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: Chaos in Partial Differential Equations December 4

Presenter: Charles Li, Institute for Advanced Study

Abstract: First I will briefly survey the results mainly on chaos in perturbed soliton equations which include soliton lattice,

(1+1)-dimensional soliton equations, and (1+2)-dimensional soliton equations. Then, I will focus on perturbed nonlinear

Schroedinger equations. I will present the linking between Darboux transformations, Floquet discriminants, and singular

foliations. Next, I will briefly survey the homoclinic orbit theorem without proof. Finally, I will prove the horseshoe

theorem, thereby, the existence of chaos.

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: The Dynamics of twist and writhe in bacterial filaments December 7

and tendril perversion in climbing plants

Presenter: Michael Tabor, University of Arizona

Abstract: A number of filamentary structures in biology can be modeled as thin elastic rods with the Kirchhoff equations

providing an effective but challenging mathematical model. A combination of linear and nonlinear stability analyses is used

to explain how the twist in a rod is converted to writhe (spatial deformation). These results are helpful in explaining the

self-assembly dynamics of the bacterial filaments of Bacillus subtilis, and the helix hand reversal (perversion) exhibited by

the tendrils of climbing plants.

Topic: Eigenvalue moments of the Schroedinger operator: a sharp December 8

one-dimensional bound

Presenter: Elliot Lieb, Princeton University

Abstract: In this talk I will first review the theory and current situation for bounds on the negative eigenvalues of the

Schroedinger operator. In most cases the sharp constants are unknown, and this situation presents opportunities for

research. In the case of the one-dimensional Schroedinger equation, however, some of the sharp constants were actually

known for some time and conjectures existed for the remaining cases. Very recently, one of these conjectures has been

proved (jointly with D. Hundertmark and L. Thomas) and, therefore, it is more likely than ever that the remaining

conjectures are correct. The proof is not difficult but requires a bit of unconventional analysis. The paper can be viewed

on the Los Alamos archive at http://xxx.lanl.gov/abs/math-ph/9806012 .

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

Random Field Ising Model

Presenter: Nicolas Sourlas, Ecole Normale Superiere

Topic: Positivity of vector bundles in Arakelov geometry December 8

Presenter: Jean-Benois Bost, Institute for Advanced Study

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

Presenter: Mark Mineev-Weinstein, Los Alamos National Laboratory

Topic: Combinatorics, representations, and vector bundles on flag varieties December 9

Presenter: Arun Ram, Princeton University

Abstract: History has taught us that the geometry of the flag variety can (and should) be used to understand the

representations of semisimple Lie groups. In recent work with H. Pittie we have reversed the picture and shown how we

can use representation theory and some recent combinatorial techniques of P. Littelmann to attack geometric problems!

I will describe this combinatorics and explain how and why this combinatorics yields very precise information both for

representation theory and for the geometry of flag varieties.

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: Principal G-bundles over T^2 and T^3 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: Differentiability of Lipschitz functions on metric measure spaces December 11

Presenter: Jeff Cheeger, Courant Institue

Topic: The hunt for the Bellman function: Control theory December 14

methods in harmonic analysis.

Presenter: Sergei Treil, Michigan State

Topic: A hypergraph version of Dirac's theorem December 17

Presenter: Endre Szemeredi, Rutgers University

Abstract: If in a 3-uniform hypergraph on n vertices, every edge is in at least n/2 hyperedges (triangles), then the

hypergraph contains a Hamiltonian cycle of triangles. Maybe the methods used in the proof are of some interest. (Joint

work with A. Rucinski and V. Rodl.)

Topic: TBA December 17

Presenter: Yanyan Li, Rutgers University

Topic: TBA December 17

Presenter: Tamas Hausel, Institute for Advanced Studies