Current as of 4-21-2000

Topic: Nonexpanding maps, Busemann functions and multiplicative ergodic theory April 20

Presenter: Anders Karlsson, Yale University

Abstract: First, we consider nonexpanding maps of proper metric spaces. We prove a result that generalizes Wolff-Denjoy type theorems in complex analysis. Second, we consider random products of nonexpanding maps of nonpositively curved spaces. In a joint work with Margulis, we obtain that almost every trajectory lies on sublinear distance from a geodesic ray. This result generalizes

Oseledec's theorem on random products of matrices, and has further applications to bounded harmonic functions on discrete groups and Brownian motion on compact manifolds.

Topic: The blow up locus of harmonic maps and heat flows April 20

Presenter: Jiayu Li, Institute of Mathematics, CAS, Beijing

Abstract: We analyze the blow-up locus of harmonic maps and the heat flows for harmonic maps. We find it related to minimal submanifolds and mean curvature flows. Using the blow up formula for the blow up set and the limiting map, We prove that the blow up set for triholomorphic maps is stationary.

Topic: Pseudoholomorphic curves in symplectisations and some April 20

global problems in contact geometry

Presenter: Casim Abbas, University of Pennsylvania

Topic: The convex minorant of random walks and Brownian motion April 21

Presenter: Toufic Suidan, Princeton University

Abstract: The convex minorant of a symmetric random walk process turns out to be a key tool in solving several interesting physics problems. We briefly describe one such problem and proceed to compute several useful statistical properties of the convex minorant. The computations will be elementary but not standard. We relate these calculations to Brownian motion.

Topic: The Structure of Topologically Closed Classes of Trees April 21

Presenter: Neil Robertson, Ohio State University

Topic: Isoperimetric inequalities on compact manifolds April 21

Presenter: Olivier Druet, University Cergy-Pontoise

Date: Friday, April 21, 2000, Time: 3:00 pm, Location: Fine 314

Topic: TBA

Presenter: Tom Branson, University of Iowa

Date: Friday, April 21, 2000, Time: 4:00 pm, Location: Fine 314

Topic: Global existence for quasilinear wave equations outside April 24

of star-shaped obstacles

Presenter: Chris Sogge, John Hopkins University

Abstract: In this joint work with M. Keel & H. Smith we prove that the global existence theorem of Christodoulou and Klainerman for quasilinear wave equations satisfying the null condition holds in the setting of Dirichlet-wave equations outside of star-shaped obstacles. We use an adaptation of Christodoulou's conformal method. The main ingredients are an energy estimate that is related to classical decay estimates of Morawetz and also a pointwise estimate that is related to recent global Strichartz estimates obtained by Hart Smith and the speaker.

Topic: 0-1 Laws for Single Molecules April 24

Presenter: Bud Mishra, Courant Institute, New York University

Abstract: Single molecule methods (e.g., optical mapping, molecular combing, fluorescent flow cytometry, ion channels, etc.) for genomics and proteomics rely on the statistical properties of a large number of identical molecules. We will use ideas from probabilistic methods to show existence of 0-1 laws governing the behavior of the group of molecules and how we exploit it in devising powerful algorithmic and automation tools to create restriction maps and sequence information from parsimonious and noisy data from single DNA molecules. The set of tools underlying our "Computational Optical Mapping Project" have been used in making clone maps (BACS and cosmids, Y-DAZ locus), microbial genomic maps (P. falciparum, D. radiodurans, E. coli, etc.), and a partial human genome map.

Topic: Polynomial-time algorithms to learn mixtures of gaussian distributions April 25

Presenter: Sanjeev Arora, CS Dept, Princeton University

Abstract: Mixtures of gaussian (aka normal) distributions are distributions in which x% of the points are from one gaussian, y% from a 2nd gaussian etc. Such distributions arise in many situations. To give an example, human heights and weights are usually distributed according to a (truncated) gaussian but this gaussian is different for different ethnic groups and for males and females. Thus height data for the US population may be viewed as a mixture of gaussians. Such mixture models also arise in in AI, computer vision, speech recognition etc.

In the learning problem, data generated from a mixture of gaussian is given and we are required to learn the component gaussians. It is an open problem in statistics to give a provably efficient algorithm for this problem; even the case of 2 gaussians is open. The classic EM heuristic for the problem does not always perform well in practice.

