# Vanderbilt MathematicsColloquiaSpring 2002

Colloquia are listed in reverse chronological order. The
top of the list is subject to change, since more colloquia
are still being planned.
All colloquia are held at 4:10p.m.
in 1431 Stevenson Center unless otherwise noted.

Our colloquia, as well as our
seminars and other activities, feature
speakers not only from our own department but also from other
departments all over the world.
For further information on activities in the department,
you may also consult our
weekly
calendar
and
past
calendars
.

Friday, May 24th, 12:45p.m., SC 1320.
A. M. W. Glass, of the University of Cambridge.
Catalan’s Conjecture.
In 1844, Catalan asked if the only two consecutive whole numbers that
are proper powers are 8 & 9. In 1976, using the Fields Medal work of
Alan Baker, Tijdeman proved that the number of solutions to
xp-yq=1 was indeed finite, and subsequent
work has put severe restrictions on their size. In 1996, using
Stickleberger elements, Preda Mihailescu improved prior algebraic work
of Inkeri et al to show that two strong congruences must hold, whence
transcendence theory gives that m=min{p,q} >
105
and M=max{p,q} < m2 (so M≠ 1
(mod m)). By considering subgroups and quotients of groups of
units in the cyclotomic field for Mth roots of unity,
Mihailescu has a most ingeniously scheme to show that
M>m2.
I will sketch the background and Mihailescu’s anticipated solution.
(Hosts: Matt Gould and Peter Jipsen.)

Thursday, April 25th.
Raul Curto, of the
University of Iowa.
Truncated moment problems: A survey of recent results.
Let g(2n):
g00,
g01,
g10,…,
g0,2n,…,
g2n,0
be a given set of complex numbers, with
g00 > 0
and gji
=
gij
for all i,j. The truncated complex moment problem entails
finding necessary and sufficient conditions for the existence
of a positive Borel measure m,
supported in the complex plane, such that
gij=

zizj
dm  (0
i+j
2n). We first
describe briefly some classical approaches to (full and
truncated) moment problems, in one or several variables.
We discuss the Hamburger, Stieltjes, Hausdorff,
and Toeplitz MP, and the work of Riesz, Haviland, Fuglede,
and others. We then present a new operator-theoretic approach,
based on matrix positivity and extension, centered around
Smul’jan’s criterion for positivity of 2×2-operator
matrices. In this approach, the structure of an associated
moment matrix M(n)
M(n)[g] plays a
fundamental role. For instance, when M(n) is flat
(meaning that rank M(n) = rank M(n
1)), then the truncated moment problem always admits
a unique representing measure m
[g], which has precisely rank M(n)
atoms. Our techniques allow for a concrete description of
the support and densities of m
[g].

There is a close connection between the existence of
representing measures supported in a prescribed algebraic
variety and the presence of corresponding dependence relations
in the columns of the moment matrix M(n). If
g
representing measure m,
then M(n) is positive, recursively generated, and
card V(g)
rankM(n), where
V(g) is the variety associated
to g. We show how to solve the
moment problem for ZZ=A1+BZ+C
Z+DZ2, D
0, where positivity and
recursiveness are not sufficient for a representing measure.

For the quadratic MP (n=1) and singular quartic MP
(n=2 and detM(2)=0), a complete description of necessary
and sufficient conditions for the existence of representing
measures can be formulated concretely in terms of the
initial data. For the singular quartic MP, we show that rank
M(2)-atomic measures exist in case the moment problem is
subordinate to an ellipse, a parabola, or a non-degenerate
hyperbola, but the minimal measures for certain degenerate
hyperbola problems may require more than rank M(2) atoms.
Finally, we present applications to the classical
(Host: Dechao Zheng.)

Monday, April 22th.

Université d’Aix Marseille II
and
Vanderbilt University.
On the Baum-Connes conjecture.
An important object of study in representation theory of locally compact
groups is the reduced C*-algebra of a group. It contains all information
about the irreducible unitary representations weakly contained in the
regular representation.

