Week
of October 2 - October 4, 2002
Statistical Mechanics Seminar
Topic: Stationary Nonequilibrium States and the Continuing Quest for a Proof of Fourier's Law
Presenter: Joel Lebowitz, Rutgers University
Date: Wednesday, October 2, 2002, Time: 2:00 p.m., Location: Jadwin 343
Abstract:
I will present a selective overview of the current state of our knowledge
(more precisely of our ignorance) regarding the derivation of Fourier's
Law, J( r) = -kappa gradient T( r); J the heat flux, T the temperature
and kappa, the heat conductivity. This law is empirically well tested for
both fluids and crystals, when the temperature varies slowly on the microscopic
scale, with kappa an intrinsic property which depends only on the system's
equilibrium parameters, such as the local temperature and density. There
is however at present no rigorous mathematical derivation of Fourier's
law and ipso facto of Kubo's formula for kappa, involving integrals over
equilibrium time correlations, for any system (or model) with a deterministic,
e.g. Hamiltonian microscopic evolution. Recent numerical and some
analytical developments will also be discussed.
Department Colloquium
Topic: Compatibility, microgeometry and materials
Presenter: John Ball, Oxford University and IAS
Date: Wednesday, October 2, 2002, Time: 4:30 p.m., Location: Fine Hall 314
Abstract:
Please
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Discrete Mathematics Seminar
Topic: Extremal set-systems with restricted k-wise intersections
Presenter: Benny Sudakov, Princeton University
Date: Thursday, October 3, 2002, Time: 2:15 p.m., Location: Fine Hall 224
Abstract:
Please
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Arnold's Diffusion Seminar
Topic: Survey of results on Arnold diffusion
Presenter: Jeff Xia, Northwestern University
Date:
Thursday, October 3, 2002, Time: 3:00 p.m., Location: Fine Hall 110
Joint Princeton/IAS Number Theory Seminar
Topic: Special values of Rankin L-functions
Presenter: Hui Xue, IAS
Date: Thursday, October 3, 2002: Time: 4:30, Location: Fine Hall 322
Abstract:
Explicit formulas are given for the central special values of twisted L-functions
associated to Hilbert modular forms.
Geometric Analysis Seminar
Topic: The Dirichlet problem for minimal surface systems
Presenter: Mu-Tao Wang, Columbia University
Date:
Friday, October 4, 2002, Time: 3:00 p.m., Location: Fine Hall 314
Week
of October 7 - October 11, 2002
Analysis Seminar
Topic: Proof of the Double Bubble Conjecture
Presenter: Frank Morgan, Williams College
Date: Monday, October 7, 2002, Time: 4:00 p.m., Location: Fine Hall 314
Abstract:
The Double Bubble Conjecture says that the familiar double soap bubble
is the least-area way to enclose and separate two given volumes of air.
I'll discuss the proof, the latest results, and open questions.
PACM Colloquium
Topic: Wave localization and guidance in photonic bandgap structures
Presenter: Fadil Santosa, University of Minnesota
Date: Monday, October 7, 2002, Time: 4:00 p.m., Location: Fine Hall 214
Abstract:
Photonic bandgap structures are anticipated to play an important role in
the development of devices for optical communication. These nano-structures
are made of material with periodic index of refraction. Defects are introduced
to guide and manipulate light. In this talk, the speaker will provide an
introduction to photonic bandgap structures, and the mathematical tools
needed to analyze them. Major ideas are illustrated with numerical examples.
Some of the open problems will be described.
Celestial Mechanics Seminar
Topic: Capture Processes and Chaotic Motions in Celestial Mechanics Part II
Presenter: Edward Belbruno, PACM
Date: Tuesday, October 8, 2002, Time: 11:00 a.m., Location: Fine Hall 1001
Abstract: We
discuss two types of topics, which present some recent results. One is
on a proof which unifies two types of seemingly unrelated capture processes
in the three-body problem. This is accomplished by proving that a hyperbolic
invariant set exists on a stability transition region called a weak stability
boundary. This boundary has
interesting applications, and the hyperbolic invariant set is related to
unbounded(parabolic motion). The set more generally is proven to exist
by Xia. In another unrelated topic we prove that Aubrey-Mather sets exist
in the three-body problem near special periodic orbits.
