The Millican Lectures are made
possible through the generosity of Mr. Olin Moore Millican (1904 -
1999) who established the Roy McLeod Millican Memorial Fund (an
endowment) in honor of his brother.
We present a joint work
with L. Rempe in which we give a very elementary proof of previously
obtained results. Not only it simplifies considerably the original
proofs but our approach also applies to the more general class of
Ahlfors Island mappings.
Let D be a fundamental set in a a
Banach space X. Greedy algorithms provide an intuitively
appealing method for approximating a given vector by
a linear combination of vectors in D. The convergence of these
algorithms in Hilbert space is well understood. We consider some
natural generalizations of the Hilbert space algorithms to the Banach
space setting and examine their convergence properties with respect to
either the norm or the weak topologies.
Recruiting and Retaining PhD
Students in the Mathematical Sciences.
Langlands Functoriality and Howe
Correspondences: An Introduction and Three Examples.
The century-old empirical observation called Benford's Law (BL) states
that the significant digits (mantissas) of many real datasets are
logarithmically distributed (e.g., more than 30% of the leading decimal
digits will be 1, and fewer than 5% will be 9), rather than uniformly
distributed as might be expected. This talk will briefly review some of
the basic probabilistic theory underlying BL, mention recent empirical
evidence, and then focus on the surprising ubiquity of BL in classical
deterministic sequences and dynamical systems. For example, it has long
been known that the powers of 2, Fibonacci and Lucas numbers, and (n!)
all follow BL. Recent developments show that iterations of large
classes of common functions (including all polynomials, power,
exponential, and trigonometric functions, and compositions thereof),
geometric Brownian motion (hence many stock market models), and many
classical ODE's and numerical algorithms such as Newton's method,
all produce BL distributions.
Applications of these theoretical results to practical problems of
fraud detection, analysis of round-off errors in scientific
computations, and diagnostic tests for mathematical models will be
mentioned, as well as several open problems in dynamical systems,
probability, number theory, and differential equations. The talk will
be aimed for the non-specialist
_______________________________________________
Parallel Computation, the Data
Vortex and Dynamical Systems.
Coke Reed is a mathematician who has made substantial contributions to
the subject of dynamical systems. He received a PhD from the
University of Texas and was Professor at Auburn University.
At Auburn he spent summers at
IDA (Institute for Defense Analyses) at Princeton. Coke left
Auburn to become the first non administrative researcher at MCC in
Austin. He left MCC to become full-time at IDA. While at
IDA he began to develop ideas on computer design and eventually left
there to found Interactic, a very innovative company devoted to the
development of parallel computation. He has since moved
Interactic from Princeton to Austin. His work on parallel
computation is based on his ideas from dynamical systems. His
talk will start with an introduction to the problems of parallel
computation, particularly the growing disparity between processing
speed and communication time. It will end with a view of the near
future in parallel computation. His talk will be of general interest.
Nov. 16, 2007
Riemann's 1859 paper
Abstract: In 1859
Riemann's paper on what we now call the Riemann zeta function appeared.
I will report on the main ideas appearing in this famous work. It is
also the paper in which the Riemann hypothesis is first formulated.
March 30, 2007
Idris Assani
(University of North Carolina at Chapel Hill)
On A. Zygmund's
di®erentiation conjecture
Abstract
March 9, 2007
Pramod Achar (LSU in
Baton Rouge)
Springer correspondences for
dihedral groups
Let G be a reductive
algebraic group, and let W be its Weyl group. (For example, if G =
GL(n), then W is the symmetric group.) A recurring theme in
representation theory is the fact that many deep ideas and
sophisticated structures attached to G are accessible via fairly
elementary calculations in terms of W. Weyl groups themselves are
fairly well-understood---they are all crystallographic finite Coxeter
groups, which have been studied since at least the 1930's -- so this
means we can really "get our hands on" abstract things like perverse
sheaves on the unipotent variety of G.
Now, suppose we start with a
group W that's not a Weyl group of anything, but is close: perhaps a
non-crystallographic Coxeter group, or even a complex reflection group.
Many representation-theoretic calculations still make sense, and the
results have some shocking properties (various compatibility,
integrality, and positivity conditions that are all explained by G in
the Weyl group case). It looks as though we're studying the
representation theory and geometry of ``nonexistent'' algebraic groups!
I will discuss various results in this vein, in particular for the case
where W is a dihedral group. This is joint work with A.-M. Aubert.
Friday March 2, 2007
Tomi Karki (University of
Turku, Finland)
Word combinatorics with
similarity relations
Abstract: We
consider words, i.e., strings over a finite alphabet together with a
similarity relation induced by a compatibility relation on letters.
This notion generalizes that of partial words. Especially, we study
periodicity. We introduce three types of periods, namely global,
external and local relational periods, and we compare their interaction
properties by proving variants of the theorem of Fine and Wilf for
these periods.
Friday, Feb. 9, 2007
Zoltán Buczolich
(Eotvos University, Hungary)
Abstract
Feb. 7, 2007
Hiroki Sumi (University of
Osaka, Japan)
Title: The space of
postcritically bounded 2-generator polynomial semigroups with
hyperbolicity
Abstract:
Nov. 17, 2006
George Androulakis
(University of South Carolina)
The Ramsey result of Gowers on
partitions of block sequences of a Banach space.
Abstract: The classical
Ramsey theorem refers to partitions of the set of all subsets of the
integers with two elements. The corresponding Ramsey result for
partitions of all finite normalized block sequences of a Banach space
was proved by T.W. Gowers in 2002 and has a game theoretical
formulation. The original proof was very involved. I will present an
easy to understand proof of this result. This Ramsey result is very
important because it yields the well known Gowers' dichotomy which was
used to solve the famous homogeneous Banach space problem of S. Banach.
