317 Milner Building
9:30 - 10:30
Speaker: D. Burago of Pennsylvania State University
Title: Boundary rigidity, volume minimality, and minimal
surfaces in L_{infinity}: a survey. Abstract.
10:30-10:45 Coffe Break
10:45 - 11:45
Speaker: O. Musin of University of Texas at Brownsville
Title: Positive definite functions in distance geometry. Abstract.
11:45 - 2:00 Lunch Break
2:00 - 3:00
Speaker: M. Sapir of Vanderilt University
Title: On asymptotic properties of groups. Abstract.
3:10 - 4:10
Speaker: A. Dranishnikov of University of Florida
Title: On macroscopic dimension. Abstract.
4:30 - 5:30
Speaker: V. Nekrashevych of Texas A&M University
Title: Hyperbolic duality. Abstract
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 nonâ€“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.