Joint Analysis-Arnold's Diffusion Seminar *** Please note special date, time and location
Topic: Spatial Heteroclinics and homoclinics for Allan-Cahn model equation
Presenter: Paul Rabinowitz, University of Wisconsin - Madison
Date:
Tuesday, October 8, 2002, Time: 1:00 p.m.,
Location: Fine Hall 214
Algebraic Geometry Seminar
Topic: Rational families of vector bundles on curves
Presenter: Ana-Maria Castravet, IAS
Date:
Tuesday, October 8, 2002, Time: 4:30 p.m., Location: Fine Hall 322
Mathematical Physics Seminar
Topic: Mass Renormalization and Energy Level Shift in Non-Relativistic QED
Presenter: Christian Hainzl, University of Munich
Date: Tuesday, October 8, 2002, Time: 4:30 p.m., Location: Jadwin A06
Abstract:
Using the Pauli-Fierz model of non-relativistic quantum electrodynamics,
we calculate the binding energy of an electron in the field of a
nucleus of charge Z and in presence of the quantized radiation field.
We consider the case of small coupling constant \alpha, but fixed
Z\alpha and ultraviolet cut-off \Lambda. We prove that after renormalizing
the mass the binding energy has, to leading order in \alpha, a finite limit
as \Lambda goes to infinity; i.e., the cut-off can be removed. The expression
for the ground state energy shift thus obtained agrees with Bethe's formula
for small values of Z\alpha, but shows a different behavior for bigger
values.
Statistical Mechanics Seminar
Topic: Introduction to some new ideas in non-equilibrium statistical mechanics
Presenter: David Ruelle, IHES, France
Date: Wednesday, October 9, 2002, Time: 2:00 p.m., Location: Jadwin 343
Abstract:
Nonequilibrium statistical mechanics has recently been idealized as the
study of a certain differentiable dynamical system. This raises a
number of physical and mathematical questions which will be discussed.
In particular we shall study the response of a system to a change of forces
acting on it.
Department Colloquium
Topic: Nonlinear PDE's in Conformal Geometry
Presenter: Sun-Yung A. Chang, Princeton University
Date: Wednesday, October 9, 2002, Time: 4:30 p.m., Location: Fine Hall 314
Abstract: Second order elliptic equations have always been an important tool in the study of problems in geometry. In this talk, I will describe some recent efforts to extend the role played by second order elliptic operators to some higher order ones; some natural curvature functions related to these operators and some geometric implications.
Discrete Mathematics Seminar
Topic: TSP cuts that do not follow the template paradigm
Presenter: Vasek Chvatal, Rutgers University
Date: Thursday, October 10, 2002, Time: 2:15 p.m., Location: Fine Hall 224
Abstract:
Please
click here to view abstract
Topology Seminar
Topic: Surface singularities with homology 3-sphere links
Presenter: Walter Neumann, Columbia University and Barnard
Date: Thursday, October 10, 2002: Time: 4:00, Location: Fine Hall 314
Abstract:
The Casson Invariant Conjecture says that the signature of the Milnor
fiber of a complete intersection singularity with homology sphere link
should be eight times the Casson invariant of the link. It suggests that
the Milnor fiber is a topologically natural coboundary for the link.
Recent progress includes a conjectural classification of the 3-manifolds,
singularities, and Milnor fibers to which this applies, and a proof of
the conjecture for a largish subclass of these.
Joint Princeton University/IAS Number Theory Seminar
Topic: TBA
Presenter: Joel Bellaiche, IAS
Date:
Thursday, October 10, 2002: Time: 4:30, Location: Fine
Hall 322
Geometric Analysis Seminar
Topic: Conformal maps, convex hulls and Kleinian groups
Presenter: Chris Bishop, SUNY at Stony Brook
Date:
Friday, October 11, 2002, Time: 3:00 p.m., Location: Fine Hall 314
Week
of October 14 - October 18, 2002
AnalysisSeminar
Topic: Dyadic models for the equations of fluid motion, and an example of turning models into reality
Presenter: Natasa Pavlovic, Princeton University
Date: Monday, October 14, 2002, Time: 4:00 p.m., Location: Fine Hall 314
Abstract:
This talk will be about dyadic models for the equations of fluid motion.
In the talk we shall introduce a dyadic model for the Euler and the Navier-Stokes
equations with hyper-dissipation. For the dyadic Euler equations we prove
finite time blow-up. In the context of the dyadic Navier-Stokes equations
with hyper-dissipation we prove finite time blow-up in case when the degree
of dissipation is sufficiently small. Also we shall discuss a partial
regularity result for the dyadic Navier-Stokes equations with hyper-dissipation,
which could be turned into a result for the actual Navier-Stokes equations
with hyper-dissipation. This talk is based on joint work with Susan
Friedlander and Nets Katz.
