Colloquia, Fall 1998

Vanderbilt Mathematics
Fall 1998

December 4. Uri
, Professor of Mathematics and director of
Dana Center
at University of
Texas at Austin
. Strengthening K-12 mathematics: the role
of university mathematics departments.
Discussion of the
dramatic improvements in Texas public school students’
performance on state and national mathematics examinations; the
role that the UT Austin Mathematics Department has played in
bringing about these improvements; and describe new projects
aimed at strengthening upper-level and advanced courses taught at
the high school level in the state of Texas.


December 3.
Professor Dani Wise, of
Cornell University.
Negative Curvature and Residual Finiteness. Gromov
revolutionized infinite group theory by treating groups as
geometric objects. From this viewpoint, the most prominent class
of groups are the word-hyperbolic groups, which are characterized
by being coarsely negatively curved. … In the last 10
years, word-hyperbolic groups have received a tremendous amount
of attention, and many of their most important properties have
been understood. For instance, Sela has shown that the
isomorphism problem is decidable for torsion-free word-hyperbolic
groups. One striking problem which remains open is whether or
not every word-hyperbolic group is residually finite. A group G
is called residually finite, provided that for each non-trivial
element g in G, there is a finite quotient G > G’, such that
g’ is not trivial. … Some well known groups which are
residually finite include surface groups, and polycyclic groups
and more generally linear groups. Also, the fundamental group of
any compact 3-manifold satisfying Thurston’s geometrization
conjecture is residually finite. … It is widely believed that
there do exist word-hyperbolic groups which are not residually
finite. However, if these groups do exist, they have been hiding
from everybody for a long time! … In my talk, I will
survey the evidence for and against the residual finiteness of
word-hyperbolic groups. I will describe examples of
non-residually finite groups which are non-positively curved; and
I will discuss some of my more recent work which demonstrates
that there is indeed a connection between negative curvature and
residual finiteness.


December 2. A.Yu.Ol’shanskii. The algorithmic complexity of
the word problem for groups and the asymptotic behavior of their
Cayley complexes.
Any group G can be described by
generators and relations. The Dehn (isoperimetric) function f(n)
of a finitely presented group, on the one hand, has a clear
geometric meaning, and on the other hand, f(n) is the upper
bound for the number of “elementary transformations” (which are
given by the relations of G ), one needs to reduce any word W
of length at most n to the empty word, if W = 1 in G. … We
are going to investigate the relation between the time complexity
of the algorithmic word problem for groups and the behavior of
Dehn functions for groups. In particular, a joint work of J.-C.
Birget, A.Ol’shanskii, E.Rips and M.Sapir (1998) gives a
geometric characterization of the groups in which the word
problem is decidable in a polynomial time by a non-deterministic
Turing machine. … There is a similarity in the asymptotic
growth of Dehn functions and the growth of distortion functions.
The latter measure the distortion of the geodesic metric of a
group (or of a Riemanian manifold) G under its embedding into
another group (manifold) H . We formulate a theorem
(Ol’shanskii, 1997) that answers to M.Gromov’s question on
possible behavior of distortion (and associated with them length)
functions for group embeddings.


December 1. Professor
Murray Eden
, MIT. Can Mathematics Do
Anything for Biology and Medicine?
Mathematics as it is used
in common discourse is a very poorly defined concept. It means
quite different things to different groups; for example, school
children studying arithmetic or math professors proving theorems.
A distinction can be made between “doing” mathematics and
“applying” mathematics. Dedekind, Hardy and Wiles “did”
mathematics. Most scientists and engineers use mathematics; that
is, functions and equations developed earlier by mathematicians
who had no purpose other than to develop mathematics. But some,
like Newton, Gauss, Maxwell, Shannon (Name your favorite here.)
created new mathematics, having a particular set of real world
problems in mind. … The physical sciences and
engineering, throughout most of their history, have progressed by
the interaction of new observations, experiments , technology,
with theory and new mathematics. Quantum mechanics, relativity,
statistical mechanics, information theory are some examples. …
But the life sciences and especially clinical medicine are
different. They appear to resist mathematical description except
for rather trivial applications. (“Trivial” is also not too well
defined but in this instance illuminates my prejudice.) The talk
will attempt to explain why this is so and will attempt to
predict where “new” mathematics might furnish insights into
problems arising in biology.


November 30. Professor
Eliyahu Rips
, of Hebrew University,
. The JSJ Decomposition of Finitely Presented Groups
and its Generalizations.
A review of the work of Sela, Sela,
and Rips; Danwoody and Sageev; and Papasoglu and Fujiwara.


November 19. Stefan
E. Schmidt
, of the University of Mainz,
, and ATT Laboratories, Research Division, Florham Park,
New Jersey. On Codes and Rings. The importance of finite
rings for coding theory was highlighted by A. R. Hammons, P. V.
Kumar, A. R. Calderbank, N. J. A. Sloane, and P. Sole, who
proved in 1994 that prominent non-linear codes — such as Kerdock
and Preparata codes — have a linear representation over the
4-element integer residue ring. … With a mix of theory and
examples, my talk will serve as an introduction to the vibrant
and exciting field of codes and rings: For example, I will give
new descriptions of known codes (like the Golay code) via matrix
ring representations. I will also sketch aspects of finite
metric algebra, by pointing out how the extendability of
isometries between codes is connected with Frobenius rings.
Finally, I will construct new small linear codes over the
integers mod 4, which turn out to be optimal.


