As of November 15-17, 2000
Statistical Mechanics Seminar Wednesday 2:00 Jadwin 343
Topic: Correlation functions in the 1D Holstein-Hubbard quasi-crystal model November 15
Presenter: Vieri Mastropietro, Rome University
Departmental Colloquium Wednesday 4:00 Fine 314
Topic: Scaling limits of random processes and the outer boundary of planar November 15
Brownian motion
Presenter: Oded Schramm, Microsoft Research and Wiezmann Institute
Abstract: Consider a random walk on the square grid in the plane: a particle is placed at the origin, and at each step moves to a random vertex adjacent to the current position, with all choices having equal probability. If one performs this process on finer and finer grids, and rescales time appropriately, the process converges to a random path, which is called Brownian motion. Perhaps surprisingly, Brownian motion is more symmetric than the random walk: it has rotational symmetry. In fact, it is conformally invariant, which is a more general kind of symmetry.
A more complicated process is critical percolation, where each edge of the square grid is deleted with probability 1/2, independently, and the connectivity properties of the resulting graph are studied.
It is an outstanding challenge to understand what happens to critical percolation and similar processes when the mesh of the grid tends to zero. Many of these processes are believed to display conformal invariance in the limit, but this is mostly unproven. Under the assumption of conformal invariance we give a complete description of the scaling limit of critical percolation and several other models. The description is based on a process that we call Stochastic Loewner Evolution (SLE). The SLE process describes a randomly growing set by specifying the conformal map to the complement of the set. The conformal map is obtained by solving a random differential equation. There's one free parameter $\kappa>0$ in the description of SLE.
In joint work with Greg Lawler and Wendelin Werner, we prove that many properties of planar Brownian Motion are the same as those of SLE with $\kappa=6$. This is then used to answer several problems regarding planar Brownian Motion. In particular, we prove Mandelbrot's conjecture stating that the Hausdorff dimension of the outer boundary of planar Brownian Motion is 4/3.
The talk will assume no prior specialized knowledge. The plan is to describe some of the random processes, explain and motivate the construction of SLE, and explain the relation with planar Brownian Motion.
Ergodic Theory and Statistical Mechanics Seminar Thursday 2:00 Fine 401
Topic: TBA November 16
Presenter: Alex Eskin, University of Chicago
Geometry Seminar Thursday 3:00 Fine 110
Topic: Induction theorems in algebra and topology November 16
Presenter: Jesper Grodal, Institute for Advanced Study
Abstract: The plan is to give 4 talks. The first talk will be an overview talk where I'll explain the basics of homology decompositions. A homology decomposition is an expression of the classifying space BG of a group G as a homotopy colimit of classifying spaces of proper subgroups. I will explain how they relate to induction theorems in group theory and can be used to calculate group cohomology. I will also explain how they can be used to classify group action on spaces, and maybe examine the case of a sphere in more detail. In the next talks I will elaborate on these issues, also depending on the interests of the participants.
Princeton/IAS/Rutgers Nonlinear Theory Seminar Thursday 4:00 Fine 214
Topic: Invariant measures for randomly forced nonlinear PDEs and the problem November 16
of turbulence
Speaker: Sergei Kuksin, Heriot-Watt/Steklov
Princeton/IAS Number Theory Seminar Thursday 4:15 Room 101
Topic: Rational points close to an arc of a curve November 16 at IAS
Presenter: Martin Huxley, University of Wales, Cardiff
Topology Seminar Thursday 4:30 Fine 314
Topic: Mapping class groups and rigidity November 16
Presenter: Richard Wentworth, Johns Hopkins University
Geometry Seminar Friday 4:00 Fine 314
Topic: Sharp Sobolev-Poincare inequalities on compact Riemannian manifold November 17
Presenter: Emmanuel Hebey, Universite de Cergy-Pontoise
Week of November 20 - 24, 2000
Analysis Seminar Monday 4:00 Fine 314
Topic: TBA November 20
Presenter: Haim Brezis, Université de Paris VI and Rutgers University
Algebraic Geometry Seminar Tuesday 4:30 Fine 322
Topic: Is M_{g,n} a Mori dream space (mod p)? November 21
Presenter: Sean Keel, University of Texas
Math Physics Seminar Tuesday 4:30 Jadwin A06
Topic: The PCT theorem and local observables November 21
Presenter: Jakob Yngvason, University of Vienna
Statistical Mechanics Seminar Wednesday 2:00 Jadwin 343
Topic: The monodromy matrix, the adiabatic WKB method and the spectrum November 22
of quasi-periodic operators on the real line
Presenter: Frederic Klopp, University of Paris
Week of November 27 - December 1, 2000
Analysis Seminar Monday 4:00 Fine 314
Topic: Harmonic analysis on the infinite symmetric group November 27
Presenter: Alexei Borodin, University of Pennsylvania
PACM Colloquium Monday 4:30 Fine 224
Topic: Non-uniform structures in granular and gas-solid flows November 27
Presenter: Sankaran Sundaresan, Chemical Engineering, Princeton University
Statistical Mechanics/Mathematical Physics Seminar Wednesday 2:00 Jadwin 343
Topic: Nonperturbative analysis of a Model Quantum System under Time November 29
Periodic Forcing: Generic and Exceptional Cases
Presenter: Ovidiu Costin, Rutgers University
Abstract: We analyze the time evolution of a one-dimensional quantum system with an attractive delta function potential whose strength is subjected to a time periodic (zero mean) parametric variation $\eta(t)$. The amplitude of $\eta$ is unrestricted. We show that for generic $\eta(t)$, which includes the sum of any finite number of harmonics, the system, started in a localized state, gets fully ionized (the particle leaves the bound state) as time grows indefinitely, irrespective of the magnitude or frequency (resonant or not) of $\eta(t)$. There are however exceptions to full ionization which while very non-generic include rather simple explicit functions. For these $\eta(t)$ the system evolves to a nontrivial localized stationary state which is related to eigenfunctions of the Floquet operator.
