Colloquia, Fall 1999
Guoliang Yu (University of Colorado at Boulder).
Large scale geometry of groups and topology of manifolds.
In the last two decades the subject of large scale geometry
has come to the fore of several areas of mathematics. The idea behind
the subject is very simple: ignore the small scale irregularities of
geometric spaces and concentrate on their larger and hopefully more
fundamental features. The tools of large scale geometry have found
applications to ordinary, small scale geometry and topology of manifolds.
In this talk I will give an introduction to some of the ideas and
problems in this area. I will try to make my talk accessible to graduate
University of Virginia.
Shape Optimization, Shells, and Applications.
In many industrial processes the performance of the system highly depends
on the geometric configuration, because of different physical phenomena.
Given a PDE defined in a bounded domain, one is interested to know what is
the perturbation of the solution of that PDE under small perturbations of
the domain. The shape derivative turns out to be an interesting tool for
this. We shall recall the the basic methods concerning the shape derivative
and also explain why the situation is more complicated in the hyperbolic
case and how we can recover some results by using the hidden regularity.
A great increase in the development of flexible structures has taken
place within the space and aeronautic industries during the last
decade, with the emergence of micro-satellites. In response to their
specific constraints, the engineers have proposed a concept known as
passive control. We shall apply the shape optimization techniques
described previously to one example: the Carpentier’s joint, which is
a shell with some geometric properties.
There are parts of this work that joint with Jean-Paul Zolesio (CNRS, France)
and parts with Jean-Paul Marmorat (Ecole des Mines de Paris, France).
The automorphism groups of relatively free groups and algebras.
We consider countably infinite groups and algebras from the title.
There exists a natural topology on the automorphism group G in this case
defined by stabilizers of finite subsets. We get a structure of Polish
space on G. We can say about probability 1 or 0 for some subsets of G.
We prove that “most” n-tuples of elements of G freely generate free
subgroups, and that there are “huge” free subgroups of G. We study
the so called small index property for G and establish this property
in some interesting cases. A number of algebraic properties of G follow
from such a property.
Mimmo Iannelli, of the
Dipartimento di Mathematica, Universit� di Trento,
The optimal control of an age-structured epidemic model.
Epidemics models of SIS type are used to describe those diseases that do
not impart immunity and may become endemic.
Although in their O.D.E. version these models are very simple, they can,
however, explain the major features of this class of epidemics. Moreover,
when age structure is taken into account, these models can show complex
dynamics, including existence of periodic solutions arising by bifurcation
from a stable steady state.
After introducing the age structured SIS models I will present a method to
approach the problem of finding an optimal strategy for controlling the
Xiaoman Chen, of the
University of Colorado at Boulder
Smooth Algebras of Groupoid C*-Algebras.
The problem of studying topological spaces is equivalent to the
problem of studying the commutative algebras of all continuous
functions on the topological spaces. For this reason
non-commutative algebras are considered as “non-commutative
spaces” in Alain Connes’ non-commutative geometry. Such
non-commutative spaces naturally arise in geometry, topology and
In this talk, we will discuss a class of non-commutative spaces
associated to groupoid. Groupoid can be considered as a notion of
symmetries. Interesting examples of groupoids include group action
and foliations. In particular we will discuss the “Smooth
structure” of such non-commutative spaces.
School of Mathematics,
Georgia Institute of
Tutte’s Edge 3-Coloring Conjecture.
Tutte conjectured in 1966 that every 2-connected cubic graph with no
minor isomorphic to the Peterson graph is edge 3-colorable. The
conjecture implies the Four Color Theorem by a result of Tait.
In the first part of the lecture, I will discuss related results and problems.
In the second part, I will outline a proof of Tutte’s conjecture obtained in
joint work with N. Robertson, D.P. Sanders and P.D. Seyour.
Sergei Ivanov, of the
University of Illinois at
Recognizing the 3-Sphere.
A modification of Rubinstein-Thompson’s algorithm
to recognize the 3-sphere will be discussed.
This modification is closely related to geometric
group presentations of the fundamental groups of 3-manifolds
and might be more available for nonspecialists in 3-dimensional
topology and for practical implementation that could
be used for disproving the Poincaré conjecture.
Department of Mathematics,
Equations and first order formulas in free and fully
residually free groups.
I will describe algebraic varieties over finitely generated fully
residually free groups and discuss the so-called implicit function
theorem for free groups. These are important tools in the solution
(joint with A. Myasnikov) of the Tarski problem for free groups.
University of Szeged, Hungary.
