Topic: Diffusion of General Data on Non-Flat Manifolds via Harmonic February 23 (top floor)

Maps Theory: The Direction Diffusion Case

Presenter: Guillermo Sapiro, University of Minnesota

Abstract: In a number of disciplines, directional data provides a fundamental source of information. A novel framework for isotropic and anisotropic diffusion of directions is presented in this paper. The framework can be applied both to denoise directional data and to obtain multiscale representations of it. The basic idea is to apply and extend results from the theory of harmonic maps, and in articular, harmonic maps in liquid crystals. This theory deals with the regularization of vectorial data, while satisfying the intrinsic unit norm constraint of directional data. We show the corresponding variational and partial differential equations formulations for isotropic diffusion, obtained from an $L_2$ norm, and edge preserving diffusion, obtained from an $L_p$ norm in general and an $L_1$ norm in particular. In contrast with previous approaches, the framework is valid for directions in any dimensions, supports non-smooth data, and gives both isotropic and anisotropic formulations. In addition, the framework of harmonic maps here described can be used to diffuse and analyze general image data defined on general non-flat manifolds, that is, functions between two general manifolds. We

present a number of theoretical results, open questions, and examples for gradient vectors, optical flow, and color images.

We will conclude the talk showing examples of other applications where the smoothing and interpolation of directional data is needed. In particular, we will show examples on automatic image inpainting, that is, the replacement of objects in images.

Topic: The Metaplectic Group, Harmonic Oscillators, Transformation of February 23

Theta Functions, Representation Theory, Orthogonal Polynomials,

and Multivariate Statistics.

Presenter: John Stalker, Princeton University

Abstract: For some time I have been interested in the connections between the classical mechanics of harmonic scillators, the corresponding quantum systems, the Heisenberg group, the inhomogeneous metaplectic group, the Schwarz class of functions and tempered distributions, and the transformation law for theta functions. All of this material is well-known, but there doesn't seems to a be a single source that puts all the connections together. My original interest in these matters came from a problem in singular perturbation theory in quantum mechanics. Recently, through some work of Bert Kostant, I realized that there are further connections with the representation theory of the universal cover of the etaplectic group and with multivariate statistics. There should also be a connection with some known families of symmetric orthogonal polynomials. Siddharta Sahi and I are currently trying to understand that connection. That sounds like rather a lot, and I may have to skip a few of the more interesting digressions, but I hope to get through the essentials of all of it.

Topic: Newton Interpolation Polynomials and Growth of number of February 24

periodic points for prevalent diffeomorphisms (joint with B.Hunt).

Presenter: Vadim Kaloshin, Princeton University

Abstract: We shall describe a new general approach to attact a class of problems about generic properties of dynamical systems. This approach develops a new perturbative technic based on perturbation of dynamical systems by Newton Interpolation Polynomials. As the by-product this approach gives that for any $\delta>0$ the number of periodic points of a prevalent diffeomorphism $f$ of a compact manifold $M$ satisfy $$ \#\{x \in M: f^n(x)=x\}\leq \exp(C n^{1+\delta}) for some C>0. $$ This result is the opposite to the result of the author which says that on a Baire generic set of diffeomorphisms the number of periodic points can grow arbitrarily fast.

Topic: An equation of Monge-Ampere type in coformal geometry, and February 24

four-manifolds of positive Ricci curvature

Presenter: Paul Yang, Princeton University

Topic: Properties of projections of fractal sets from infinite into finite February 25

dimensional spaces" (joint with B.Hunt) (Note earlier time)

Presenter: Vadim Kaloshin, Princeton University

Topic: Temperley-Lieb algebras and Four Color theorem February 25

Presenter: Robin Thomas, Georgia Institute of Technology

Abstract: The Temperley-Lieb algebra T_n with parameter 2 is the associative algebra over Q generated by 1, e_0, e_1,..., e_n, where the generators satisfy the relations e_i^2=2e_i, e_ie_je_i=e_i if |i-j|=1 and e_ie_j=e_je_i if |i-j|>1. We use the Four Color Theorem to give a necessary and sufficient condition for certain elements of T_n to be nonzero. It turns out that the characterization is, in fact, equivalent to the Four Color Theorem. This is joint work with L.H.Kauffman.