This is a joint work with S. Dilworth and N. Kalton.
Oct 27, 2006
Kathrin Bringmann (Univ.
of Wisconsin and Univ. of Minnesota)
Freeman Dyson's "Challenge
for the Future": The mock theta functions.
In his last letter to Hardy,
Ramanujan defined 17 peculiar functions which are now referred to as
his mock theta functions. Although these mysterious functions have been
investigated by many mathematicians over the years, many of their most
basic properties remain unknown. This inspired Freeman Dyson to proclaim
"The mock theta-functions give us
tantalizing hints of a grand synthesis still to be discovered. Somehow
it should be possible to build them into a coherent group-theoretical
structure, analogous to the structure of modular forms which Hecke
built around the old theta-functions of Jacobi. This remains a
challenge for the future."
Freeman Dyson
1987, Ramanujan Centenary
Conference
Here we announce a solution to
Dyson's "challenge for the future" by providing the "coherent
group-theoretical structure" that Dyson desired in his plenary address
at the 1987 Ramanujan Centenary Conference.
In joint work with Ken Ono, we
show that Ramanujan's mock theta functions, as well a natural
generalized infinite class of mock theta functions may be completed to
obtain Maass forms, a special class of modular forms. We then use these
results to prove theorems about Dyson's partition ranks.
In particular, we shall prove the
1966 Andrews-Dragonette Conjecture, whose history dates to Ramanujan's
last letter to Hardy, and we shall also prove that Dyson's ranks
`explain' Ramanujan's partition congruences in an unexpected way.
September 22, 2006
Anne Zdunik (University of
Warsaw)
Dynamics of meromorphic maps;
measures and dimensions.
Abstract: In last years there
have been a growing interest in the dynamics of meromorphic maps in the
complex plane. We shall discuss various geometric (dimensions) and
dynamical aspects of the structure of invariant sets and invariant
measures for a large class of entire and meromorphic mappings. Some
natural open questions will be also presented.
Thursday, July 20th
Ty Thompson-CO School of
Mines
Solution Estimates for the Ginzburg-Landau
Superconductivity Model on Thin Disks
April 21, 2006
V. S. Varadarajan (UCLA)
Symmetry and Supersymmetry
Abstract:
Supersymmetry is a discovery of the physicists in the 1970's which
extends the classical notion of symmetry to the world of elementary
particles and their fields. Its origins lie in the dichotomy of this
world of particles into bosons and fermions. Supersymmetries are
transformations between these two kinds of particles. Physicists
believe that any theory unifying all the fundamental forces will be
supersymmetric. Mathematically, supersymmetry involves a
generalization of geometry in which the local coordinates include the
usual commuting coordinates and additional anticommuting coordinates.
The automorphisms of such supermanifolds are supersymmetries. In
this talk I shall present an elementary account of the basic concepts
and attempt to link the mathematics with physics. Very little technical
background is needed for understanding the talk.
April 14, 2006
Professor Steven Hurder, University of
Illinois at Chicago
Title: Foliations - a
playground for topology and dynamics
Abstract: A
foliation is a way to partition a manifold into a regular system of
smaller components, the leaves, which are immersed submanifolds. A
decomposition into 1-dimensional submanifolds is just a traditional
dynamical system, while a partition into submanifolds of dimension
larger than one incorporates many ideas from traditional topology, and
yet is often an invitation to chaos. The motivation for the study
of foliations is that they arise in many different settings, as they
are used to solve problems in geometry, dynamical systems, analysis and
even physics. One of the beauties of their study is that to understand
them often requires equal measures of ideas from geometry, topology,
and dynamical systems, and the most interesting and often difficult
questions ask how these aspects play together. This talk will
give an introduction to the topic, and present some recent results
about the dynamics and topology of foliations.
April 7, 2006
Hales (CCR in La Jolla)
Title: Jordan
Decomposition in Integral Group Rings
Abstract: Let A be a
square matrix with rational entries. Then A can be written as the sum S
+ N where S and N also have rational entries, S is semisimple, N is
nilpotent, and S and N commute. This representation is unique, and is
called the Jordan decomposition of A. It can be considered as a
coordinate-free, and ambient-field-free, version of the usual
Jordan canonical form for matrices. This decomposition (in its
multiplicative version) is particularly useful in the study of
algebraic groups. If G is a finite group and u is an element of the
rational group ring Q[G], i.e. u is a linear combination of group
elements with rational coefficients, then there is an analogous
decomposition: u = s + n where s and n lie in Q[G], s is semisimple, n
is nilpotent, and s and n commute (this representation is also
unique). Consider the integral versions of these decompositions:
if the matrix A has integer entries, need S and N have integer entries?
if the element u in Q[G] has integer coefficients, need s and n have
integer coefficients? We give complete answers to these questions. The
multiplicative version of the integral group ring question is much more
subtle, however - for this we give a complete answer when G has 2-power
order, and partial results in the general case.
