Topic: Zeros of Graph-counting Polynomials November 25

Presenter: David Ruelle, I.H.E.S., France

Topic: A Fourier Analysis approach to translational tilings. November 30

Presenter: Michael Koluntzakis, Urbana-Champaign

Topic: Coupling Computational Fluid Dynamics and Numerical November 30

Optimization Methods for Aircraft Design

Presenter: Luigi Martinelli, MAE Department and PACM Princeton University

Abstract: The definition of the aerodynamic shapes of modern aircraft relies heavily on computational simulation to enable

the rapid evaluation of many alternative designs. Wind tunnel testing is then used to confirm the performance of designs

that have been identified by simulation as promising to meet the performance goals. The use of computational simulation

to scan many alternative designs has proved extremely valuable in practice, but it still suffers the limitation that it does

not guarantee the identification of the best possible design. To ensure the realization of the true best design, the ultimate

goal of computational simulation methods should not just be the analysis of prescribed shapes, but the automatic

determination of the true optimum shape for the given measure of performance. This is the underlying motivation for the

combination of computational fluid dynamics with numerical optimization methods.

Following the lead of Jameson, we selected to approach the problem using the framework of the mathematical theory for

the control of systems governed by partial differential equations. In this view the wing is regarded as a device to produce

lift by controlling the flow, and its design is regarded as a problem in the optimal control of the flow equations by changing

the shape of the boundary.

In the past three years I contributed to the development and implementation of this approach for designs in

three-dimensional viscous flow. I will present an overview of this approach using the compressible Reynolds Averaged

Equations as the mathematical model of the flow, give a detailed account of the numerical building blocks

which make our approach computationally feasible, and discuss some illustrative designs.

Topic: Statistical Mechanics and the Eigenvalue Density of Random Matrices December 2

Presenter: Michael Kiessling, Rutgers University

Topic: Enumerating Periodic Three-Dimensional Tilings December 3

Presenter: Daniel Huson, Princeton

Abstract: The topology and symmetries of periodic tilings of simply connected spaces can be described combinatorially in

terms of so-called Delaney-Dress symbols. This observation is the foundation of what one might call "combinatorial tiling

theory".

In two dimensions, this approach has given rise to theorems and also efficient algorithms for systematically enumerating

periodic tilings of the euclidean plane, sphere and hyperbolic plane.

Classifying periodic three-dimensional tilings is much more difficult. The main problem can be stated as follows: Given a

(compatible) triangulation of a three-dimensional orbifold, is the orbifold Euclidean? Whereas a general and efficiently

computable solution to this problem seems out-of-reach, Olaf Delgado Friedrichs (Bielefeld) has developed an approach

that works well in practice.

Based on this, we have addressed the problem of systematically enumerating all tile-transitive tilings of three-dimensional

Euclidean space by cubes, octahedra or tetra. The tilings we considered were face-to-face, but the tiles were not

necessarily regular. We claim there exist precisely 11, 3, or 9 topological types of such tilings by cubes, octahedra, or

tetra, respectively. Also, we have partial results for tilings by dodecahedra or icosahedra.

Topic: Determinantal formulas for the correlation functions and the infinite December 3

symmetric group

Presenter: A. Borodine, University of Pennsylvania

Topic: A super-rigid non-arithmetic group: A counter example to December 3

Platonov conjecture

Presenter: A. Lubotzky, Hebrew University

Abstract: The celebrated theorem of Margulis says that lattices (=discrete subgroups of finite covolume) in higher rank

Lie groups are super-rigid and arithmetic. Platonov conjectured that any linear super-rigid group is of arithmetic type. The

conjecture could have nice applications. We present counter examples which shows that our understanding of linear

groups is still far from being satisfying. The idea of the construction is based on ideas applied to answer Grothendieck's

problem on maps between pro-finite completions as well as hyperbolic groups. (joint work with Hyman Bass)

Topic: TBA December 3

Presenter: Haim Brezis, Rutgers University and Universite Paris VI

Topic: An explicit construction of an automorphic descent map from December 3

self-dual GL(N)-modules to modules on classical groups

Presenter: Steven Rallis, Ohio State University

Topic: Gauge Theory, TQFT's, and the Braid Groups December 3

Presenter: H. Elmar Winkelnkemper, University of Maryland

Topic: Spectral geometries for the writhing of knots and the helicity of vector fields December 4

Presenter: Dennis Deturck, University of Pennsylvania

Abstract: The writhing number of a smooth curve in 3-space is the standard measure of the extent to which the curve

wraps and coils around itself. It is important for molecular biologists in the study of knotted DNA and of the enzymes

which affect it. The helicity of a smooth vector field defined on a domain in 3-space is the standard measure of the extent

to which the field lines wrap and coil around one another. It is important in fluid mechanics, magnetohydrodynamics and

plasma physics.