Topic: Analytic torsion and hyperbolic manifolds February 25

Presenter: W. Mueller, Univ. of Bonn and IAS

Topic: Rational Billiards, volumes of fundamental domains, and February 25

the Siegel-Veech constants

Presenter: Alex Eskin, University of Chicago

Topic: Propagation of singularities for the wave equation on conic manifolds February 28

Presenter: Jared Wunsch, Columbia University

Abstract: When a singularity of a solution to the wave equation on a riemannian manifold reaches a point of conic

singularity of the metric, it undergoes a mixture of dispersive and geometric propagation first described by

Kalka-Menikoff and Cheeger-Taylor. New notions of boundary wavefront set permit a simpler, more conceptual

approach to the problem, and associated positive commutator methods broaden the class of manifolds on which we can

work. This is joint work with Richard Melrose.

Topic: Non-holonomic and piecewise-holonomic mechanical systems February 28

Presenter: Philip Holmes, Department of Mechanical and Aerospace Engineering

and Program in Applied and Computational Mathematics, Princeton University

Abstract: Nonholonomic (velocity dependent) constraints can lead to asymptotically stable motions in certain conservative mechanical systems; the Chaplygin sleigh is a canonical example. In studying models for legged locomotion, piecewise-holonomic constraints (due to intermittent foot placements) are typical. The resulting hybrid dynamical systems include flows along a smooth vectorfield punctuated by impulsive jumps governed by discrete `collision maps.' They may be viewed as generalisations of billiards-type problems. Such systems can also exhibit partial asymptotic stability, even while conserving total energy. I will describe joint work with Michael Coleman (Cornell University) and John Schmitt (Princeton University) on a discrete sister to the Chaplygin sleigh, and on a simple model for rapidly running insects, which illustrate this phenomenon. Tea will be served at 3:45 p.m. in 204 Fine Hall

Topic: Moduli of stable maps with group action February 29

Presenter: Valery Alexeev, University of Georgia

Topic: Conformal maps and the Whitham equations March 1

Presenter: I. Krichever, Columbia University

Abstract: The Whitham equations are a core stone of the perturbation theory of the soliton equations. They are deeply connected with structures of topological quantum field theories (WDVV equations), and with the Seiberg-Witten solution of N=2 supersymmetric gauge models. Recently, it was discovered that special solutions of the Whitham equations describe conformal maps.

Topic: Newton Interpolation Polynomials and Growth of number of periodic March 2

points for prevalent diffeomorphisms (joint with B.Hunt).

Presenter: Vadim Kaloshin, Princeton University

Abstract: We shall describe a new general approach to attact a class of problems about generic properties of dynamical systems. This approach develops a new perturbative technic based on perturbation of dynamical systems by Newton Interpolation Polynomials. As the by-product this approach gives that for any $\delta>0$ the number of periodic points of a prevalent diffeomorphism $f$ of a compact manifold $M$ satisfy $$ \#\{x \in M: f^n(x)=x\}\leq \exp(C n^{1+\delta}) for some C>0. $$ This result is the opposite to the result of the author which says that on a Baire generic set of diffeomorphisms the number of periodic points can grow arbitrarily fast.

Topic: TBA March 2

Presenter: Daniel Tataru, Northwestern University

Title: On a class of algebraically defined graphs March 3

Speaker: Felix Lazebnik, University of Delaware

Abstract: Let $F^n$ denote be the $n$-dimensional vector spaces over a field $F$. For $n\ge 2$ and each $i=1,2,\ldots, n-1$, let $f_i: F^{2i}\to F$ be a function of $2i$ variables. We consider a bipartite graph whose vertex partitions $P$ and $L$ are copies of $F^n$ with $p = (p_1,p_2,\ldots, p_n)\in P$ and $l = (l_1,l_2,\ldots, l_n)\in L$ being joined by an edge if and only if the following $n-1$ equalities are satisfied: $$\eqalign{& l_2 + p_2 = 1(p_1,l_1)\cr &l_3 + p_3 = f_2(p_1,l_1, p_2,l_2)\cr ldots\ldots\ldots\ldots\ldots\ldots\ldots\cr&l_n + p_n = f_{n-1}(p_1,l_1, p_2,l_2, \ldots, p_{n-1},l_{n-1})\cr}$$ For particular fields $F$ and particular functions $f_i$'s, the families of graphs defined this way (or slightly modified) posses many remarkable properties. They are concerned with forbidden cycles, girth, graph homomorphism, eigenvalues, edge-decompositions of complete graphs and complete bipartite graphs, and some Ramsey type problems. In this talk we survey some published results, and present several new ones.