March 31, 2006
Nat Thiem (Stanford University)
Title: Hecke algebras
in combinatorial representation theory
Abstract: Hecke algebras
appear as valuable tools in many areas of mathematics including
algebra, geometry, number theory, and physics. This talk explores
how Hecke algebras can interpolate between the representation theory of
groups of Lie type (such as the general linear group over a finite
field) and the natural combinatorics associated with finite reflection
groups (such as the symmetric group). After reviewing some of the
fundamental questions in combinatorial representation theory, we will
explore various definitions of Hecke algebras, describing how they
``feel" like reflection groups but ``see" representations of larger
groups. While many of the results and techniques are more general, this
talk will largely focus on the fundamental example involving the
symmetric group and the the general linear group.March 24, 2006
Ben Miller (UCLA)
Coordinatewise decomposition
of Borel functions
Abstract: We will discuss
a variety of descriptive set-theoretic questions that have strong
connections to their ergodic-theoretic counterparts. Our main focus
will be on the following sort of question: Suppose that $S
\subseteq \mathbf{R} \times \mathbf{R}$ is a Borel subset of the plane
and $f : S \rightarrow \mathbf{R}$ is a Borel function. Under what
circumstances are there Borel functions $u,v : \R \rightarrow \R$ such
that $f(x,y) = u(x) + v(y)$?
February 24, 2006
Coefficient dynamics
Mike Keane (Wesleyan University)
Abstract: In this
lecture, I describe an old method in a new setting which may be useful
for the study of symbolic sequences obtained by expanding numbers
either in continued fractions or in bases. After an introduction to the
method, I intend to sketch an earlier result of mine which describes a
possible way in which Gauss discovered the statistical distribution of
partial quotients of continued fractions, some two hundred years ago.
Then, using these ideas, I present what seems to be a new proof of an
old theorem of Lagrange, stating that quadratic algebraic numbers have
eventually periodic continued fraction expansions. Finally, I would
like to discuss the conjecture that there are no irrational algebraic
numbers belonging to the classical Cantor set, and present a new
conjecture using dynamics which, if true, would lead to a partial
solution of this problem.
Feb. 17, 2006
HongYang Chao
(Sun Yat-Sun University, China).
Abstract: Iimage enhancement technology is always a hot topic in the
area of digital image processing. This talk is mainly about improved
histogram equalization. We will introduce a new method and will
show how to improve the outlooking of image data from a digital camera,
i.e. CCD data analysis. In addition, we will talk about why we have to
do video compression and how people do it. Including some recent
developments.
February 10, 2006
Peter
Massopust, (Tuboscope Pipeline Services, Houston)
Data and Image
Analysis in Pipeline Inspection
Abstract: One of
the main tasks of the pipeline integrity industry is the detection and
classification of defects in pipelines. There is a wide variety of
defects but they can be put into essentially three major groups: (i)
defects due to corrosion, (ii) defects generated by mechanical damage,
and (iii) cracks created by stresses in the pipe wall. A general defect
may belong to more than one group.
The main mathematical problems
encountered when analyzing, enhancing, and evaluating data and images
collected by inspection tools yield a plethora of beautiful and
interesting mathematics ranging from partial differential
equations and special functions
to wavelet, curvelets, and splines. In this talk, we will highlight
some of these problems and their solutions and point out communalities
with other data and image types such as biomedical measurements.
Jan. 20th, 2006
Edward Odell, (UT
Austin)
Ramsey
theory and Banach spaces
Abstract:
Ramsey theorems have a certain general flavor.
A structure is (usually) finitely colored. A certain monochromatic
substructure of a certain type and size is sought. This is then shown
to hold true if the original structure is sufficiently large, sometimes
with added restrictions as to the nature of the coloring. In this
expository talk we will discuss some Ramsey theorems and their impact
on problems in the geometry of Banach spaces and vice versa.
Ramsey theory and
Banach spaces
Fall
2004 - Fall 2005
Tuesday, Dec. 6th, 2005
Tony Zettl
(Northern Illinois University)
Eigenvalues
of Sturm-Liouville Problems
Abstract:
We discuss eigenvalues and eigenfunctions of classical regular
self-adjoint Sturm-Liouville problems. Then indicate extensions to some
nonclassical cases: non-self-adjoint, indefinite, singular.
December 2, 2005
Walter Bergweiller
(University of Kiel, Germany)
Fixed points and periodic
points of quasiregular maps
Let f,g be a
quasiregular self-maps of d-dimensional Euclidean spacewith an
essential singularity at infinity, where d>1. We will discuss two
results concerning such mappings. The
first result, due to Heike Siebert, says that if n>1, then f has
infinitely many periodic points of period n. This means that the n-th
iterate has infinitely many fixed points that are not fixed points of
any k-th iterate where k<n. The second result says that the
composite function f(g) has infinitely many fixed points. The
proofs are based on normal family arguments, and thus we shall also
discuss normal family analogues of the above results. The results
above where known before if d=2 and the functions f,g are holomorphic
(and they had been conjectured by Baker and Gross for this case). The
present approach leads to new and simpler proofs for this special case.November
4, 2005
Voker Mayer (University
of Lille, France)
Thermodynamical
formalism for meromorphic functions of finite order
We present
joint work with Mariusz Urbanski in which we make available, for a wide
class of meromorphic functions, one of the main tools for the geometric
study of the Julia set. Namely, based on
Nevanlinna theory, we proof that the thermodynamical formalism is valid
for very general hyperbolic meromorphic functions of finite order
(including exponential, sine and tangent families). Then we give
geometric applications such as Bowen's formula: the Hausdorff dimension
of the radial Julia set is given by the (only) zero of the topological
pressure. The Question of Concreteness of Cofinitary Subgroups of the
Infinite Symmetric Group.
October 21, 2005
Kathrin Bringmann
(University of Wisconsin at Madison)
On Mock Theta functions and a
Conjecture of Dragonette and Andrews.
This is a joint work with Ken Ono.