November 12. Gerhard
X. Ritter
of the University of
. Neural Networks based on Lattice Algebra. The
theory of artificial neural networks has been successfully
applied to a wide variety of pattern recognition problems. In
this theory, the first step in computing the next state of a
neuron or in performing the next layer neural network computation
involves the linear operation of multiplying neural values by
their synaptic strengths and adding the results. Thresholding
usually follows the linear operation in order to provide for
nonlinearity of the network. In this paper we discuss a novel
class of artificial neural networks, based on lattice algebra, in
which the operations of multiplication and addition are replaced
by addition and maximum (or minimum), respectively. By taking the
maximum (or minimum) of sums instead of the sum of products,
computation is nonlinear before thresholding. As a consequence,
the properties of these networks, which are also known as
morphological neural networks, are drastically different than
those of traditional neural network models. … In this talk, we
briefly describe the classical models of neural network
computation and then define the underlying theory of neural
networks based on lattice algebra. We will show that associative
and bidirectional associative memories based on lattice algebra
are generally superior to traditional associative memories in
both storage capacity and perfect recall of noise corrupted
patterns. Additionally, we define the notion of perceptrons based
on the lattice model and compare their performance with the
performance of traditional perceptrons.


November 10. Professor Emeriti Richard Arenstorf, of Vanderbilt
University. Where Is The Gap In This ‘Proof’ of the Twin-Prime
One of the oldest conjectures in mathematics
states that there exists infinitely many pairs of primes whose
difference is to (the twin primes), and there is now overwhelming
computational evidence for its truth. … I will describe the
hopeful, but not yet successful attack on affirming this
conjecture, using methods from analytic number theory.


November 5. Professor
John Conway
, of the University of Tennessee –
. Non-Abelian Approximation. This talk will
present an exposition of part of the theory of approximation in
B(H), the space of operators on a separable Hilbert space (hence
the word “non-abelian in the title). The talk will begin with a
description of an approximate Jordan form found by Apostol
Morrel. This will be applied to obtain a complete description of
the set of operators that can be approximated by operators having
a square root or logarithm. … This talk will be completely
expository with no new results. It should be accessible to
graduate students who know basic functional analysis.


October 29. Serge
of the Hong Kong
University of Science and Technology
(Visiting Lecturer,
Vanderbilt University). Reliably Connected Networks. In
constructing networks, we require reliable connectedness in
addition to the usual requirment of reliability (i.e., the higher
the connectivity, the more reliable the network). Two nodes are
called reliability connected if they are joined by a reliable
path. For given n and m, it is an open problem to
construct a communication network on n nodes having a
given value of connectivity and as few edges as possible and in
which any pair of nodes, with one exceptional node, are
m-reliably connected. We have settled this problem only
for m = 1. (Refreshments afterward in
SC-1425.)(There will be no Graph Theory Seminar this week; it is
replaced by this colloquium.)


October 27. Karlheinz
of the University of Connecticut,
Storrs, CT (Visiting Lecturer at Vanderbilt University).
Non-Uniform Sampling of Band-Limited Functions. For
numerical algorithms a finite-dimensional model with
trigonometric polynomials turns out to be efficient. In the
limit the infinite-dimensional theory of non-uniform sampling of
band-limited functions can be recovered. It is shown how the
numerical practice can be reconciled with the sampling theory of
band-limited functions. As an additional benefit new theorems
for the approximation of entire functions from finitely many
samples are obtained.


October 22. Professor Archil
of Ohio University,
Athens, OH. On the Heat Equation with a Singular
We study the Sobolev smoothing properties of
Schrodinger semigroups, corresponding to the heat equation with a
Kato class potential. The local smoothing results are sharp and
they have applications to the smoothness problem for the
Schrodinger eigenfunctions. The techniques of proof of the
smoothing theorems involve a new version of the Leibniz rule for
fractional derivatives as well as a new characterization of the
Kato class.


October 13. Chris
of Georgia
Institute of Technology
. Modulation Spaces and Trace-Class
Modulation spaces are function spaces that
quantify the joint localization in both time and frequency of
functions or distributions. We will discuss importance and
nature of time-frequency localization. We will show how
operators mapping the Hilbert space
L2(Rd) into itself can be
realized by time-frequency superpositions — this is the classic
“pseudodifferential operator calculus.” Each pseudodifferential
operator is determined by a symbol function that is defined on
R{2}. The time-frequency characteristics of
the symbol translate into spectral properties of the operator.
In particular, we present some new results giving sufficient
conditions on the time and frequency decay of a symbol that imply
that the corresponding pseudodifferential operator is
trace-class, Schatten-class, or bound on


October 8. Gabriele Gühring, of the University of
, Germany. A principle of linearized
stability for a semilinear nonautonomous delay equation.
On a
Banach space X we study the abstract delay equation

(RSNP)   dw(t)/dt = Bw(t)+K(t,wt),  t>s>0,
ws = g
The operator
B is assumed to be the generator of a
C0-semigroup on X, and K is a
nonlinear function mapping
R+ × C([-1,0],X)
into a Banach space F larger than X. Assuming that
K is differentiable with respect to the second variable,
we derive exponential stability of a solution of (RSNP) from
exponential stability of the zero solution of the linearized
equation. The result will be applied to a partial differential
delay equation.


September 21. Professor Abdelaziz Rhandi of the University of
Marrakesh. On a functional analytic approach for transition
semigroups on L2.
By using only analytic tools we
prove positivity of the transition semigroup associated formally
with the stochastic differential equation

DX(t) = (AX(t)+F(X(t)))dt+Q1/2dW(t),
   X(0)=x,    t > 0,
in the case
where FUCB(H,H). As a consequence we
obtain the existence of an invariant measure of the above
stochastic equation.


September 17. Professor
Jaroslav Jezek
, of Charles University. Big
A finite lattice is said to be big if it is a
maximal sublattice of an infinite lattice. In a joint work with
J.B.Nation and R.Freese, we find 145 minimal big lattices and
prove that a finite lattice is big if and only if it contains one
of them as a sublattice. Outlines of the proof will be given.

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