Geometric Analysis and its Applications

A Conference in Honor of Edgar Feldman


Cristina Abreu-Suzuki, CUNY Graduate Center student,,
Contributed Talk,
Sat 2:00-2:25pm

Distance Manipulators

Among several geometric properties of Riemannian manifolds which are preserved under Distance Manipulators (a.k.a. rough isometries as defined by M. Kanai , 1985) one will find upper bound on volume growth, isoperimetric dimensions and Sobolev constants, Liouville property, transience of Brownian motion and Harnack inequalities. As a main tool we show that for maximal rank onto mappings between Riemannian manifolds with bounded geometry, under assumptions on the fibers and assumptions on the subspaces of horizontal vectors, there exists a Distance Manipulator between the domain and the product of the base manifold and any fixed fiber of the domain. From there one can make analytic assumptions about the behavior of the base manifold and a fixed fiber, in order to conclude the appropriate result for the domain, such as the geometric properties cited above.

Fabiana Cardetti, University of Connecticut,
Contributed Talk,
Sat 4:30-4:55 pm

Local controllability on SL(2,R)

We define linear control systems on Lie Groups and present some of the latest results on local controllability of this type of systems. In particular, we present the results on the special linear group, SL(2,R).

Isaac Chavel, City College, CUNY,,
Plenary Talk,
Sun 12:00-12:30 pm

A Survey of Feldman's Work

Jeff Cheeger, Courant Institute, NYU,,
Plenary Talk,
Sunday 1:45-2:30 pm

Collapsing and noncollapsing of Einstein $4$-manifolds

We will discuss joint work with Gang Tian concering sufficiently collapsed Einstein 4-manifolds. As one consequence, we show that if the global volume has definite lower bound and the Einstein constant is normalized to equal 3, then then the manifold cannot be very collapsed.

Mohammad Ghomi, Georgia Tech and Penn State,,
Contributed Talk,
Sat 10:15-10:40 am

Immersions and Embededdings with Curvature Constraints

The first result in this area is a theorem of Feldman who showed that closed curves without inflection points in Euclidean 3-space have exactly two regular homotopy classes. Later, Gluck and Pan proved that knots without inflection points are isotopic through such knots provided that they are isotopic through general knots, and have the same self-linking number. The proof of the latter result follows from other theorems of Gluck and Pan on deformations of surfaces with boundary and positive curvature, which they had obtained by explicit constructions. In this talk we show how to use Gromov's h-principle, specifically the holonomic approximation theorem, to generalize these results to submanifolds with prescribed signs of principal curvatures and homotopy type of principal directions. Also, we show how convex integration techniques can be used to find knots with constant curvature or torsion in each isotopy class, and construct isotopies which keep the curvature or torsion of these knots constant.

Maria Gordina, University of Connecticut,,
Plenary Talk,
Sat 3:00-4:15 pm

Riemannian geometry in infinite dimensions and Gaussian/heat kernel measures

First we will show how stochastic differential equations can be used to construct a Gaussian measure on an infinite dimensional group. An example of such a group is a space of orthogonal infinite matrices. It is known that properties of such Gaussian measures are related to the geometric properties of these infinite dimensional groups viewed as Riemannian manifolds. We will describe some new results on the geometry of these groups, and explain how it is related to the analytic properties of the Gaussian measures in infinite dimensions.

Allen Gorin, AT&T Laboratories - Speech Research,,
Contributed Talk,
Sat 1:30-1:55pm

Semantic Information Processing of Spoken Language - How May I Help You?

The next generation of voice-based user interface technology enables easy-to-use automation of new and existing communication services, achieving a more natural human-machine interaction. By natural, we mean that the machine understands what people actually say, in contrast to what a system designer expects them to say. This approach is in contrast with menu-driven or strongly-prompted systems, where many users are unable or unwilling to navigate such highly structured interactions. AT&T's 'How May I Help You?' (tm) technology shifts the burden from human to machine, wherein the system adapts to peoples' language, as contrasted with forcing users to learn the machine's jargon. We have developed algorithms which learn to extract meaning from fluent speech via automatic acquisition and exploitation of salient words, phrases and grammar fragments from a corpus. In this talk I will describe the speech, language and dialog technology underlying these nationally deployed systems.

Alexander Grigor'yan, Imperial College London,,
Plenary Talk,
Sunday 5:15-6:00 pm

Heat kernels and function theory on fractal-like metric spaces

Evans Harrell, Georgia Institute of Technology,,
Contributed Talk,
Sat 5:30-6:00 pm

Eigenvalue gaps and mean curvature for Schrodinger operators on immersed manifolds.

Commutator relations are used to investigate the spectra of Schrodinger Hamiltonians,

H = -&Delta + V(x)
acting on functions of a smooth, compact d-dimensional manifold Md immersed in R&nu where &nu &ge d+1. Here &Delta denotes the Laplace-Beltrami operator, and the real-valued potential--energy function V(x) acts by multiplication. The manifold Md may be complete or it may have a boundary, in which case Dirichlet boundary conditions are imposed.

It is found that the mean curvature of a manifold poses tight constraints on the spectrum of H. Further, a special algebraic role is found to be played by a Schrodinger operator with potential proportional to the square of the mean curvature:

Hg := -&Delta + g h2,
where &nu = d+1, g is a real parameter, and h := &kappa1+ ... + &kappad , with &kappaj, j = 1 ... d denoting the principal curvatures of Md. For instance, each eigenvalue gap of an arbitrary Schrodinger operator is bounded above by an expression using H1/4. The ``isoperimetric" parts of these theorems state that these bounds are sharp for the fundamental eigenvalue gap and for infinitely many other eigenvalue gaps.

