Topic: Capacity of Quantum Channels February 14

Presenter: Mary Beth Ruskai, University of Massachusetts, Lowell

Abstract: In recent years the possibility of quantum computation has attracted considerable attention. However, quantum particles can also be used to transmit information; moreover, quantum communication is more feasible experimentally. As with classical communication, one can expect that signals sent using quantum particles will encounter some form of noise. This gives rise to a number of mathematical questions. The first is to find an appropriate model for noise in a quantum channel. One of the others is to extend Shannon's "noisy coding theorem" to quantum channels. However, the theory is much richer because multiple uses of the channel can involved entangled, as well as product, signals and decoding devices. This talk will give an overview of this subject for non-specialists. Properties of the von Neumann entropy, introduced over 20 years before the Shannon entropy, play an important role. The talk will conclude with some recent results and open questions.

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: Combinatorial Nullstellensatz February 16

Presenter: Peter Keevash, Princeton University

Abstract: I'll present highlights from a paper by Noga Alon that elegantly proves a variety of results by defining appropriate polynomials and applying a version of the Nullstellensatz.

Topic: Periodizations of functions February 19

Presenter: Oleg Kovrijkine, Institute for Advanced Study

Topic: Performance Measurements in the Internet February 19

Presenter: Paul Barford, Wisconsin University

Abstract: Accurate and representative measurement is an essential component in the development of models which are meant to describe a system. Within the context of the Internet, in addition to modeling, measurements can also used for real-time decision making such as determining where to route client requests or when performance of a server falls below a specified threshold. Measurements are commonly used for these and other applications however the task of making measurements in the Internet which are accurate and representative is fraught with difficulty. In this talk, we will present an overview of Internet measurement techniques and Internet measurement projects in both research and commercial settings. We will present data from our current study which highlights differences between the two principal techniques for measuring Internet performance: passive capture of packet traffic and active probing of the network. We will also describe the challenges associated with interpreting measurement data and present a means for applying critical path analysis (CPA) to analyze TCP transactions. CPA enables the precise set of events which determine delay to be extracted. We present initial results of applying CPA to Internet data transfers which show how latency in the principal components of an end-to-end transaction contribute to the overall file transfer delay.

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

Abstract: The Bethe lattice is bipartite, and hence can be easily shown to display period-two long-range-order. We illustrate how the Bethe lattice generically displays higher-period ordered phases by explicitly finding the period-three regions of stability for the spinless Falicov-Kimball model in the infinite-coordination-number limit. Surprisingly, we discover that the while the phase transition to the period-two phase is second order, the transition to a period-three phase is always first-order and occurs at very low temperatures. We also find the presence of "bound-states" in the spectrum, as the interacting density of states develops delta-function contributions for the period-three phase. Hence, even though higher-period phases do exist, they only occur at very low temperatures and in a small region of phase space.

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: What numbers can be represented by a quadratic form? February 22

Presenter: Jonathan Hanke, Rutgers University

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: Ideal Membership in Polynomial Rings over the Integers: Kronecker's Problem February 27

Presenter: M. Aschenbrenner, University of Illinois at Champaign-Urbana

Topic: Exponential convergence to nonequilibrium stationary states in classical February 28

statistical mechanics

Presenter: Luc Rey-Bellet, University of Virginia

Topic: The longest paths in random matrices : connection to the eigenvalues of February 28

random matrices

Presenter: Jinho Baik, Princeton University

Abstract: In the classical central limit theorem, the sum of i.i.d. random variables has the same limiting fluctuation : the Gaussian distribution. Now we make an M by N matrix with entries of i.i.d. random variables, and take a maximum of sums of entries along a class of up/right paths. Then we ask the question of the limiting fluctuation as M and/or N tend to infinity. This type of problem arises in for example, queueing theory and interacting particle systems. There are a few examples for which one can compute the limiting distribution directly, and it turned out that the role played by the Gaussian distribution in the classical central limit theorem is now played by the largest eigenvalue of a random Hermitian matrix taken from the Gaussian unitary ensemble. We will discuss topics centered around this relation between the longest path in a random matrix and the largest eigenvalue of a random Hermitian matrix.

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: Segregation problems in binary fluids March 7

Presenter: Rafelle Esposito, University of Rome

Topic: Gross-Zagier formula with characters March 8

Presenter: Shou-Wu Zhang, Columbia University

Topic: TBA March 9

Presenter: Tian-Jun Li, Princeton University

Topic: Rank-two vector bundles over a general curve: A conjecture of March 12

Bertram-Feinberg-Mukai

Presenter: H. Clemens, University of Utah

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: Fluctuations for stochastic lattice gases March 14

Presenter: Rosanna Marra, University of Rome

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: Holomorphic disks and invariants for 3-manifolds and smooth 4-manifolds April 4

Presenter: Zoltan Szabo, Princeton University

Abstract: We will introduce and study topological invariants for closed 3-manifolds and smooth 4-manifolds. The 3-manifold construction uses Heegaard diagrams and a version of Lagrangian Floer homology. The 4-manifold invariant uses the previous construction, a pairing on Floer-homology and a handle decomposition of the 4-manifold. We will also present some applications in three and 4-manifold topology. This is a joint result with Peter Ozsvath.

Topic: TBA April 6

Presenter: Daniel Pollack, University of Washington

Topic: TBA April 13

Presenter: Guan Bo, University of Tennessee

Topic: TBA April 23

Presenter: John Hopfield, Molecular Biology, Princeton University

Topic: TBA April 27

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