Approximation by Partial Sums of Fourier Series on the One-
and Two-Dimensional Torus.
We consider only 2p
periodic functions (in each variable). For classes of
functions with convergent Fourier series, the problem of estimating
the rate of convergence has always been of interest. The classical
Dirichlet-Jordan theorem assures the convergence everywhere of
the Fourier series of a function of bounded variation. The
classical Young test states the the conjugate series to the Fourier
series of a function f of bounded variation converges at a
point x if and only if the conjugate function
~f exists at x.
In this survey, we present a concise account of the quantitative versions
of the above mentioned classical results as well as of their
extensions from single to double Fourier series or conjugate series,
respectively. The estimates are stated in a greater generality, by
introducing rectangular oscillation of a function of two variables
over a rectangle. The notion of bounded variation for a function of
two variables is meant in the sense of Hardy and Krause.
of the Department of Mathematics,
Tennessee Tech University.
Unique continuation for solutions to systems of partial differential
Results on unique continuation for solutions to partial differential
are of importance in many areas of applied mathematics, in particular in
control theory and inverse problems. The unique continuation problem can
be formulated as follows. Considering a solution to a homogeneous partial
differential equation in a bounded domain with zero Cauchy data on the
boundary, can one conclude that the solution vanishes everywhere in the domain?
The classical results on unique continuation are Holmgren’s theorem and
Hörmander’s theorem. Holmgren’s theorem requires analytic coefficients of
the differential operator. That makes it impractical for many applications.
On the other hand, Hörmander’s theorem is valid only for scalar equations
and its results are optimal only for second order equations. Its proof is
based on a certain type of weighted energy estimate which was introduced by
Carleman. Recently, these estimates have led to uniqueness results for
solutions to higher order equations and to systems of partial differential
Applications often yield operators with time-independent coefficients. In
1995, D. Tataru proved a sharp uniqueness result for equations with time
independent coefficients using a new type of Carleman estimate. We will
show that Tataru’s result can be extended to some systems and higher order
equations and present new uniqueness results for the Kirchhoff plate equation
and a thermoelastic system.
of the Department of Statistics,
Mathematical Modeling of Evolution of Repeat-DNA:
Humans versus Chimps.
Microsatellite DNA is composed of tandem repeats of simple motifs of length
2-6 nucleotides. Because of its high mutation rate it provides a convenient
tool for timing of evolutionary events on the time-scale on the order of
100,000 generations. The split between human and chimp lineages has occurred
about 5 million years ago. Examination of the same microsatellite DNA
sequences in both species reveals differences that might be arguably explained
by differences in mutation rates, population demography and/or biased sampling.
We build a mathematical model of evolution of microsatellite DNA, taking into
account the processes mentioned above. The model involves a stochastic process
of coalescence, i.e., convergence of lineages of sampled chromosomes to a
common ancestor (an outline of mathematical problems and methods related to
coalescence will be presented in the talk). We apply the model to the known
data on human and chimp microsatellites and conclude that the only plausible
explanation seems to be a much higher mutation rate in humans compared to
University of Colorado, Boulder.
Volume growth and positive scalar curvature.
Gromov conjectures that certain large Riemannian manifolds
cannot have positive scalar curvature. This is a special case of
a general principle that a macroscopically large Riemannian manifold
cannot be microscopically small. In this talk, I will discuss how the
higher index theory of elliptic differential operators can be used to
prove this principle when the Riemannian manifold has subexponential
volume growth. This is joint work with G. Gong.
University of Colorado, Boulder.
The Novikov conjecture and geometry of groups.
A fundamental problem in the topology of high-dimensional manifolds is
the Novikov conjecture. Roughly speaking the Novikov conjecture states
that manifolds are rigid at a certain infinitesimal level. In this talk,
I will explain the Novikov conjecture, why it is interesting, and how it
is related to certain aspects of geometric group theory. I will try to
make this talk accessible to graduate students.
Bojan Mohar, of
University of Ljubljana.
From the Four Color Problem to the Graph Minor Theorem.
Topological graph theory has its roots in the Four Color Problem
and in the combinatorial topology. However, it was established as
a separate branch of Graph Theory only in the seventies after
Ringel and Youngs solved the Heawood Map Color Problem. The theory
about graphs on surfaces culminated in the late eighties with the
Robertson- Seymour theory on graph minors.
In the talk, the results mentioned above will be explained, some
of their consequences in graph theory, topology and theoretical
computer science will be outlined, and some directions of the
current research will be presented.