We solve the classical problem of obtaining
formulas for $N_e(n)$ (resp. $N_o(n)$), the number of partitions of an
integer $n$ with even (resp. odd) rank. Thanks to Rademacher's exact
formula for the partition function, this problem is equivalent to that
of obtaining a formula for the coefficients of the mock theta function
$f(q)$, a problem with its own long history dating to Ramanujan's last
letter to Hardy. Little was known about this problem until Dragonette
in 1952 obtained asymptotic formulas. In 1966, G. E. Andrews refined
Dragonette's results, and conjectured an exact formula for the
coefficients of $f(q)$. By constructing a weak Maass-Poincare series
whose ``holomorphic part" is $q^{-1}f(q^{24})$, we prove the
Andrews-Dragonette conjecture, and as a consequence obtain the desired
formulas for $N_e(n)$ and $N_o(n)$.
September 16th, 2005
Bart Kastermans
(University of Michigan)
The Question of Concreteness
of Cofinitary Subgroups of the Infinite Symmetric Group.
A question often encountered in mathematics is
how concrete certain types of objects can be. This is a question with
immediate intuitive meaning, but with just this intuitive meaning not
subject to study (certainly not in case of a negative answer). In this
talk I will explain how it translates to a precisely stated logical
question.
The second component of this talk is about
maximal cofinitary groups. These are certain subgroups of the infinite
symmetric group. The difficulties in constructing these groups are
often of a combinatorial nature.
In this talk I will introduce maximal cofinitary
groups. Then give some of their basic properties, and show how these
lead to interesting questions. I will explain the question of concrete
such groups, how to study it, and give the results so far known. Then
I'll show some of the ingredients in constructing them, and end with
some open questions.
September 9, 2005
Hiroki Sumi (Osaka
University, Japan)
Random dynamics of polynomials
and devil's-staircase-like functions in the complex
plane
We consider random dynamics of polynomials on the complex
plane. More precisely, let $tau $ be a Borel probability measure
in the space of polynomials and we consider i.i.d. random dynamical
systems on the Riemann sphere such that at every step we choose a
polynomial according to the distribution $/tau .$ Let $T(z)$ be the
probability of tending to infinity from the initial value $z.$ Suppose
that the support of $tau $ is compact, the postcritical set of
semigroup $G$ generated by the support of $tau $ is bounded in the
complex plane, and the Julia set of $G$ is disconnected. Then we
show that the function $T$ on the Riemann sphere has the following
properties. (1) $T$ is continuous on the Riemann sphere.
(2) $T$ on the Riemann sphere has the following properties. (1)
$T$ is continuous on the Riemann sphere. (2) $T$ varies only on
the Julia set of $G$. (3) $T$ has some monotonicity
property. Hence $T$ is like the devil's staircase.
May 5, 2005
Genevieve Walsh, UT Austin
"Which 3-manifold is the
universe?"
The surface of the earth is a 2-manifold, which means that it is
locally modelled on $R^2$. In fact, we know exactly what 2-manifold it
is, the 2-sphere. Similarly, our universe is a 3-manifold, meaning
locally modelled on $R^3$. In this talk, we will explore the various
possibilities for the 3-manifold that is our universe.
April 29, 2005
Charles Holton, UT Austin
"The C*-algebraic Rohlin property for shifts of
finite type"
The Rohlin property for an automorphism of a C*-algebra can be thought
of as a noncommutative topological analogue to the Rohklin Lemma of
ergodic theory. We briefly describe a construction of a C*-algebra and
an automorphism from a shift of finite type. We then expound at great
length on the definition and significance of the Rohlin property, and
how one can deduce it from an "approximate" Rohlin property. The main
result is that the Rohlin property holds for an automorphism
constructed from a shift of finite type, and hence the C*-crossed
produce of the algebra and the automorphism is determined up to
automorphism by K-theoretic data.
April 1, 2005
Julie Hartmann, University of Heidelberg
"Galois Groups of Linear
Differential Equations"
Galois theory is the study of polynomial equations and their
solvability by means of their symmetry groups, the so-called Galois
groups. This theory was generalized by Picard, Vessiot and later
Kolchin to linear differential equations. The differential Galois
groups are matrix groups (more precisely, linear algebraic groups)
acting on the solution space. The talk will give an introduction to the
subject and report on recent developments, with particular focus on the
inverse problem, i.e., the question which matrix groups occur as
symmetry groups of linear differential equations. The main result we
present is that over a rational function field with algebraically
closed field of constants of characteristic zero, every linear
algebraic group occurs as the Galois group of a linear differential
equation.
April 7, 2005
Henk Bruin, University of Surrey
"Renormalization for
piecewise rotations on the circle"
Interval translation maps
(which generalize interval exchange transformations) can have
interesting Cantor attractors. In this joint work with Serge
Troubetzkoy (Luminy), we studied the occurrence and geometry of such
attractors for maps of the circle consisting of two rotations. The main
tool is a renormalization operator which acts on the parameter space as
an infinite alphabet Horseshoe map with a neutral fixed point. A
symbolic coding of the parameter determines the Hausdorff dimension and
(non)unique ergodicity of the attractor.
March 28, 2005
Ioana Ghenciu, University
of Wisconsin, River Falls
"Complemented Spaces of
Operators, Dunford-Pettis and Gelfand-Phillips Properties"
March 21, 2005
Ozlem Imamoglu, ETH Zurich
"Representation of integers
as sums of squares, old and new results"
The problem of finding explicit formulas for the number or
representations of an integer as a sum of squares has a long history.
After briefly introducing the problem and its history, I will report on
some recent results.
December 14, 2004
Dr. Hiroaki Terao, Tokyo Metropolitan
University
December 10th
Dr. Giorgio Fusco,
Università di L'Aquila, Italy
"Numerical Experiments and Conjectures on the
Dynamics defined by some Singularly Perturbed Non-Convex Functionals of
the Gradient."
We consider a class of non-convex functional in one space dimension.