The Baum-Connes conjecture, first stated about twenty years ago,
proposes a way to calculate the K-theory of the reduced C*-algebra of
a group G by mainly topological methods. More precisely, the conjecture
asserts that this K-theory group is isomorphic to the K-homology of the
classifying space for proper actions of G.

By now the conjecture has already been proved for large classes of
groups: Lie groups, reductive p-adic groups, amenable groups, etc.
I will discuss in this talk the statement of the conjecture and also
examples and methods of proof in some known cases of the conjecture.
(Host: Guoliang Yu.)

Friday, April 19th.
Jerry Kaminker, of the
IUPUI.
Noncommutative geometry and solid state physics.
Noncommutative geometry, as introduced by Alain Connes, has
found several applications in physics. One of the most successful has
been in solid state physics. This has largely been due to the program
developed by Jean Bellissard. Recently one of the main conjectures in
the area, the “gap labeling conjecture”, was resolved by three different
groups of workers. This talk will start with an introduction to some
aspects of noncommutative geometry, explaining its motivation and basic
applications. The way it fits very well with the physics of solids will
then be discussed and a sketch of the proof of the gap labeling
conjecture presented. The latter is joint work with Ian Putnam.
(Host: Guoliang Yu.)

Thursday, April 18th.
M. Zuhair Nashed, of the
University of Delaware.
Variational inequalities, nonsmooth calculus, and Newton-like
methods for ill-posed problems: Un ménage à trois.

Newton’s method is one of the most widely used algorithms for finding
approximate solutions of nonlinear operator equations F(x) = 0.
The method and the (Kantorovich) theory for its convergence require
the existence and bounded invertibility of the Fréchet derivative
of the operator F. The goal of this talk is to describe a theory
for Newton-like methods when the derivative does not exist or when
the derivative has no bounded inverse or bounded generalized inverse.
Along the way we visit variational inequalities and discuss their
role in minimization of nonsmooth functionals. We also introduce a
new concept of “differentiability” for nonsmooth operators and use it
to formulate a new Newton-like method. Finally, we give applications
to bounded-variation regularization and nonsmooth ill-posed problems.
(Host: Akram Aldroubi.)

Tuesday, April 16th.
Yuri Muranov of Vitebsk State Technological University,
Belarus.
Splitting of homotopy equivalence along submanifolds.
Let X be a submanifold of a manifold Y. A homotopy
equivalence f:M –> Y splits along the submanifold X
if f is homotopic to a map g which is transversal to
X, and the restrictions of g to a transversal preimage
of X and to its complement are homotopy equivalences. If a
map f is homotopic to a homeomorphism then, obviously,
this map splits along any submanifold. The corresponding obstruction
groups were introduced by Wall.
The splitting methods are effectively applied for computation of maps
in surgery exact sequence and for solution of the oozing problem
(the problem of realizing elements of Wall groups by normal maps of
closed manifolds). We describe geometrical and algebraic aspects of the
splitting problem and relations of this problem with surgery theory.
(Host: John Ratcliffe.)

Tuesday, April 9th.
Stephen Smale, of

University of California at Berkeley
.
Evolution of language.

A mathematical model is presented which helps to understand how
languages are formed. A theorem in this setting is the convergence
to a common language under a hypothesis on linguistic encounters.
(Host: Akram Aldroubi.)

Thursday, April 4th.

Laurent Pujo-Menjouet
, of
McGill University
and the Université de Pau.
Asymptotic behavior of a singular transport equation modeling
cell division.

This paper analyses the behavior of the solutions of a model of cells
that are capable of simultaneous proliferation and maturation.
This model is described by a first-order singular partial differential
system with a retardation of the maturation and a time delay. Both
delays are due to cell replication. We prove that uniqueness and
asymptotic behavior of solutions depend only on cells with small
maturity (stem cells).
(Host: Glenn Webb.)

Thursday, March 21st.
Martin Kochol, of the Slovak Academy of Sciences, Bratislava,
and the Georgia Institute of
Technology
.

Superposition — a method for constructing graphs without
nowhere-zero flows.