PACM Colloquium
Topic: Theoretical and Numerical Analysis for Some Non-linear Interface Problems
Presenter: Zhilin Li, North Carolina State University
Date: Monday, October 14, 2002, Time: 4:00 p.m., Location: Fine Hall 214
Abstract:
Non-linear partial differential equations with discontinuity in the coefficient
have many applications. In this talk, I will focus on two different non-linear
interface problems. One is the potential equation for magneto-rhological
(MR) fluid that contains metal particles. The permeability is discontinuous
across the interface between the fluid and the particles. The second problem
is the weighted minimal surface problem. We have generalized the Sneil's
law for optical path to the three dimensional weighted minimal surface
problem. Numerically, we use the substitution method to solve the non-linear
PDE. Since the coefficient of the potential equation depends on the
gradient of the solution, we use the maximum preserving immersed interface
method coupled with multigrid solvers to solve the linearized problems.
Numerical examples will also be presented.
Statistical Mechanics Seminar
Topic: Random graphs with a positive density of triangles
Presenter: J.P. Eckmann, University of Geneva
Date: Wednesday, October 16, 2002, Time: 2:00 p.m., Location: Jadwin 343
Abstract:
In a study with E. Moses (Weizmann) we discovered that the graph of the
World Wide Web and many other graphs occurring in the Sciences have a density
of triangles which is much higher than what can be expected for a random
graph. As a consequence, P. Collet and I studied the "statistical mechanics"
of subsets of random graphs with a positive density of triangles. This
corresponds to a theory of extremely large deviations. If time permits,
I will also describe a few aspects of the work on the WWW.
Discrete Mathematics Seminar
Topic: TBA
Presenter: David Gamarnik, T.J. Watson Research Center, IBM
Date:
Thursday, October 17, 2002, Time: 2:15 p.m., Location: Fine Hall
224
Topology Seminar
Topic: Simple closed curves in surfaces
Presenter: Nancy Hingston, the College of New Jersey
Date:
Thursday, October 17, 2002: Time: 4:00, Location: Fine
Hall 314
Geometric Analysis Seminar
Topic: On some conformally invariant fully nonlinear equations
Presenter: Yanyan Li, Rutgers University
Date:
Friday, October 18, 2002, Time: 3:00 p.m., Location: Fine Hall 314
Week
of October 21 - October 25, 2002
AnalysisSeminar
Topic: The solution of the square root problem of Kato
Presenter: Steven Hofmann, University of Missouri at Columbia
Date:
Monday, October 21, 2002, Time: 4:00 p.m., Location: Fine Hall 314
PACM Colloquium
Topic: Theory and Computation of Resonances of Photonic Microstuctures
Presenter: Michael Weinstein, Bell Laboratories
Date: Monday, October 21, 2002, Time: 4:00 p.m., Location: Fine Hall 214
Abstract:
Photonic crystal waveguides are a class of optical waveguides with novel
transverse microstructure. A great deal of tunability of optical properties
is achieved through variations in the geometry of microfeatures, their
distribution and refractive index contrasts. We discuss a multiple scale
approach to the study of photonic crystal waveguides. Energy escapes from
the core due to a combination of propagation and tunneling. Of central
importance are leaky modes (resonance states) and their associated complex
effective indices (scattering resonance poles). The leading order theory
agrees with classical homogenization theory, describing an effective homogeneous
medium with dielectric properties given by an appropriate averaging of
the refractive index profile. We compute the first non-trivial correction,
which takes into account the microstructure, and find that the higher order
homogenization expansion gives very good agreement with full simulations
by Fourier and multipole methods. The higher order expansion is crucial
for estimation of leakage rates; in various examples of physical interest,
the leading order (homogenization) gives a substantial underestimation.
Finally, a rigorous justification of the homogenization expansion of scattering
resonances is obtained as part of a resonance perturbation theory for an
appropriate "preconditioned" Lippmann-Schwinger equation.
Department Colloquium
Topic: Algebraic Geometry of Nash Equilibria
Presenter: Bernd Sturmfels, UC at Berkeley
Date: Wednesday, October 23, 2002, Time: 4:30 p.m., Location: Fine Hall 314
Abstract:
We present an algebraic approach to Nash equilibria in game theory, along
the lines of Chapter 6 in my new book "Solving Systems of Polynomial
Equations" (http://math.berkeley.edu/~bernd/cbms.html). The set of all
Nash equilibria of an N-person game is a real algebraic variety, which
is typically a finite set. A combinatorial formula for its expected cardinality
was given by the economists McKelvey and McLennan in 1997. We show how
to find all Nash equilibria using computational algebraic geometry, and
we present the Universality Theorem (proved by Ruchira Datta in 2002) which
states that every real algebraic variety arises from a game with three
players.
