

Topic: TBA March 9

Presenter: Tian-Jun Li, Princeton University

Topic: TBA March 12

Presenter: Mei-Chi Shaw, University of Notre Dame

Topic: TBA March 12

Presenter: Rich Mclaughlin, University of North Carolina

Topic: Families of singular rational curves March 13

Presenter: S. Kebekus, Bayreuth

Topic: TBA March 26

Presenter: Linda Rothschild, University of California-San Diego

Topic: Stochastic Growth Models on Lattices and Trees March 26

Presenter: Thomas Liggett, University of California, Los Angeles

Abstract: For the past thirty years, probabilists have studied a number of stochastic growth models that were motivated by problems in physics and biology. One of the most important of these is known as the contact process -- growth occurs as the result of "contact" with existing individuals. Such models often exhibit phase transitions, and this is the feature that leads to most of our interest in them. Until a decade ago, the contact process was studied almost exclusively on Euclidean lattices, leading to a rather complete theory in that context. Since then, it has been discovered that the behavior of the process can be quite different on exponentially growing structures such as homogeneous trees. In particular, the phase structure is richer than it is in the lattice case. In this lecture, I will briefly describe the most important results about the contact process on Z^d, and then the contrasting results for the process on a tree. I will then discuss a variant of the contact process on a tree that has the appealing property that the critical value for the phase transition can be computed explicitly. One of the ingredients in the computation is a collection of combinatorial identities satisfied by the d-ary Catalan numbers.

Topic: TBA March 27

Presenter: J. Starr, MIT

Topic: Some mathematical challenges from materials science March 28

Presenter: J. Taylor, Rutgers University

Topic: TBA March 30

Presenter: Vincent Moncrief, Yale University

Topic: TBA April 2

Presenter: Steve Wainger, University of Wisconsin

Topic: TBA April 2

Presenter: Eric Vanden-Eijnden, CIMS, New York University

Topic: Basic facts about wavelets that the cognoscenti are sure they know April 3

but I have doubts about this

Presenter: Guido Weiss, Washington University

Topic: TBA April 6

Presenter: Daniel Pollack, University of Washington

Topic: TBA April 13

Presenter: Guan Bo, University of Tennessee

Topic: TBA April 27

Presenter: Joel Hass, Institute for Advanced Study and University of California at Davis