The gradient flow associated to such functional is ill posed. Therefore
we regularize by perturbing with a small higher order term. We discuss
various nonlinear phenomena of the regularized dynamics for small value
of the perturbative parameter. In particular we describe three well
separated time scales and formation and evolution of interfacies. We
also discuss the possibilty of defining a notion of weak solution for
the original unperturbed ill posed problem. Our approach is both
theoretical and numerical.
October 15, 2004
Dr. Eugen Mihailescu,
Institute of Mathematics of the Romanian Academy
"Thermodynamic
formalism and higher dimensional complex dynamics"
In this talk we will review some of the most important features of
dynamics in several complex variables. This can be done for
automorphisms in C^n or for endomorphisms in P^n. We will focus mainly
on the case of holomorphic endomorphisms and give some constructions,
methods and results. Thermodynamic formalism is used in this setting to
estimate the Hausdorff dimension of certain fractal sets. A new notion
of inverse pressure will be introduced in the general continuous case
(no holomorphicity required) and will be applied to dimension
estimates.
October 8, 2004
Dr. Ted Slaman, UC
Berkeley
"Measures and Their Random
Reals"
I will begin with an overview of the effective randomness of an
infinite sequence, including a discussion of the equivalent
characterizations by Martin-L\"of in terms of measure and by Kolmogorov
in terms of descriptive complexity. Then, I will discuss joint work
with Jan Reimann, University of Heidelberg, in which we examine reals
which are random for measures other than the standard Lebesgue measure.
As an example, I will go into some detail about the following theorem.
Theorem (joint with Reimann) For
$X\in 2^{\omega}$, the following conditions are equivalent.
1. There is a probability
measure $\mu$ on $2^\omega$ such that $X$ is not a $\mu$-atom and $X$
is random relative to $\mu$.
2. $X$ is not recursive.
Fall 2003
- Spring 2004
April 30, 2004
Dr. Debastien Ferenczi,
IML-CNRS, Marseille, France
"Substitutions on an Infinite
Alphabet"
Abstract: A
substitution on an alphabet A is an application from A to the set of
finite words on A; under mild conditions, it has an infinite sequence
as a fixed point, and the shift on its closed orbit gives a
substitution dynamical system. When A is finite, these are compact
topological dynamical systems, which are generally minimal and uniquely
ergodic, and finite rank measure-theoretic dynamical systems. We study
here a few examples where A is infinite: the infini-Bonacci system,
which keeps most properties of the finite case; cases where the matrix
is positive recurrent, where we have still a finite measure-preserving
ergodic system; and the drunkard substitution, where we get an infinite
measure- reserving system for which we can prove ergodicity. Though the
rank is not finite, Rokhlin stacks are still the main tools of our
study.
April 23, 2004
Dr. Ron Solomon, Ohio State University
"Finite Simple Groups from
Galois to Aschbacker"
The concept "groupe simple" was introduced by Galois in 1832. A
concerted effort to classify all finite simple groups was proposed by
Holder in 1892, but it did not really take flight until the 1950's. The
project seems at last to have been completed by Aschbacher and Smith
this year. Along the way, many insights were gained about the structure
of finite groups, and several amazing new mathematical objects were
discovered. Many questions have been answered, but several mysteries
remain. This talk will touch on some of the highlights and mention some
open questions.
April 2, 2004
Dr. Allen Butler, Wagner Associates
"The Mathematics of Data
Fusion"
In recent years, tremendous strides have been made in the improvement
of existing and the development of new, more powerful, sensor systems.
The result is a tidal wave of data which threatens to overwhelm the
user, rather than assist her. The process of automatically filtering,
aggregating and extracting the desired information from multiple
sensors and sources is an emerging technology, commonly referred to as
Data Fusion. In this talk, I will show how a wide variety of
mathematical techniques are applied in this new discipline. I will
begin with a discussion of the state estimation problem determining the
current position and velocity of an object based on a set of discrete
observations (e.g. radar tracking of an aircraft). I will discuss a
number of filtering techniques, including the a-b filter, the classic
Kalman Filter, the Extended Kalman Filter and the Unscented Kalman
Filter. I will then discuss the data association problem given a set of
observations or measurements taken over a period of time, determine
which ones originate from the same real-world object. I will consider
both real-time solution techniques and batch techniques such as those
based on the Expectation-Minimization algorithm. Finally, I will
conclude with a discussion of Data Fusion Measures of Performance,
attempting to answer the question, How do you grade a system that
produces probability distributions for answers?
March 9, 2004
Dr. Ralph Chill, Ulm University
"Around the Lojasiewicz-Simon
inequality and its applications to PDEs"
The talk will start with an easy proof that bounded and global
solutions of finite-dimensional gradient systems converge to a steady
state if the underlying energy satisfies the Lojasiewicz inequality. It
turns out that the same proof works for the semilinear heat equation or
the semilinear wave equation. Motivated by this, we study theoretical
aspects of the Lojasiewicz inequality (gradient inequalities), its
generalization to infinite dimensions, we show how to recover known
convergence results and we indicate how to prove similar results for a
variety of PDEs and for the steepest descent method.
March 5, 2004
Xinfu Chen, University of
Pittsburgh
"Interfacial Dynamics and
Free Boundary Poblems"
Interfacial phenomena are commonplace in physics, chemistry, and in
various other fields. They occur whenever a medium is present that can
exist in at least two different states and there is some mechanism that
generates or enforces a spatial separation between these two states.
The separation boundaries are then called interfaces of free
boundaries. Generally speaking, the study of interfacial phenomena can
be grouped into two categories:
(i) free boundary models in which states
(phases) are described by binary valued phase indicator functions and
free boundaries are hypersurfaces where the phase indicators switch
their values.