Departmental Colloquium Wednesday 4:00 Fine 314
Topic: TBA November 29
Presenter: H. Hofer, Courant Institute and Princeton University
Algebraic Geometry Seminar Thursday 4:30 Fine 322
Topic: TBA November 30
Presenter: V. Shokurov, Johns Hopkins University
Geometry Seminar Friday 4:00 Fine 314
Topic: TBA December 1
Presenter: Wang Guo-Fang, Max Planck Institute
Week of December 4 - 8, 2000
Analysis Seminar Monday 4:00 Fine 314
Topic: d-bar-regularity for weakly pseudoconvex domains in compact Hermitian December 4
symmetric spaces with respect to invariant metrics
Presenter: Yum-Tong Siu, Harvard University
PACM Colloquium Monday 4:30 Fine 224
Topic: TBA December 4
Presenter: Salvatore Torquato, Chemistry, Princeton University
Discrete Math Seminar Tuesday 2:15 Fine 224
Topic: The Theta Body and Partitionable Graphs December 5
Presenter: Bruce Shepherd, Lucent Technologies
Algebraic Geometry Seminar Tuesday 4:30 Fine 322
Topic: The moduli space of cubic surfaces is complex hyperbolic December 5
Presenter: Jim Carlson, University of Utah
Departmental Colloquium Wednesday 4:00 Fine 314
Topic: Billiards in Rational Polygons and Moduli Spaces of Holomorphic Differentials December 6
Presenter: Alex Eskin, University of Chicago
Abstract: A polygon is called rational if all angles are rational multiples of pi. It turns out that the problem of counting periodic billiard trajectories on such a polygon can be reduced to a certain dynamical problem of the moduli space of pairs (M,w) where M is a Riemann surface, and w is a holomorphic 1-form on M. Even though it is not locally homogeneous, this moduli space is analogous in many ways to the moduli space of Euclidean lattices SL(n,R)/SL(n,Z). I will discuss these constructions and some associated problems in combinatorial enumeration.
Geometry Seminar Friday 4:00 Fine 314
Topic: Ricci Curvature, Minimal Volumes, and Seiberg-Witten Theory December 8
Presenter: Claude Le Brun, SUNY Stony Brook
Week of December 11 - 15, 2000
PACM Colloquium Monday 4:30 Fine 224
Topic: Brownian Dynamics Methods for the Solution of Complex Polymeric December 11
Flows Based on Kinetic Theory Models: Early (CONNFFESSIT) and
More Recent (Configuration Field) Approaches
Presenter: Antony N. Beris, University of Delaware
Abstract: In the past, closed-form continuous models have been used for the solution of complex (i.e. multidimensional and/or time dependent) flow problems involving polymer solutions or melts. However, those models involve closure approximations that result in unpredictable errors in the complex flows that they are employed. Nevertheless, the presence of one or more internal variables makes the dimensionality of the microscopic problem prohibitively large to allow for a direct solution of the microscopic equations (such as those arising from kinetic theory) in even the simplest multidimensional flow problems. Fortunately, in the last decade new methodologies has emerged that allow for such solutions to be obtained at significantly less computational cost through the coupling of a stochastic solution for the polymer chain configuration to a more traditional macroscopic finite-element or spectral flow approximation of the momentum and continuity equations. In this presentation, after reviewing the first of these approaches (called CONNFFESSIT) which has first been developed by Laso and Oettinger, we will discuss the more recently developed (by Hulsen and co-workers) configuration fields approach that result in even more reduced computational requirements.
Math Physics Seminar Tuesday 4:30 Jadwin A06
Topic: Coherent and Dissipative Electronic Transport in Aperiodic Media December 12
Presenter: Jean V. Bellissard, Universite Paul Sabatier, Toulouse, France and Institut Universitaire de France