In this joint work with Jason Cantarella, Herman Gluck and Mikhail Teytel, rough upper bounds for the writhing number

of a knot in terms of its length and thickness and for the helicity of a vector field terms of its energy and the geometry of its

domain, are provided in terms of 4/3 power growth laws. Sharp upper bounds for helicity are obtained by formulating

the question as a spectral problem for a compact selfadjoint operator on an appropriate Hilbert space of vector fields, in

the spirit of Arnold's study of the asymptotic Hopf invariant on closed, orientable 3-manifolds.

We will explore the topology and geometry of vector fields that maximize helicity for given energy among all

divergence-free vector fields tangent to the boundary of a domain in 3-space. This leads naturally to the isoperimetric

problem: maximize helicity among all divergence-free vector fields of given energy, defined on and tangent to the

boundary of all domains of given volume in 3-space. It turns out that the round ball is NOT the maximizing domain. We'll

end with a discussion of why we expect the optimal domain to be singular.

Topic: Stable Polytropic Galaxies in Stellar Dynamics December 7

Presenter: Yan Guo, Brown University

Topic: Entropy and the Second Law of Thermodynamics December 8

Presenter: Elliot Lieb, Princeton University

Abstract: The second law is one of the few really fundamental physical laws (in the sense that no deviation, however tiny,

is permitted). As such it ought to be derivable from some mathematical axioms which, once they are clearly stated,

require no additional "physical" input. A precise formulation of this kind would also have useful pedagogical value and

would clarify what is actually needed to make the second law work. Many physicists and mathematicians, both 19th and

20th century, have sought such a formulation, with varying success. For one thing, almost all discussions rely on the

calculus as the relevant analytic tool; these local considerations rarely give rise to convexity which, as Gibbs understood, is

one of the key required ingredients. Another problem concerns temperature. It is usually inserted as a known physical

quantity, but it ought to be derivable, as a theorem from the basic principles, i.e., it should come at the end of the

discussion rather than at the beginning.

The second law is equivalent to the existence of the state function, "entropy" with certain well defined properties. In

recent work with Jakob Yngvason a set of simple, unambiguous axioms have been found that ultimately give rise to an

entropy function with all the properties required by thermodynamics, e.g., monotonicity, additivity, concavity,

differentiability---the latter serving to define temperature.

A brief summary of this material is in Notices of the Amer. Math. Soc. v. 45, 571-581 (1998). No prior knowledge of

thermodynamics is needed for this lecture.

Topic: Universality in Disordered Systems: The Case of the December 8

Random Field Ising Model

Presenter: Nicolas Sourlas, Ecole Normale Superiere

Topic: Integrability and selection in non-linear interface dynamics December 9

Presenter: Mark Mineev-Weinstein, Los Alamos National Laboratory

Topic: Percolation and Collision December 10

Presenter: Peter Winkler, Bell Labs

Abstract: Suppose two tokens are taking simple random walks on the same (finite, connected, undirected) graph G. A

"schedule demon" (left over from a 1990 asynchronous distributed computing problem) wishes to push them as far as

possible along their pre-ordained paths without a collision, just by exercising the privilege of deciding at each moment

which token moves next.

The "clairvoyant demon" conjecture says that if the demon is not unlucky and the graph is sufficiently complex, and he

knows where the tokens are going infinitely far into the future, then he can keep them apart forever. This conjecture

remains open.

The "fickle" demon cannot see the future but has a more powerful feature: she can take moves back. Both problems can

be formulated rather neatly as dependent percolation problems. We show why standard percolation methods fail here

but, at least for the fickle demon, a novel approach does the trick. (Our approach is quite different from the independent

proof presented by Bela Bollobas at IAS on Nov 16, joint work with Paul Balister and Alan Stacey.)

Topic: TBA December 10

Presenter: John Morgan, Columbia University

Topic: Intersecting a curve with algebraic subgroups of a multiplicative group December 10

Presenter: Enrico Bombieri, Institute for Advanced Study

Topic: TBA December 17

Presenter: Yanyan Li, Rutgers University

Topic: TBA December 17

Presenter: Tamas Hausel, Institute for Advanced Studies