Topic: Long-time evolution in general relativity and geometrization of 3-manifolds March 3

Presenter: Michael Anderson, SUNY Stony Brook

Abstract: We will discuss some surprising relations between the geometrization of 3-manifolds (Thurston conjecture) and issues in general relativity. The relation comes from examining the long-time asymptotics for the evolution of space (i.e. space-like hypersurfaces) under the vacuum Einstein equations. The detailed relationship between these topics is completely conjectural, and involves very hard issues for the vacuum Einstein evolution. Thus, we will discuss some of these conjectures, and present a few initial results giving perhaps some credence to these relations.

Topic: Periodic complexes and group actions March 6

Presenter: Alejandro Adem, University of Wisconsin at Madison

Topic: The Challenge of Complex Systems: How will Science be Changed? March 6

Presenter: Brian D. Josephson, Cavendish Lab, Cambridge University

Abstract: It is only gradually becoming recognised that complex systems are more than complicated versions of ordinary systems or even chaotic systems. They have their own laws and their own kinds of regularities, and instead of reductionistic derivations we have to think, in this context, in terms of interrelated emergent patterns.

These ideas have been well established by workers such as Robert Rosen, but even more interesting are possibilities opened up by the fact that current science may be only an approximation that ignores and smooths out the details of a deeper underlying structure with the nature of an organised complex system. Some frequently reported anomalies may find rationalexplanations in such terms.

Topic: How to put guesswork back into computing March 6 Fine Hall

Presenter: Alexandre J. Chorin, University of California, Berkeley

Abstract: Many problems in science are described by equations whose solutions are too complicated to be solved reliably on any computer; the question is what is the best one can do in such circumstances. One often has some idea about a family of possible outcomes of a computation, and I will explain how such knowledge can be used to find a most likely solution given the limitations on computing power. It turns out that often the most mathematically likely solution looks very unlikely to the naked eye. The reason is related to uncertainty principles that are well understood in physics; I will give examples and show how the paradoxes can be resolved.

Topic: Modularity of elliptic curves March 7

Presenter: B. Conrad, Harvard University

Topic: TBA March 8

Presenter: G. Tian, M.I.T.

Topic: TBA March 9

Presenter: Frank Merle, Departement de Mathematiques

Ecole Normale Superieure, France

Topic: Random Walks and the Gittins Index March 10

Presenter: Peter Winkler, Bell Labs

Abstract: Let $G$ be a fixed finite graph with a distinguished target node, and suppose that two tokens reside initially at nodes $x$ and $y$ of $G$. At each tick of a clock you may select either token, which then takes a uniformly random step to a neighboring node. Your object is to get one token to the target in minimum expected time. Say "$x>y$" if your correct strategy begins with selecting the token at $x$. If $x>y$ and $y>z$, is $x>z$?

Topic: Existence results for some fully non-linear elliptic March 10

equations on Riemannian manifolds

Presenter: Jeff Viaclovsky, University of Texas and M.I.T.

Topic: Stochastic Navier Stokes Equations and Wiener Chaos March 16

Presenter: B.L. Rozovskii, University of Southern California, Los Angeles

Abstract: In this talk we are concerned with fluid dynamics described by stochastic flows of diffeomorphisms. Stochastic Euler and Navier-Stokes equations will be derived from the conservation laws of mass and momentum. Well-posedness of these equations shall be discussed. A Wiener chaos expansion of the velocity field will be presented and formulas for the statistical moments of this field will be derived.

Topic: Global secular dynamics in the planar three-body problem March 23

Presenter: Jacques Fejoz, Northwestern University