Current as of 4-12-2000

Topic: Strong form of Poincare duality April 13

Presenter: Edgar Brown, Brandeis University

Topic: Can One Hear the Shape of a Bell? April 14

Presenter: Jade Vinson, Princeton University

Abstract: On a planar domain or a Riemannian surface, the eigenvalues of the Laplacian carry a lot of geometric information: the area, the perimeter, the genus, and the lengths of some (sometimes all) closed geodesics. Do the eigenvalues carry enough information to recover the domain or surface (the inverse spectral problem)? In at least some cases, no: we'll see two planar domains with the same spectrum. In some cases, yes or almost. We now consider the inverse spectral problem for surfaces of revolution, and review known results by Bruning/Heintze and Zelditch. The speaker then presents a reconstruction algorithm for surfaces of revolution and conjectures that it almost always succeeds. The almost-sure success is proven for a toy problem.

Topic: Noncommutative gauge theory and Kontsevich deformation quantization April 14

Presenter: Peter Schupp, University of Munich

Topic: Permanents, mixed volumes and a generalization of a theorem of Rado April 14

Presenter: Alex Samorodnitsky, Institute for Advanced Study

Abstract: We present a deterministic polynomial-time algorithm that computes the mixed discriminant of an n-tuple of positive semidefinite matrices to within a multiplicative factor of n^{n^2}. The main idea is an appropriate definition and application of scaling. This generalizes the approach of Linial, Samorodnitsky and Wigderson to deterministic approximation of the permanent.

As a corollary, we obtain a deterministic polynomial-time algorithm that computes the mixed volume of n convex bodies in R^n to within an error which depends only on the dimension. This answers a question of Dyer, Gritzmann and Hufnagel. A 'side benefit' is the following version of Rado's theorem on existence of a linearly independent transversal: Let A_1,...,A_n be subsets of R^n. Assume, that there exists an e > 0, such that for any k-subset S of {1...n} there are k vectors in \cup_{i\in S} A_i which span a k-dimensional box of volume > e^k. Assume also, that the vectors in A_1,...,A_n are of length at most l. Then there is a choice of vectors a_1 in A_1 ... a_n in A_n spanning an n-dimensional box of volume at least (e/ln)^{n^2}. This is joint work with Leonid Gurvits.

Topic: Bubbling and Quantization for Critical Points of April 14

Ginzburg-Landau Functionals

Presenter: Changyou Wang, University of Kentucky.

Abstract: We consider a family of solutions to the E-L equations of the Ginzburg-Landau functionls. We prove that for almost all points on the energy concentration set the density function of the energy can be quantified as the finite sum of energies of harmonic maps from $S^2$. As a byproduct, we obtain an improved version of energy identity for approxiamted harmonic maps in $2$-dimension, with tension fields bounded in $L^p$ for any $p>1$. This is a joint work with F.H. Lin.

Topic: Global Wellposedness of KdV for Rough Data April 17

Presenter: Jim Colliander, UC Berkley

Abstract: Techniques introduced by Bourgain were applied and extended by Kenig, Ponce and Vega to the Korteweg-deVries equation leading to a sharp local-in-time result in the standard Sobolev spaces H^s, s>-3/4. It is conjectured that local wellposedness implies global wellposedness for KdV. I plan to survey the wellposedness theory of KdV, motivate considering the initial value problem with rough data and describe recent progress towards the global-in-time conjecture.

Topic: OSTWALD RIPENING: The effect of the Geometry April 17

of the Distribution

Presenter: Nicholas Alikakos, University of Tennessee

Abstract: We consider a two-phase system in 3d . We are interested in the coarsening of the spatial distribution, driven by the reduction of interfacial energy, and limited by diffusion as described by the quasi static Stefan free boundary problem. We address the regime where the one phase covers only a small fraction of the total volume, and consists initially of many disconnected Components (particles). In this situation mass diffuses from the vicinity of the smaller particles towards the larger, a phenomenon known as Ostwald Ripening. In the early 60's Lifshitz, Slyosov, and Wagner separately, formally derived an evolution for the distribution of the particle radii. We present a refinement of their theory, which takes into account the geometry of the spatial distribution and appears to agree qualitatively better with experiments.

Topic: Abelian varieties over number fields with good reduction everywhere April 18

Presenter: R. Schoof, University of Rome II and Harvard University

Topic: C*-Algebras and the Novikov Conjecture April 19

Presenter: N. Higson, Pennsylvania State University

Abstract: I will give an introductory account of some of the pros and cons of attacking the Novikov conjecture (and related problems in manifold theory) using C*-algebra methods. To its credit, C*-algebra theory has helped prove some of the most general known theorems about the Novikov conjecture. But the limitations of the C*-algebra method are now becoming apparent, and it is unclear if further real progress can be made using it.

Topic: Pseudoholomorphic curves in symplectisations and April 20

some global problems in contact geometry

Presenter: Casim Abbas, University of Pennsylvania

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: TBA April 24

Presenter: Chris Sogge, John Hopkins University

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

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: 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

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: "New" geometry and topology of orbifolds April 27

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

Topic: TBA April 28

Presenter: Daniel Burns, University of Michigan

Topic: TBA May 2

Presenter: K. Conrad, Ohio State University