Nowhere-zero flow problems in graphs are dual to graph coloring
because, by Tutte, a planar graph is k-colorable iff its dual has a
nowhere-zero k-flow (its edges can be oriented and assigned values
1,…,k so that for every vertex, the sum of the values of the incoming
edges equals the sum of the outcoming ones). Graphs without
nowhere-zero k-flows are called k-snarks. In particular, snarks are
nontrivial cubic 4-snarks (by nontrivial we mean cyclically
4-edge-connected and with girth at least 5). Snarks present an
important family of graphs, because many conjectures about graphs can
be reduced on them. Among the most interesting belong the 5-flow
conjecture of Tutte (every bridgeless graph has a nowhere-zero 5-flow)
and the cycle double cover conjecture (every bridgeless graph has a
family of circuits containing each edge twice).

We present a method for constructing graphs without nowhere-zero
k-flows. Using this method we obtain several results regarding
nowhere-zero flows. Primarily we construct new families of snarks. The
most interesting is the construction of snarks with arbitrary large
girth, which disproves a conjecture of Jaeger and Swart that every
snark has girth at most 6. (Note that if this conjecture would be true,
it would imply the 5-flow and cycle double cover conjectures). We also
present new results about the 3-flow conjecture of Tutte (every graph
without 1- and 3-edge cuts has a nowhere-zero 3-flow) and show that
this is equivalent with seemingly stronger or weaker statements.
(Host: Mark Ellingham.)

Monday, March 18th.

Dietmar Bisch, of
the

University of California, Santa Barbara
.
Entanglement – the spooky action at a distance.

Entanglement is a feature of quantum mechanics, which does
not exist in classical physics. It expresses a correlation of two
subsystems of a quantum physical system which appears naturally as
soon as the commutative algebras of functions in classical physics
are replaced by non-commutative algebras of operators (matrices) in
quantum physics. Einstein called this phenomenum “spooky action at
a distance”. Entanglement is believed to be related to what speeds
up a quantum computer and is currently the subject of intense study in
quantum information science.

I will discuss entanglement and show how it can be used to transmit
quantum information on a classical channel (“quantum teleportation”).
If time permits I will present some of the proposals of how to quantify
entanglement, most of which have the flavor of entropy-like quantities
in operator algebras.
(Host: Guoliang Yu.)

Friday, March 15th.
Alexei Myasnikov of CUNY, New York.
The Andrews-Curtis conjecture and black box groups.
If G is a group and V_k(G) is the set of all
ktuples of elements in G which generate G
as a normal subgroup, then the Andrews-Curtis graph
Delta_k(G) of G is the set V_k(G) (as
the set of its vertices) in which two elements are connected by
an edge iff one of them can be obtained from another by an
elementary transformation (Nielsen transformations and conjugation
of one of the components). These objects appear naturally in the
Andrews-Curtis conjecture in algebraic topology and in the theory
of black box groups in probabilistic group theory.

In my talk I am going to discuss some recent results on these
subjects based on study of Andrews-Curtis graphs of various
groups.
(Host: Mark Sapir.)

Thursday, March 14th. (Biomathematics Colloquium)

Mount Sinai School of Medicine.
Computational analysis of a biological signaling network.
Signaling networks receive and process information to control the
function of cellular machines. The MAP-kinase 1,2/protein kinase C
system is one such network that regulates many cellular machines,
including the cell cycle machinery and autocrine/paracrine factor
synthesizing machinery. We used a combination of computational analysis
and experiments in NIH-3T3 fibroblasts to understand some of the design
principles of this controller network. We find that the growth factor
stimulated MAP-kinase 1,2/protein kinase C network can operate as both a
monostable as well as a bistable system. At low concentrations of
MAP-kinase phosphatase the system exhibits bistable behavior, such that
brief stimulus results in sustained MAP-kinase activation. The
MAP-kinase induced increase in the levels of MAP-kinase phosphatase
moves the network to a monostable state, where it behaves as
proportional response system responding acutely to stimulus, but
incapable of sustained responses. Thus the MAP-kinase1, 2/protein
kinase C controller network is flexibly designed and MAP-kinase
phosphatase is the locus of flexibility.
(Host: Emmanuele DiBenedetto.)