Discrete Mathematics Seminar
Topic: Set systems with restricted intersections
Presenter: Jacques Verstraete, Microsoft Research
Date: Thursday, October 24, 2002, Time: 2:15 p.m., Location: Fine Hall 224
Abstract:
Please
click here to view abstract
Topology Seminar
Topic: TBA
Presenter: Igor Rivin, Temple University
Date:
Thursday, October 24, 2002: Time: 4:00, Location: Fine
Hall 314
Geometric Analysis Seminar
Topic: TBA
Presenter: Simon Brendle, Princeton University
Date:
Friday, October 25, 2002, Time: 3:00 p.m., Location: Fine Hall 314
Week
of November 4 - November 8, 2002
Analysis Seminar
Topic: Purely Nonlinear Instability of Minimal Energy Standing Waves
Presenter: Andrew Comech, University of North Carolina
Date: Monday, November 4, 2002, Time: 4:00 p.m., Location: Fine Hall 314
Abstract:
For a variety of nonlinearities, the nonlinear Schroedinger equation is
known to possess localized quasistationary solutions ("standing waves").
We prove that in the generic situation the standing wave of minimal energy
among all other standing waves is unstable. This case was falling out of
the scope of the classical paper by Grillakis, Shatah, and Strauss on orbital
stability of standing waves. An interesting feature of the problem is the
absence of (exponential) instability in the linearized system; in this
sense, the resulting instability is ``purely nonlinear''. Essentially,
the instability is caused by higher algebraic degeneracy of zero eigenvalue
in the spectrum of the linearized system. The result can be generalized
to abstract Hamiltonian systems with U(1) symmetry.
PACM Colloquium
Topic: The Forced van der Pol Equation: New Insights on an Old Problem
Presenter: John Guckenheimer, Cornell University
Date: Monday, November 4, 2002, Time: 4:00 p.m., Location: Fine Hall 214
Abstract:
The forced van der Pol equation is the original example of chaos in dynamical
systems. It is also an example of relaxation oscillations, periodic motions
with short and fast time scales. This lecture describes joint work, primarily
with Kathleen Hoffman and Warren Weckesser, to understand bifurcations
of dynamical systems with multiple time scales. Using the forced
van der Pol equation as a case study, we illustrate how canards - solutions
that track unstable slow manifolds - play an important role in both bifurcations
and chaos. In many situations canards cannot be computed by solving initial
value problems, so many of the phenomena that we exhibit are missed by
typical simulation studies of multiscale systems.
Joint Algebraic Geometry and Number Theory Seminar
Topic: A Mahler formula for dynamical systems on the sphere
Presenter: Lucien Szpiro, Columbia University
Date:
Tuesday, November 5, 2002, Time: 4:30 p.m., Location: Fine Hall 322
Department Colloquium
Topic: Expander graphs - where Combinatorics and Algebra compete and cooperate
Presenter: Avi Wigderson, IAS and The Hebrew University
Date: Wednesday, November 6, 2002, Time: 4:30 p.m., Location: Fine Hall 314
Abstract:
Expansion of graphs can be given equivalent definitions in combinatorial
and algebraic terms. This is the most basic connection between combinatorics
and algebra illuminated by expanders and the quest to construct them. The
talk will survey how fertile this connection has been to both fields, focusing
on recent results. In particular, we'll explain the zig-zag product
of graphs, how it leads to an elementary combinatorial construction of
expanders, and its relation to semi-direct product in groups.
Discrete Mathematics Seminar
Topic: TBA
Presenter: Santosh Vempala, MIT
Date:
Thursday, November 7, 2002, Time: 2:15 p.m., Location: Fine Hall
224
Topology Seminar
Topic: TBA
Presenter: Peter Ozsvath, Columbia University
Date:
Thursday, November 7, 2002: Time: 4:00, Location: Fine
Hall 314
Joint Princeton University/IAS Number Theory Seminar
Topic: On the sign of Kloosterman sums
Presenter: Philippe Michel, IUF and Univ. Montpellier II
Date:
Thursday, November 7, 2002: Time: 4:30, Location: TBA
Geometric Analysis Seminar
Topic: TBA
Presenter: Hubert Bray, MIT
Date:
Friday, November 8, 2002, Time: 3:00 p.m., Location: Fine Hall 314
Week
of November 11 - November 15, 2002
AnalysisSeminar
Topic: TBA
Presenter: Victor Nistor, Pennsylvania State University
Date:
Monday, November 11, 2002, Time: 4:00 p.m., Location: Fine Hall 314
PACM Colloquium
Topic: The Level Set Method-what's in it for you?