(ii) continuum models in which states are
described by smooth functions which experience large gradients in
places called interfacial regions.
In the ideal limit of a continuum model, a
smooth phase indicator function becomes binary valued, the thickness of
an interfacial region becomes zero, and the interfacial region becomes
a free boundary. In this talk, a few examples will be given to
illustrate both models and their relationships.
February 13, 2004
Zoltan Buczolich, Department of Analysis,
Eotvos Lorand University, Budapest, Hungary.
"An L1 Counting Problem in Ergodic Theroy"
January 30, 2004
Paula Cohen, Texas A&M University
"Hyperbolic Distribution Problems"
Siegel in 1932 and Schneider in 1937 obtained the first results on the
transcendence and linear independence of periods of classical doubly
periodic functions. This led to the first results on the transcendence
at algebraic points of 1-variable complex functions invariant with
respect to discrete groups of fractional linear transformations
(modular groups). We discuss some problems that arise in the modern
outgrowth of this work. These problems motivate the study of
distribution problems for certain families of points on modular
varieties, in particular questions of equidistribution. The talk will
be accessible to the general audience.
September 19, 2003
Christian Wolf, Wichita State University
"Measures of maximal dimension for
hyperbolic diffeomorphisms"
We discuss the existence of ergodic measures of
maximal Hausdorff dimension for hyperbolic sets of surface
diffeomorphisms. This is a dimension- heoretical version of the
existence of ergodic measures of maximal entropy. The crucial
difference is that while the entropy map is upper-semicontinuous, the
map $\nu\mapsto\dim_H\nu$ is neither upper-semicontinuous nor
lower-semi\-continuous. This forces us to develop a new approach, which
is based on the thermodynamic formalism. Remarkably, for a generic
diffeomorphism with a hyperbolic set, there exists an ergodic measure
of maximal Hausdorff dimension in a particular two-parameter family of
equilibrium measures.
October 23, 2003
Benedikt Lowe, University of Amsterdam
"Blindfolding stochastic opponents in
infinite games; or: How do you win infinite stochastic games if you can
only train against dummies?"
Take a set A of infinite strings of zeros and ones and play the
following infinite game: infinitely often, players I and II play single
0-1 bits and thus produce an infinite string of zeros and ones. We say
that player I wins if this string lies in A, otherwise player II wins.
A winning strategy in such a game is a procedure
to choose the next move that guarantees a win for the player using it
against all counterstrategies. Peculiarly, in order to test whether a
given strategy is winning, you only need to test it against fixed
strategies, i.e., counterstrategies that don't react to your moves.
This property is lost if you move from strict
winning strategies in the above sense to strategies that win with
probability one. In this case, you could have a strategy that
guarantees a win with probability one against all passive opponents,
but loses with probability one against some active opponents. In this
talk, we shall discuss consequences of this and means of dealing with
it.
November 14, 2003
Brian Conrey (American Institute of Mathematics)
Title: "Random matrix theory and the Riemann
zeta-function"
In 1972 a chance meeting between Hugh Montgomery and Freeman Dyson
first led people to suspect that there was a relationship between the
statistics of the zeros of the Riemann zeta-function and eigenvalues of
random matrices. This relationship was developed over the years,
notably through data found by Andrew Odlyzko. In 1998 a deeper
connection was discovered between the value distribution of the
zeta-function and the distribution of values of characteristic
polynomials of these random matrices. Today we have an amazing set of
parallels between the Riemann zeta-function (also families of
L-functions) and unitary matrices (also orthogonal and symplectic
matrices). In this talk we will describe some of these connections. We
will focus especially on the random matrix side of this duality, and
discuss some of the (elementary) techniques that prove the elegant
theorems on this side. The talk will be aimed at a general audience.
November 21, 2003
Ion Mihai, University of Bucharest, Romania
"Kaehler Manifolds and Their
Submanifolds"
The most important class of complex manifolds are the Kaehler
manifolds. They are manifolds endowed with a special type of metric,
the Kaehler metric. The complex n-space C^n (with the Euclidean
metric), the complex torus T^n (with metric induced by the Euclidean
metric on C^n), the complex projective space (with the Fubini- Study
metric), the complex Grassmannian, the unit complex disk (with the
Bergmann metric) are all examples of Kaehler manifolds.
However, there are interesting almost Hermitian
manifolds which do not admit Kaehler metrics. It is known that the
6-dimensional sphere S^6 carries a distinguished non- aehler nearly
Kaehler structure. There are cohomological obstructions to the
existence of Kaehler metrics on compact complex manifolds. For
instance, it can be shown that the Calabi-Eckmann manifolds (in
particular, the Hopf manifolds) cannot admit any Kaehler metric.
Special classes of submanifolds of a Kaehler
manifold can be defined according to the behavior of their tangent
spaces under the action of the complex structure of the ambient space.
We mention here: complex submanifolds, totally real submanifolds, slant
submanifolds, CR-submanifolds.
In this expository talk we will introduce and
comment on some of the objects named above.
December 9, 2003
Mario Roy, Concordia University, Canada
Title: "On how potential theory sometimes pays
back complex analysis"
Abstract: We will discuss two
instances in which potential theory repays its debt to complex analysis
in the form of applications to the theory of dynamical systems. We will
first glance over some properties of the Julia set of rational
functions, and then establish parallels for the attractor of iterated
function systems. All pertinent notions will be introduced and
exemplified during the talk.