Tuesday, March 12th.

Laurent Baratchart
of
INRIA Sophia-Antipolis.

Some extremal problems arising in frequency identification and
deconvolution for recovering Hardy functions from incomplete
boundary values.

We consider the problem of L2 or
L¥
approximating a function on a subarc of the unit circle (or on a
subinterval of the imaginary axis) by the trace of a Hardy function
satisfying pointwise or norm constraints on the remaining of the
circle (or the imaginary axis). This generalization of Carleman’s
recovery problem exhibits connections with Hankel and Toeplitz
operators, whose spectral theory allows one to derive error
estimates as well as explicit computational
schemes. These can be used in several practical situations of
engineering science.
(Host: Ed Saff.)

Thursday, March 7th.
Joachim Cuntz of the
Mathematisches Institut, Universitaet Muenster.

K-homology for the Heisenberg commutation relations.
A very prominent quantum space is represented by the so called
Weyl algebra which is generated by two elements satisfying the
Heisenberg commutation relations. Until recently it was not
known how to define and compute some of the standard invariants
of noncommutative geometry for this space. We describe a new
theory which does exactly that.
(Host: Guoliang Yu.)

Friday, March 1st,
3:10-4:00p.m.,
1307 Stevenson Center

Konstantin Rybnikov of
Cornell University.

Gain graphs and their applications to convex polyhedra and splines.
A gain graph (G,h,H) is a homomorphism h from the free
group on the edges of a graph G to some group H; it
is called balanced if all closed walks of G lie in
the kernel of h. I’ll explain a test, formulated in terms
of the binary cycle space of G, that can be used to
detect if (G,h,H) is balanced for some choices of H,
e.g. the abelian case.

I am going to show a few examples of gain graphs
arising from discrete geometry. In the 1860s Maxwell
described a relationship between equilibrium stresses
in a plane framework and polyhedral surfaces projected
on this framework. In the most simple case, Maxwell’s
correspondence can also be interpreted in terms of
lifting a tiling of the plane to a spatial surface.
Lifting a tiling of Rd to a convex surface, tangent
to a paraboloid, appears to be a powerful technique
in geometry of numbers (Voronoi, 1908) and
computational geometry (Brown 1978, Edelsbrunner
1986).

I’ll present various criteria for a tiling of Rd or,
more generally, a PL-manifold in Rd, to be the vertical
projection of a convex d-surface. These criteria lead
to improvements of algorithms determining whether a
given tiling can be regarded as the projection of a PL-surface.
Gain graphs can also be used to obtain some topological
results on the dimension of the space of splines over
a non-simplicial tiling of a domain in Rd.

Some of the discussed results are
joint with S. Ryshkov and T. Zaslavsky.
(Host: Paul Edelman.)

Thursday, February 28th.
Zlil Sela, of the

Hebrew University of Jerusalem
.
Diophantine geometry over groups and the elementary theory of
a free group.

We study sets of solutions to equations over a free group,
projections of such sets, and the structure of elementary sets defined
over a free group. The structure theory we obtain enable us to answer
some questions of A. Tarski’s, and classify those finitely generated
groups that are elementary equivalent to a free group. Connections
with low dimensional topology, and a generalization to general
hyperbolic groups and their elementary classification will also be
discussed.
(Host: Mark Sapir.)

Thursday, February 21st.

, of

SUNY-New Paltz
.
Universal varieties of algebras.

For a variety of algebras (equational class), the notions
of universal in the categorical sense and universal in
the quasivariety sense will be considered.

The two notions will be compared and their relationship
illustrated by different examples, including varieties of bounded
lattices.
(Host: Matthew Gould.)

Wednesday, February 20th.

Amos Ron
, of the

.
Wavelet frames: The power of redundant representation.
One of the hallmarks of the IT revolution is the rapid increase in
connectivity and data acquisition capabilities. Questions concerning the
effective processing of this avalanche of data is becoming a top
national priority, fueled even further by the recent increase in
military and security needs.