Presenter: Stanley Osher, University of California, Los Angeles
Date: Monday, November 11, 2002, Time: 4:00 p.m., Location: Fine Hall 214
Abstract:
The level set method for capturing moving fronts was introduced in 1987
by Osher and Sethian. It has proven to be phenomenally successful as a
numerical device. For example, typing in "Level Set Methods" on Google's
search engine gives roughly 3200 responses. Applications range from
capturing multiphase fluid dynamical flows, to special effects in Hollywood
to visualization,image processing, control, epitaxial growth, computer
vision and many more. In this talk we shall give an overview of the numerical
technology and a few applications.
Algebraic GeometrySeminar
Topic: TBA
Presenter: James McKernan, Santa Barbara
Date:
Tuesday, November 12, 2002, Time: 4:30 p.m., Location: Fine Hall 322
Discrete Mathematics Seminar
Topic: TBA
Presenter: Mike Saks, Rutgers University
Date:
Thursday, November 14, 2002, Time: 2:15 p.m., Location: Fine Hall
224
Topology Seminar
Topic: TBA
Presenter: Mikhail Khovanov, UC Davis
Date:
Thursday, November 14, 2002: Time: 4:00, Location: Fine
Hall 314
Joint Princeton University/IAS Number Theory Seminar
Topic: TBA
Presenter: Jonathan Pila, University of Melbourne and IAS
Date:
Thursday, November 14, 2002: Time: 4:30, Location: TBA
Geometric Analysis Seminar
Topic: TBA
Presenter: Emmanuel Hebey, Université de Cergy-Pontoise and
Olivier Druet, Université de Cergy-Pontoise
Date:
Friday, November 15, 2002, Time: 3:00 p.m., Location: Fine Hall 314
Week
of November 18 - November 22, 2002
Analysis Seminar
Topic: TBA
Presenter: Christoph Thiele, UCLA
Date:
Monday, November 18, 2002, Time: 4:00 p.m., Location: Fine Hall 314
PACM Colloquium
Topic: An Eulerian Method for Multiphase Computations of the Schrodinger Equation
Presenter: Xiantao Li, PACM, Princeton University
Date: Monday, November 18, 2002, Time: 4:00 p.m., Location: Fine Hall 214
Abtract:
We present a new numerical method for the computation of the semiclassical
limits of the Schrodinger equation. We first use Wigner transform technique
to derive a Vlasov equation in the phase space, and then find its solution
in the multiphase regime. By taking moment closure, we obtain the
multiphase equations in the physical space. The numerical procedure follows
the solution of the Vlasov equation, but only operates in the physical
space, which offers great efficiency and simplicity. In addition, we will
show some other application of this technique, such as the multivalued
solutions of the Euler Poisson system in Klystron.
Algebraic GeometrySeminar
Topic: One dimensional families of Calabi-Yau threefolds
Presenter: Charles F. Doran, Columbia University
Date:
Tuesday, November 19, 2002, Time: 4:30 p.m., Location: Fine Hall 322
Joint Princeton University/IAS Number Theory Seminar
Topic: TBA
Presenter: Gautam Chinta, Brown University
Date:
Thursday, November 21, 2002: Time: 4:30, Location:
Fine Hall 322
Week
of December 2 - December 6, 2002
AnalysisSeminar
Topic: TBA
Presenter: Mihaela Iftime, Northeastern University
Date:
Monday, December 2, 2002, Time: 4:00 p.m., Location: Fine Hall 314
PACM Colloquium
Topic: TBA
Presenter: Natalia Berloff, University of Cambridge
Date:
Monday, December 2, 2002, Time: 4:00 p.m., Location: Fine Hall 214
Discrete Mathematics Seminar
Topic: TBA
Presenter: Peter Winkler, Bell Labs
Date:
Thursday, December 5, 2002, Time: 2:15 p.m., Location: Fine Hall
224
Topology Seminar
Topic: TBA
Presenter: Yair Minsky, SUNY at Stonybrook
Date:
Thursday, December 5, 2002: Time: 4:00, Location: Fine
Hall 314
Joint Princeton University/IAS Number Theory Seminar
Topic: TBA
Presenter: Ling Long, IAS
Date:
Thursday, December 5, 2002: Time: 4:30, Location:
Fine Hall 322
Week
of December 9 - December 13, 2002
Joint Princeton University/IAS Number Theory Seminar
Topic: TBA
Presenter: Yuval Flicker, Ohio State University
Date:
Thursday, December 12, 2002: Time: 4:30, Location:
Fine Hall 322