Fall 2002 -
Spring 2003
November 15, 2002
Anthony Quas, University of Memphis
Title: "Arrow's Impossibility Theorem"
Abstract: Arrow's Impossibility Theorem states that in an election with
3 or more candidates, there is no voting satisfying a small number of
basic fairness requirements. In spite of this, many voting systems are
used with a wide variety of properties. Here, we focus on the
requirement of monotonicity: that the more votes you get, the more
likely you are to win. Surprisingly, a fairly popular voting system
does not have this property. We will discuss the probability that
unfairness of this type arises in the single transferable vote system.
December 9, 2002
Sergey Yuzvinsky, University of Oregon
Title: "Topological robotics on hyperplane
complements"
Abstract: We will start by
defining a new simple invariant of topological spaces - topological
complexity (TC). If one views a space X as the configuration space of a
robot, then TC(X) describes roughly the complexity of a motion planning
algorithm for the robot. We'll discuss the property of the invariant,
and in particular, a lower bound for it coming from the ring theory. In
the case when X is the complement of a complex hyperplane arrangement,
the ring in question is defined by the underlying matroid---it is the
Orlik-Solomon algebra of the matroid. Thus the problem of computing
TC(X) includes the problem of computing a matroid invariant. We'll
compute this invariant for the braid arrangements where TC(X) has a
special importance because a motion on X is just a collision free
motion of several ordered points on a plane. We'll also show the values
of TC(X) for the configuration spaces of several distinct ordered
points in the higher dimensional real spaces. At the end some problems
and conjectures will be formulated.
January 24, 2003
Volker Mayer, Lille University, France
Title: "Renormalizations and Rigidity in
Conformal Dynamics"
Abstract: We explain how
renormalization techniques can be used to obtain simple proofs of
rigidity phenomenas in holomorphic and quasiregular dynamics.
February 28, 2003
Thomas Schlumprecht, Texas A&M University
Title: "Can all the central sections of a
bigger body be smaller?"
Abstract: We present a unified
analytic solution to the following problem stated by Busemann and Petty
(1956): Let K and L be two convex and symmetric n-dimensional bodies
and assume that all the (n-1)-dimensional central sections of K have
smaller volume than the corresponding sections of L. Does it follow
that the volume of K is smaller than the volume of L?
March 6, 2003
Boris Adamczewski, Montpellier, France
Title: "An introduction to uniform distribution
modulo 1"
Abstract: The aim of this talk is
to give an introduction to the theory of uniform distribution modulo
one. We will focus on the notion of discrepancy, that is on the
quantitative aspect of this theory. We will deal in particular with
questions related to number theory and diophantine approximation.
March 13, 2003
Marco Fontelos, Universidad Rey Juan Carlos,
Spain
Title: "Singularities in fluids"
Abstract: In 1755 the Swiss
mathematician Leonard Euler wrote for the first time the differential
equations describing the motion of an inviscid (i.e. without internal
friction forces) fluid. In 1822 C. Navier and, independently, G. Stokes
introduced the viscosity term and obtained the so called Navier-Stokes
system. Despite the apparent simplicity of the equations, most of the
fundamental questions concerning them remain unsolved. It is not known,
for instance, whether or not the solutions remain smooth for all time
or form some kind of singularity. The later possibility could be
related to the appearence of chaotic motions that are observed when the
flow becomes turbulent.
In the talk we will present some results
regarding the formation of singularities in fluids focusing on three
situations: 1) the possible blow-up of the velocity field inside the
fluid,
2) the existence of a topological transition in the interface between
two fluids by which a simply connected fluid domain becomes
disconnected (with the formation of drops),
3) the formation of wavy structures via Kelvin-Helmholtz instability.
The appearance of these phenomena is very sensitive to the nature of
the fluid. We will also briefly discuss the situation when the fluids
presents some "elastic" behavior.
April 4, 2003
Mike Keane, Wesleyan University
Title: "On spontaneous emergence of opinions"
Abstract: One of the
distinguishing properties of the present scientific method is
reproducibility. In one of its guises, probability theory is based on
statistical reproduction, near certainty being obtained of truth of
statements by averaging over long term to remove randomness occurring
in individual experiments. When one assumes, as is often the case, that
events farther and farther in the past have less and less influence on
the present, the probabilistic paradigm is currently well understood
and is successful in many scientific and technological applications.
Recently, however, we have come to realize that precisely in these
applications important stochastic processes occur whose present
outcomes are significantly influenced by events in the remote past.
This behavior is not at all well understood and some of the simplest
questions remain today irritatingly beyond reach. A salient example
occurs in the theory of random walks, where there is a dichotomy
between recurrent and transient behavior. After explaining this
classical dichotomy, we present a very simple example with infinite
memory which is neither known to be transient nor recurrent. Then,
using a reinforcement mechanism due to POLYA, we explain the nature of
a particular infinite memory process in terms of spontaneous emergence
of opinions. Finally we would like to discuss briefly some of our
recent results towards understanding the recurrence-transience
dichotomy for reinforced random walks ,and indicate an application to
universal coding used in optical CD technology.
April 11, 2003
Giorgio Fusco, University di L'Aquila, Italy
Title: "A regularized Perona-Malik functional:
Some aspect of the gradient dynamics"
Abstract: We study an elliptic
regularization of the classical Perona-Malik functional. After a
suitable rescaling we characterize the Gamma-limit ot the regularized
functional. We analize the gradient dynamic associated to the
Gamma-limit and obtain some preliminary results concerning the
convergence of the dynamic corresponding to the regularized functional
to the one generated by the Gamma-limit.
April 25, 2003
AGANT Minhyong Kim, University of Arizona
Title: "The topology of algebraic surfaces and
reduction modulo p"
Abstract: We will discuss some
classical relationships between the mod p arithmetic of varieties and
their topology. Furthermore, we will mention one new result regarding
the homeomorphism type of simply-connected surfaces.