We give a new algorithm that, under reasonable "nondegeneracy" conditions learns mixtures of k gaussians in R^n. The running time is polynomial in n. This generalizes a recent result of Dasgupta (FOCS'99), which learns mixtures of "spherelike" gaussians of identical "radius."

The design of our algorithm uses elementary ideas from Brunn-Minkowski theory. (Joint work with Ravi Kannan of Yale University)

Topic: Hyperplane arrangements, cohomology and syzygies April 25

Presenter: Sorin Popescu, Columbia University

Topic: Phase Separation and the Wulff Problem in Ising-Potts Models April 25

Presenter: Agoston Pisztora, Carnegie Mellon University

Topic: On the Quantum Mechanics of Individual Systems April 26

Presenter: J. Ax, Princeton University

Abstract: Taking standard quantum mechanics (SQM) as a statistical theory, we extend the standard Hilbert space formulation to a mathematical model of the individuals which comprise the statistical ensembles of SQM. The model of two interacting systems is a singular toroidal bundle over the unit sphere in the Hilbert space of the composite system, together with a natural connection which permits the Schrodinger evolution in the sphere to be lifted to the bundle.The main mathematical innovation required is the construction of convex periodic tilings of Euclidian spaces (which is new even in 3 dimensions). These tilings descend to partitions of the toroidal fibers. The states of the subsystems are determined by which tile contains the lifted evolution. The toroidal tilings are the unique functorial convex partitions consistent with SQM. This is joint work with Simon Kochen.

Topic: Gromov's Mean Dimension April 27

Presenter: Elon Lindenstrauss, Institute for Advanced Studies

Date: Thursday, April 27, 2000, Time: 2:30 p.m., Location: Fine 110

Abstract: Recently, Gromov has introduced a new invariant for dynamical systems called mean dimension. This invariant,

originally introduced to study algebraic varieties and spaces of meromorphic functions, has found applications in

topological dynamics (including a one line answer to a question that has been open for 25 years), and is probably also

relevant to mathematical physics.

Topic: Dynamic Percolation

Presenter: A. Skorokhod

Date: Thursday, April 27, 2000, Time: 3:30 - 4 p.m., Location: Fine 110

Topic: "New" geometry and topology of orbifolds April 27

Presenter: Y. B. Ruan, University of Wisconsin at Madison

Abstract: Orrbifold appears naturally in many branches of mathematics and has been studied by mathematicians since 70’s. Traditionally, orbifolds were studied as an extension of the theory of smooth manifolds. The central theme is that if we are willing to work over the field of rational coefficients the theory of smooth manifolds can be extended to orbifold. Hence, "old" geometry and topology can be considered as part of theory of smooth manifolds. Very recently, the situation started to change where a "new" theory of geometry and topology is emerging. The motivation of the new theory is from orbifold string theory. Therefore, the "new" geometry and topology can be thought as a stringy geometry and topology of orbifolds. The mathematical motivation is follows: if we have a complex orbifold, there are two ways to desingulize the orbifold by either a resolution or a smoothing. We would like to construct a theory on orbifold to capture the information of manifolds obtained by desingulization. The core of the new theory is a new cohomology of orbifold (orbifold cohomology) introduced by Chen-Ruan. In my talk, I will try to touch many aspects of the new theory. It includes orbifold cohomology ring, discrete torsion and twisted orbifold cohomology ring, orbifold K-theory, orbifold stable map, orbifold quantum cohomology, relation to log-quantum cohomology and orbifold mirror symmetry.

Topic: TBA April 28

Presenter: Daniel Burns, University of Michigan

Topic: On discrete Schroedinger operators with potentials defined May 1

by the skew-shift (joint work with J. Bourgain and M. Goldshtein

Presenter: Wilhelm Schlag, Princeton University

Topic: TBA May 2

Presenter: K. Conrad, Ohio State University

Date: Tuesday, May 2, 2000, Time: 4:15 p.m., Location: Fine 322

Topic: TBA May 8

Presenter: Gabor Francsics, Columbia University

Date: Monday, May 8, 2000, Time: 4:00 p.m., Location: Fine 314

Topic: Towards a microscopic theory of classical liquids May 10

Presenter: Philippe Choquard, Ecole Polytechnique, Lausanne