Texas A&M University, Department of Mathematics

Workshop on

Asymptotic and Extreme Properties of Metric Spaces and Groups

April 12, 2010


317 Milner Building

Organizer: Rostislav Grigorchuk, Oleg Musin


D. Burago of Pennsylvania State University
Boundary rigidity, volume minimality, and minimal surfaces in L_{infinity}: a survey

A Riemannian manifold with boundary is said to be boundary rigid if its metric is uniquely determined by the boundary distance function, that is the restriction of  the distance function to the boundary. Loosely speaking, this means that the Riemannian metric can be recovered from measuring distances between boundary points only. The goal is to show that certain classes of metrics are boundary rigid (and, ideally, to suggest a procedure for recovering the metric).

To visualize that, imagine that one wants to find out what the Earth is made of. More generally, one wants to find out what is inside a solid body made of different materials (in other words, properties of the medium change from point  to point). The speed of sound depends on the material. One can "tap" at some points of the surface of the body and "listen when the sound gets to other points". The question is if this information is enough to determine what is inside.

This problem has been extensively studied from PDE viewpoint: the distance between boundary points can be interpreted as a "travel time" for a solution of the wave equation. Hence this becomes a classic Inverse Problem when we have some information about solutions of a certain PDE and want to recover its coefficients.  For instance such problems naturally arise in geophysics (when we want to find out what is inside the Earth by sending sound waves), medical imaging etc.

In a joint project with S. Ivanov we suggest an alternative geometric approach to this problem. In our earlier work, using this approach we were able to show boundary rigidity for metrics close to flat ones (in all dimensions), thus giving the first class of boundary rigid metrics of non–constant curvature beyond two dimensions. We were now able to extend this result to include metrics close to a hyperbolic one.

The approach is grew up from another long-term project of studying surface area functionals in normed spaces, which we have been working on it for more than ten years. There are a  number of related issues regarding area-minimizing surfaces in Riemannian manifold. The talk gives a non-technical survey of ideas  involved. It assumes no background in inverse problems and is supposed to be accessible to a general math audience (in other words,  we will not get into any technical details of the proofs).

O Musin of University of Texas at Brownsville
Positive definite functions in distance geometry

I. J. Schoenberg  proved that a function  is positive definite  in the unit sphere if and only if this function is a nonnegative  linear combination of Gegenbauer  polynomials. This fact play a crucial role in Delsarte's method for finding bounds for the density of sphere packings on spheres and Euclidean spaces.

One of the most excited applications of Delsarte's method is  a solution of the kissing number problem in dimensions 8 and 24. However, 8 and 24 are the only dimensions in which this method gives a precise result. For other dimensions (for instance, three and four) the upper bounds exceed the lower. We have found an extension of the Delsarte method that allows to solve the kissing number problem (as well as the one-sided kissing number problem) in dimensions  four.

In this talk we will discuss the maximal cardinalities of spherical  sets with few distances.  Using the so-called  polynomial method, Nozaki's theorems and Delsarte's method these cardinalities can be determined for many dimensions. This method also can be applied for bounds on s-distance sets in Hamming and Johnson spaces.

Recently, were found extensions of Schoenberg's theorem for multivariate positive-definite functions. Using these extensions  and semidefinite programming can be improved some upper bounds for spherical codes.

M. Sapir of Vanderbilt University
On asymptotic properties of groups

The isoperimetric function of a group bounds the minimal area of a disc with the given boundary loop in the Cayley complex of the group in terms of the length of the loop.  I will survey results concerning possible isoperimetric functions of groups, relations with the asymptotic cones of the group and computational complexity of the word problem.

A. Dranishnikov of University of Florida
On macroscopic dimension

Gromov introduced the notion of macroscopic dimension in order to characterize manifolds that admit a metric of positive scalar curvature in terms of the large scale geometry of their universal covers. He conjectured that for an n-manifold with positive scalar curvature the macroscopic dimension its universal cover is at most n-2. Another his conjecture was that for a rationally essential n-manifolds the macroscopic dimension of the universal cover is less than n. Thus Gromov's conjectures can be stated for classes of groups just by considering manifolds with the fundamental group from a given class of finitely presented groups.

We will discuss a recent progress on these two conjectures. In particular we prove the conjectures for some classes of groups.

V. Nekrashevych of Texas A&M University
Hyperbolic dualiity

A notion of a hyperbolic groupoid and of its dual (its boundary) will be presented. Some examples will be discussed: hyperbolic groups, hyperbolic rational functions, contracting self-similar groups and
Ruelle groupoids of hyperbolic dynamical systems.