Wavelets are widely considered to be among the most successful
contributions of the mathematical community to the theory and
applications of data processing. Most of the progress during the 1990s
was confined to non-redundant wavelet systems, primarily because their
theory was developed first.

In the last 5-6 years, a theory for wavelet frames (which are a major
type of redundant wavelet systems) was established. The theory leads to
effective constructions of finely-tuned wavelet frames, together with
fast implementation algorithms, opening thereby the door to a range of
possible applications.

The talk will be devoted to a review of this exciting development.
After highlighting the main ingredients of the theory, I will show
examples of an on-going research on applications, and will conclude
with a demo of the Wavelet IDR Framenet, a web-based
interactive software, currently under development, too.
(Host: Larry Schumaker.)

Tuesday, February 19th.

László Lipták
, of the

University of Waterloo
.
Stable set problem and the lift-and-project ranks of graphs.

We study the lift-and-project procedures for solving combinatorial
optimization problems, as described by Lovász and Schrijver, in the
context of the stable set problem on graphs. We investigate how the
procedures’ performance changes as we apply fundamental graph
operations. We give examples showing that adding, deleting, or
subdividing an edge can increase the N0– and
N-rank of a graph, and define two classes of
graphs when these and the subdivision of a star operation does not
increase the N0– and N-rank of the
underlying graph. We present a graph-minor based characterizations
of the rank of subdivisions of the complete graph Kn,
and define a class of graphs with large rank that
can be obtained from Kn using just the
stretching of a vertex operation. Finally, we provide improved bounds
for the N+-rank of graphs in terms of the number of
vertices in the graph and prove that the
subdivision of an edge or cloning a vertex operations can increase the
N+-rank of a graph.

This is joint work with Levent Tuncel.
(Host: Paul Edelman.)

Thursday, February 14th,
3:10-4:00p.m.,
1312 Stevenson Center

Dietmar Bisch, of the

University of California, Santa Barbara
.
Subfactors and symmetry.

John von Neumann discovered in the 30’s that certain algebras
of bounded operators on a Hilbert space are the natural algebras of
symmetries of quantum physical systems. These
von Neumann algebras
as they are now called can be viewed as non-commutative measure spaces
which feature many astonishing mathematical structures and have led to
rich theories, largely due to Connes, Jones and Voiculescu.

Vaughan Jones initiated in the early 80’s the
theory of subfactors,
a theory which deals with certain highly non-commutative, infinite
dimensional probability spaces. These subfactors turn out to display
a surprising rigidity phenomenon, which ultimately led Jones to the
discovery of his famous knot invariant, the Jones polynomial. A
subfactor is a functional analytical object that captures what one
might call the
generalized symmetries of the mathematical or physical situation
from which it was constructed. Analytical techniques can then be used
to decode this information and to compute with it. Surprising connections
to statistical mechanics and knot theory appear naturally.

I will present in my talk some of the basic ideas and concepts in
subfactor theory and will discuss some applications if time allows.
No prior knowledge of operator algebras is required for this talk.
(Host: Glenn Webb.)

Monday, February 11th,
2:10-3:00p.m.,
1214 Stevenson Center

Florian Pfender
, of

Emory University
.
Pancyclicity of 3-connected graphs: Pairs of forbidden subgraphs.

We say that G is {H1,…Hl}-free,
if it contains no induced copies of any of the graphs
H1,…Hl. The problem of characterizing all
families of H1,…Hl such that each “sufficiently
connected” {H1,…Hl}-free graph has some
Hamiltonian property has been studied by a number of authors.
In particular, the family of all pairs of graphs X, Y,
such that each 2-connected {X,Y}-free graph G\neq Cn
on n\geq 10 vertices is pancyclic, has been characterized by Faudree
and Gould. In this talk, I will characterize all graphs
X, Y, such that each 3-connected {X, Y}-free
graph is pancyclic.

This is joint work with R. Gould and T. Luczak.
(Host: Mark Ellingham.)

Thursday, February 7th.

Gil Strang
, of the

Massachusetts Institute of Technology
.
Filtering and signal processing.

