Colloquia, Spring 2004

Mathematics Colloquia, Spring 2004

Thursdays 4:10 pm in 1206 Stevenson Center, unless otherwise noted

Tea at 3:30 pm in 1425 Stevenson Center

February 5, 2004 

Mary Ann Horn, Vanderbilt University

Theoretical challenges arising from
the questions of controllability for elastic structures

In the context of control of elastic systems, the challenge of obtaining
rigorous theoretical results has given rise to the development of new
mathematical methods and expanded applications of known tools. This talk
will give an overview of the questions of controllability and
stabilizability for plates, cylindrical shells and three-dimensional
elasticity, as well as the issues arising when single components are
coupled into a more complex system. The need for and the utility of
high-powered mathematical tools such as microlocal analysis, Carleman
estimates and Riemannian geometry will be illustrated and examples of some
of the current challenges will be discussed.

Contact person: Guoliang Yu

February 12, 2004 

Kim Ruane, Tufts University

Groups with
specified boundary

The result I will speak about is in the spirit of two well-known
results. The first says if $G$ is a convergence group acting on the
then $G$ is virtually fuchsian (proved by Casson-Jungreis, Tukia-Gabai).
second result says if $G$ is a one-ended word hyperbolic group with
one-dimensional boundary, then that boundary is a circle, a Sierpinski
or a Menger curve (proved by Kapovich and Kleiner). If the boundary is
the first result applies to show $G$ is virtually fuchsian. If it’s a
Sierpinski carpet, then they have a theorem about the structure of the
In this case, it is conjectured that $G$ is a geometrically finite
group (obviously related to Cannon’s Conjecture concerning S^2 boundary).
Menger curve boundary is the “generic” case for a word hyperbolic group
one-dimensional boundary and a characterization of these groups is far

I wish to consider the following related general question. Suppose $G$
geometrically on a $\cat(0)$ space $X$ whose boundary is homeomorphic to
space $Y$. Given a particular space $Y$, what can you say about $X$ and
For instance, if $Y$ is a circle, then I will show that $X$ is either the
Euclidean plane and $G$ is a Bieberbach group or $X$ is the hyperbolic
plane in
which case the above result applies to say that $G$ is virtually fuchsian.
case where $Y$ is the circle is not difficult to prove, but the techniques
already show something very interesting. Next, I consider $Y$ the
of the Cantor set and show that $X$ is in fact “almost” isometric to the
product of a tree with the real line and $G$ is virtually the product of a
group and $\mathbb Z$. This case is already much more difficult than the
circle and again, the techniques used should hopefully yield some powerful

Contact person: Mike Mihalik

February 19, 2004 

Frank Quinn, Virginia Polytechnic Institute and
State University

What is a

Short names such as “manifold” or “group” tend to be attached to
central objects in an area, as identified by tradeoffs between generality
and properties. More general objects with significantly fewer properties,
such as “homology manifolds” or “semigroup”, and more special objects with
slightly more properties, such as “real analytic manifold” or “algebraic
group” are thought of as perturbations on the central object. However what
we see as the “central object” often changes as we learn more. In this
lecture I’ll trace changes in the meaning of “manifold” as it tracked
development of the subject over the last century. We may be on the verge
another major change of viewpoint, though whether the word “manifold” will
follow it this time remains to be seen.

Contact person: Bruce Hughes

February 25, 2004 

Mikhail Perepelitsa, Northwestern University

Note: Wednesday, 4:10 pm in SC 1308

Existence of solutions of two
compressible Navier-Stokes equations with uniform in time
point-wise bounds on density

We consider the two dimensional Navier-Stokes equations that
the motion of compressible, viscous, barotropic flow. The
bulk viscosity coefficient is assumed to be a function of the
of the flow that grows a least as fast as the equilibrium pressure.
prove that given periodic initial data in which the density is
and does not contain vacuum, there exists global in time weak
and the density is uniformly bounded and does not contain vacuum.