Jenny Harrison, Berkeley,
Contributed Talk,
Sun 4:30-5:00 pm

Discrete Exterior Calculus with Convergence to the Smooth Continuum

In Analysis Situs, Poincare used a simplicial complex as his basic discrete model. This has had broad appeal since a finite set of points determines the simplexes and the boundary operator has a natural, discrete definition. In over a hundred years, this approach has not yielded a full discrete theory and convergence to the smooth continuum has remained elusive. In this lecture, entirely new definitions of discrete chains are introduced in a Riemannian manifold as idealized, geometric analogues to differential forms. Reminiscent of the shift from simplicial to singular homology, an intrinsic theory of discrete calculus arises which converges to the smooth continuum. The theory lays new foundations for much of standard analysis in a way that captures infinite limiting processes--- the heart of calculus--- in terms of finite computations. Certain constructions of topology are restored to geometry at the level of chains and cochains, without passing to homology and cohomology. Examples include Poincare duality and intersection of chains. Smooth manifolds, metrics, fractals, vector fields, curvature, differential forms, foliations and measures can be discretized. Operators on forms have geometric counterparts for chains including Lie derivative, Hodge star, codifferential and Laplace, giving them physical meaning alongside the venerable boundary operator. Newly defined geometric products have natural definitions such as interior, exterior, wedge, cap, cup, slant, and convolution, with interesting commutator relations.

Note that Jenny Harrison will continue her presentation the Einstein Chair Seminar:

Mark Hillery, Hunter College, CUNY,,
Plenary Talk,
Sun 12:45-1:30pm

Quantum walks on graphs: Recent work with Edgar Feldman and Janos Bergou

Classical random walks on graphs are the basis of algorithms that solve a number of problems, for example graph connectivity and 2-SAT. Quantum algorithms, which would require a quantum computer to run on, have proven more powerful than classical algorithms for some problems. It is hoped that quantum walks on graphs, which have only recently been defined, might serve as a basis for additional quantum algorithms. These walks will be defined and some of their properties discussed. In particular, the fact that they spread faster than classical random walks suggests that they may be able to sample the structure of a graph faster than can a classical walk.

Tom Ilmanen, ETH,
Plenary Talk,
Sat 11:45-12:30

Mean curvature flow in three dimensions

Abstract: I will describe the current picture of singularity formation for mean curvature flow of surfaces in R^3, with some comparisons to Ricci flow of 4-manifolds.

Lucio Prado, CUNY Graduate Center Student,
Contributed Talk,
Sat 2:30-2:55 pm

Kelvin-Nevanlinna-Royden Criterion

The aim of this talk is to present the Kelvin-Nevanlinna-Royden criterion for p-hyperbolicity on graphs and some applications:

Regina Rotman, Penn. State University,
Contributed Talk,
Sat 10:45-11:10 am

The length of a shortest closed geodesic and the area of a minimal surface

I will talk about the upper bounds for the length of a shortest closed geodesic on a closed Riemannian manifold. I will also discuss two curvature-free upper bounds for the minimal mass of a strongly stationary 1-cycle, an object that is in some ways similar to a closed geodesic, and for the smallest area of a minimal surface. (Joint with A. Nabutovsky).

Leonid Srubshchik, Touro College and Baruch College, CUNY,,
Contributed Talk,
Sat 5:00-5:25 pm

The Energy Saddle Points and Dynamic Snap-Through of Thin Elastic Shells

The dynamic instability in the large is investigated for an elastic shell under step loading and a more general non-linear elastic continuous conservative system with Rayleigh friction and a given initial velocity. By using a potential theory analysis, the concepts of a well in topological spaces and its equilibrium stability factor introduced by A. D. Myshkis, the definitions are given of the dynamic stability of the system, the critical load of its dynamic snap-through, and the astatic critical load. The latter is determined from a stationary problem and yields a lower limit of those values of the load for which dynamic snap-through occurs. For the class of the system with potential energy of the square of the norm in an energetic space plus a weakly continuous functional , it is proved by L.S. Srubshchik and V.I. Yudovich that the existence of saddle points with negative index follows from the non-uniqueness of the stable equilibrium. At least one saddle point is found on the boundary of the well of the each stable equilibrium.

The examples of the computation of the astatic critical load for spherical, conical, and ellipsoidal shells in the case of the ambiguity of the families of unstable equilibriums are presented. It is established that a necessary condition for the dynamic snap-through (eversion) of the shell subjected to an impulsive load is the existence of the saddles of the potential energy for the same load-free shell at the boundary of a stable zero equilibrium well. For the strictly convex shallow shells the existence of the saddles is proved by the asymptotic method.

It is shortly discussed the new results (obtained by S.S. Antman and L.S. Srubshchik) about the eversion of nonlinearly elastic compressible and incompressible axisymmetric strictly convex shells within a general geometrically exact theory.

Dennis Sullivan, CUNY Graduate Center and SUNY Stony Brook,
Plenary Talk,
Sun 3:30-4:15 pm

Algebraic topology of the space of closed curves in a manifold.

Kristopher Tapp, Williams College,,
Contributed Talk,
Sat 11:15-11:40 am

Examples of quasi-positive curvature

I will describe new examples of manifolds which admit a Riemannian metric with sectional curvature nonnegative, and strictly positive at one point. These examples include the unittangent bundles of CPn, HPn and CaP2, and a family of lens space bundles over CPn. These examples are related to Wilking's recent examples of almost-positively curved manifolds.