We discuss two filters that are frequently used to smooth data.
One is the (nonlinear) median filter, that chooses the median
of the sample values in the sliding window. This deals effectively
with “outliers” that are beyond the correct sample range, and will
never be chosen as the median. A straightforward implementation of
the filter is expensive, particularly in two dimensions (for images).

The second filter is linear, and known as “Savitzky-Golay”. It is
frequently used in spectroscopy, to locate positions and peaks and
widths of spectral lines. This filter is based on a least-squares fit
of the samples in the sliding window to a polynomial of relatively
low degree. The filter coefficients are unlike the equiripple filter
that is optimal in the maximum norm, and the “maxflat” filters that
are central in wavelet constructions.

We will discuss the analysis and the implementation of both filters.
(Host: Doug Hardin.)

Thursday, January 31st.

Gui-Qiang Chen
, of
Northwestern University.
On nonlinear degenerate parabolic-hyperbolic equations.

In this talk we will discuss a well-posedness theory for solutions
in L1 to the Cauchy problem of general degenerate
parabolic-hyperbolic equations with non-isotropic nonlinearity. A
notion of kinetic solutions and a corresponding kinetic formulation
will be introduced. The notion of kinetic solutions applies to more
general situations than that of entropy solutions; and its advantage
is that the kinetic equations in the kinetic formulation are well
defined even when the macroscopic fluxes are not locally integrable,
so that L1 is a natural space on which the kinetic
solutions are posed. It includes a new ingredient, a chain rule type
condition, which makes it different from the isotropic case. Based
on this notion, we will present an effective approach to prove the
contraction property of kinetic solutions in L1,
especially including entropy solutions, among others.
(Host: Glenn Webb.)

Wednesday, January 23rd.
Eric Schechter, of

Vanderbilt University
.

“If it is raining right now then the moon is round.” That’s logical
to a mathematician, but nonsense to everyone else. Classical logic,
when presented by itself, lacks the relevance, causality,
constructivism, quantitativeness and other features that people
(even mathematicians!) are accustomed to using in their everyday
nonmathematical reasoning. Also, a single example (i.e., classical
logic) does not provide adequate illustrations for an abstract idea
(e.g., completeness). These may be some of the reasons that the
undergraduate introduction to mathematical logic. Consequently, I am
experimenting with a new curriculum that introduces several nonclassical
logics alongside the classical.

The math in this talk is not mine, nor is it new — it can be found in
research-level monographs and journals from the last few decades. What
is new is the attempt to convey this material at a much more elementary
level.

This talk is intended not just for logicians, but for general
mathematicians (including graduate students), plus a few folks from the
philosophy department who have expressed interest. Basics of logic
are sketched in the talk, so they’re not a prerequisite for attending.
The nonclassical examples may be amusing if you haven’t seen them before.

Monday, January 21st.

Vilmos Totik
, of the
University of Szeged and the
University of South Florida.

Polynomial inverse images and how to transfer results from one
interval to more general sets.

I proposed the following problem on the 1991 Schweitzer contest:
To divide an inheritance, n brothers hire an impartial judge.
Secretly however, each brother bribes the judge. The value of the
inheritance that a given brother gets strictly (and continuously)
increases in his own bribe and strictly decreases in everybody
elses bribe. Show that if the eldest brother does not give too
much to the judge, then the others can give so that the decision
will be fair.

In the talk a few other systems with similar characteristics will be
mentioned, and some basic properties of such monotone systems will
be discussed. In particular, equilibrium measures on sets of
finitely many intervals form such systems. A small modification of
the problem, in which each brother gets
a rational fraction of the inheritance no matter what the
initial judgement of the judge is, turns out to be the same problem
– when translated in the language of equilibrium measures –
as the density of polynomial inverse images of intervals among
sets consisting of finitely many intervals. This was proved in order
to transfer some polynomial inequalities from one interval to general
compact sets on R. In the talk some further applications of the density
theorem will be described that are of similar nature, namely they are
the extensions of results on compact sets that
have been known only on intervals.
(Host: Ed Saff.)

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