Current as of 4-20-2000

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.

Geometry Seminar

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

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

Date: Thursday, April 20, 2000, Time: 3:00 pm, Location: Fine Hall 322

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.

Topology Seminar

Topic: Pseudoholomorphic curves in symplectisations and some global problems in contact geometry

Presenter: Casim Abbas, University of Pennsylvania

Date: Thursday, April 20, 2000, Time: 4:30 p.m., Location: Fine 314

Graduate Student Seminar

Topic: The convex minorant of random walks and Brownian motion

Presenter: Toufic Suidan, Princeton University

Date: Friday, April 21, 2000, Time: 12:30 p.m., Location: Fine 214

Geometry Seminar

Topic: Isoperimetric inequalities on compact manifolds

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

Discrete Mathematics Seminar

Topic: The Structure of Topologically Closed Classes of Trees

Presenter: Neil Robertson, Ohio State University

Date: Friday, April 21, 2000, Time: 2:30 p.m., Location: Fine 322

Week of April 24 - 28, 2000

Analysis Seminar

Topic: Global existence for quasilinear wave equations outside of star-shaped obstacles

Presenter: Chris Sogge, John Hopkins University

Date: Monday, April 24, 2000, Time: 4 p.m., Location: Fine 314

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.

PACM Colloquium

Topic: 0-1 Laws for Single Molecules

Presenter: Bud Mishra, Courant Institute, New York University

Date: Monday, April 24, 2000, Time: 4:00 p.m., Location: Fine 224

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.

Computer Science - Mathematics Joint Colloquium

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

Presenter: Sanjeev Arora, CS Dept, Princeton University

Date: Tuesday, April 25, 2000, Time: 12:10 p.m., Location: Fine 314

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)

Algebraic Geometry Seminar

Topic: Hyperplane arrangements, cohomology and syzygies

Presenter: Sorin Popescu, Columbia University

Date: Tuesday, April 25, 2000, Time: 4:15 p.m., Location: Fine 322

Mathematical Physics Seminar

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

Presenter: Agoston Pisztora, Carnegie Mellon University

Date: Tuesday, April 25, 2000, Time: 4:30 p.m., Location: Jadwin A06

Colloquium

Topic: On the Quantum Mechanics of Individual Systems

Presenter: J. Ax, Princeton University

Date: Wednesday, April 26, 2000, Time: 4:30 p.m., Location: Fine Hall 314

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.

Ergodic Theory & Mathematical Physics

Topic: Gromov's Mean Dimension

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

Topology Seminar

Topic: "New" geometry and topology of orbifolds

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

Date: Thursday, April 27, 2000, Time: 4:30 p.m., Location: Fine 314

Geometry Seminar

Topic: TBA

Presenter: Daniel Burns, University of Michigan

Date: Friday, April 28, 2000, Time: 3:00 pm, Location: Fine 314

Analysis seminar

Topic: On discrete Schroedinger operators with potentials defined by the skew-shift (joint work with J. Bourgain

and M. Goldshtein

Presenter: Wilhelm Schlag, Princeton University

Date: Monday, May1, 2000, Time: 4:00 p.m., Location: Fine 314

Algebraic Geometry Seminar

Topic: TBA

Presenter: K. Conrad, Ohio State University

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

Mathematical Physics Seminar

Topic: Towards a microscopic theory of classical liquids

Presenter: Philippe Choquard, Ecole Polytechnique, Lausanne

Date: Wednesday, May 10, 2000, Time: 4:30PM, Location: Jadwin A06.

Analysis seminar

Topic: TBA

Presenter: Gabor Francsics, Columbia University

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