Contact person: Gieri Simonett

February 26, 2004 

Jennifer Pietenpol, School of Medicine, Vanderbilt

p53 and
Mechanisms of Cell Cycle Checkpoints

Although most cells in an adult human are quiescent or in a
non-proliferative state, specialized cells such as those of the
hematopoietic system, those that generate skin, or those that line the
gastrointestinal tract, maintain proliferation. On average, about 2
trillion cell divisions occur in an adult human every 24 hours. It is
critically important that various cell types divide at a rate sufficient
to produce the needed cells for growth and replacement. However, if any
given cell type divides more rapidly than is necessary, the normal
organization and functions of the organism will be disrupted as
specialized tissues are invaded and interfered with by the rapidly
dividing cells. Such is the course of events in cancer. Over the past
two decades, unraveling the basic molecular events controlling eukaryotic
cell cycle transitions has been an area of intense research pursuit.
Studies in a variety of organisms have identified an evolutionarily
conserved signal transduction system for controling cell cycle transitions
through regulation of the activity of key enzymes called cyclin-dependent
kinases. Further, many investigations have focused on how the signaling
pathways that mediate the cell cycle transitions are regulated and
modified after cellular stresses. Human cells are continuously exposed to
external agents (e.g., reactive chemicals and UV light) as well as
internal agents (e.g. byproducts of normal intracellular metabolism such
as reactive oxygen intermediates) that can induce cell stress. Eukaryotic
cells have evolved cell cycle machinery with a series of surveillance
pathways termed cell cycle checkpoints to ensure that cells copy and
divide their genomes with high fidelity during each replication cycle.
Cell cycle arrest after DNA damage is critical for maintenance of genomic
integrity and loss of normal cell cycle checkpoint signaling is a hallmark
of tumor cells. The ability to manipulate cell cycle checkpoint signaling
also has important clinical implications, as modulation of the checkpoints
in human tumor cells may enhance cellular sensitivity to chemotherapeutic
regimens that induce DNA damage. This presentation will focus on the
mechanics of the cell cycle as well as checkpoint signaling pathways, with
particular focus on the tumor suppressor p53, and how this knowledge may
lead to more efficient use of current anticancer therapies and the
development of novel agents. The hope is that from a detailed
understanding of these processes, more incisive approaches to cancer
treatment will evolve that exploit the molecular defects in cancer cells.

Contact person: Guoliang Yu

March 4, 2004 

Ralph Mckenzie, Vanderbilt University
Valeriote, McMaster University
Pawel M. Idziak, Jagiellonian

complexity in algebra

The generative complexity of a class $K$ of algebras is the function $G_K$
defined for positive integers $n$ by setting $G_K(n)$ equal to the number
of non-isomorphic $n$-generated algebras in $K$. We have studied $G_K$
mainly for $K$ a locally finite variety (equationally defined class of
algebras). Besides making some easy observations about the ways in which
the behaviour of $G_K$ imposes algebraic and structural constraints on
$K$, we shall outline the proof of a three-year-old result of P. Idziak,
R. McKenzie and M. Valeriote giving a precise determination of all the
locally finite varieties $K$ for which $G_K(n)$ is bounded by some
polynomial function of $n$. In the category of varieties, the directly
indecomposable locally finite varieties with polynomially many models are
the varieties of modules over finite rings of finite representation type,
and the finite matrix powers of varieties of $M$-sets with (possibly) some
constants, where $M$ is any finite group.

Contact person: Ralph Mckenzie

March 18, 2004 

Stefan Richtner, University of Tennessee,

Analytic contractions and
nontangential limits

Let f be a complex-valued holomorphic function in the open unit disc D of
the complex plane. f is said to have nontangential limit L at z, |z|=1, if
there is a fixed angle A inside D with vertex at z such that the limit of
f(z) from within A is L. A theorem of Fatou’s implies that every function
f in the Hardy space H2 has nontangential limits at a.e. point of the unit
circle, and a theorem of Beurling’s exploits these boundary values to give
precise information about the invariant subspace structure of that the
linear operator Mz, (Mz)f(z) = zf(z) on H2. If H is a Hilbert space of
analytic functions that properly contains H2, then there may be a set of
positive measure inside the unit circle such that all functions in H have
nontangential limits a.e. on this set.

In this talk I will discuss efforts to describe conditions for the
existence of such a set in terms of the norm on H and the relevance of
for the invariant subspace structure of Mz when Mz acts on H.

Contact person: Dechao Zheng

March 25, 2004 

Roger Smith, Texas A&M University

of von Neumann Algebras

In the 1940’s, Hochschild introduced cohomology groups for
algebras, and these were adapted by Kadison and Ringrose
in the 1970’s to the functional analytic setting of von Neumann
algebras, the weakly closed self-adjoint subalgebras of bounded
operators on a Hilbert space. These groups $H^n(M,X)$ are defined
in terms of a module $X$ over the algebra $M$ and can be used as
isomorphism invariants. When $X=M$, they also act as obstruction groups
whose vanishing gives structural information about the algebras.
For example, the statement that $H^1(M,M)=0$ is, in different language,
a celebrated theorem of Kadison and Sakai that derivations of
a von Neumann algebra are always implemented by elements of the algebra.
In their original work, Kadison and Ringrose conjectured that
$H^n(M,M)$ should always vanish, and proved this in a number of cases.
Further progress had to await the recently developed theory of completely
bounded maps. In this talk we will survey the current state of affairs and
describe some of our own work on this topic. The presentation will be
aimed at a general audience with very little background assumed.

Contact person: Dietmar Bisch

April 1, 2004 

Burt Ovrut, University of Pennsyvania

[Joint colloquium with Department of Physics]

NOTE: Thursday, 4:00 pm in SC
Tea at 3:30 pm in SC 6333

Worlds and Geometry

The fundamental concepts underlying superstrings and M-theory will be
reviewed. The compactification of M-theory to the observed
four-dimensional spacetime will be discussed, leading to the idea that our
universe is a “brane” floating in a higher dimensional space. To obtain
realistic particle physics on the brane, one must have the appropriate
geometric and vector bundle properties on the background compact manifold.
These provide an important link between modern mathematics and high energy

Contact person: Tom Kephart


Colloquium Chair (Spring 2004): Guoliang Yu

Back Home