December 3, 2003
Title: "Porosities and dimensions"
Esa Jarvenpaa
Abstract: I will give a short
survey on different notions of porosities and their relations to
dimensions. Dimension is a concept which describes the size of a set or
a measure. Porosity, in turn, tells how big holes there are in the set
or measure. Intuitively, it seems natural that if there are a lot of
big holes then the dimension cannot be very big. I will explain to
which extend this intuitive picture is correct.
Fall 2001 - Spring 2002
G.A. Edgar, Ohio State University
Title: "How big is a subgroup of the reals?"
October 19, 2001
Abstract: Suppose a subset of the
real line is "nice" both in a topological sense and in an algebraic
sense. How big can it be? Topological versions of nice sets could be
closed sets or especially Borel sets. Algebraic versions of nice sets
could be subgroups, or subrings, or subfields. "how big" can be asked
in several senses. One sense (Hausdorff dimension will be discussed,
ranging from Volkman in 1960 up to current results.)
November 2, 2001
Vladimir Pestov, Victoria University of Wellington, New Zealand
Title: "Asymptotic geometric analysis and
topological transformation groups"
Abstract: Asymptotic geometric
analysis (also known as geometry of large dimensions) studies various
counter-intuitive phenomena occuring in geometric structures of high
finite dimension. The framework used is the concept of a metric space
with measure (mm-space), and the theory is mainly concerned with the
phenomenon of concentration of measure on high-dimensional structures.
Among the best known manifestations of the phenomenon are the law of
large numbers (probability), Dvoretzky theorem (geometric functional
analysis), blowing-up lemma (coding theory), and many other results and
techniques cutting across mathematical sciences. The present growth of
interest in the subject is due to the work of Paul Levy, Vitali Milman,
Michel Talagrand, Mikhail Gromov, and others. Many model examples upon
which asymptotic geometric analysis is built are in fact transformation
groups, and the concentration phenomenon in the phase space leads to
dynamical phenomena, such as the existence of fixed points. Work by
mathematicians including Milman, Gromov, Glasner, Furstenberg, Weiss,
Giordano, and the present author has led to new insights into the
dynamical properties of some important` infinite-dimensional' groups
and representations in infinite-dimensional Hilbert spaces. In this
talk we will survey the basic concepts of asymptotic geometric analysis
and its applications in the context of dynamics and representation
theory.
December 14
Hiroshi Suzuki, The Ohio State University, and
International Christian University
Title: "On Weakly Distance-Regular Digraphs
Highly regular graphs and digraphs"
Abstract: This is an introduction
to distance-transitive graphs, distance-regular graphs,
distance-transitive digraphs, distance-regular digraphs, weakly
distance-transitive digraphs and weakly distance-regular digraphs. The
class of weakly distance-regular digraphs is the largest class among
them. My talk includes, historical background, basic properties,
examples, and recent results on weakly distance-regular digraphs.
January 22, 2002
Eric Sommers, University of Massachusetts at Amherst
Title: "Coherent Sheaves on the Nilpotent Cone"
Abstract: I will survey results
concerning the Grothendieck group of equivariant coherent sheaves on
the nilpotent cone of a simple Lie algebra. The Grothendieck group
(after Bezrukavnikov's proof of a conjecture of Lusztig) has two
natural bases. One is indexed by the dominant weights of the Lie
algebra; the other (which is much more difficult to construct) is
indexed by equivariant irreducible vector bundles on the nilpotent
orbits of the Lie algebra. I will discuss some of the representation
theory and geometry behind this work, as well as attempts to explicitly
describe the bijection between the two bases.
March 13, 2002
Robbert Fokkink, TU Delft, Holland
Title: "On the problems of Dido and Borsuk"
Abstract: The isoperimetric
inequality says that of all geometric figures, the disk has the largest
area compared to its circumference. This inequality has been
generalized in many directions, using techniques from all branches of
mathematics. This talk focuses on one particular version, called the
isodiametric inequality. It turns out that this inequality is related
to Borsuk's conjecture from convex geometry.
March 28, 2002
Lisa Bloomer, Middle Tennessee State University
Title: "Random Probability Measures with given
Mean and Variance"
Abstract: The
question "is there a natural way to construct random probability
measures?" has been addressed in several ways since Dubins and Freedman
proposed the question in 1963. Recently, the question was modified by
Hill and Monticino to be "is there a natural way to construct random
probability measures while incorporating given information?" I will
review the original algorithm described by Dubins and Freedman and the
algorithm described by Hill and Monticino that allows the mean of the
measure to be specified in advance. Then I will describe several
methods of choosing random probability measures while specifying the
mean and the variance.
Fall 2000 - Spring 2001
September 1, 2000
Jean-Paul Allouche, Paris: (CNRS, LRI, Orsay,
France)
Title: "Non-integer bases, iteration of
continuous functions, and an arithmetic fractal"
Abstract: In a recent paper
appeared in American Mathematical Monthly, V. Komornik and P. Loreti
study the real numbers q in (1,2) such that the number 1 has a unique
expansion in base q. They prove in particular that there exists a
smallest such number q, say t, and that the base-t expansion of 1 is
essentially the Thue-Morse sequence. We prove how this result is a
reformulation of a result by M. Cosnard and the author (1983) that was
obtained in the framework of iterations of unimodal continuous
functions. Furthermore we confirm a conjecture of Komornik and Loreti
on the irrationality of t by proving that this number is actually
transcendental (the proof is easy up to using a theorem of Mahler). We
conclude by exhibiting an arithmetic fractal related to the above
questions.
September 8, 2000
Hiroaki Terao (Tokyo Metropolitan University)
Title: "Double Coxeter arrangements, the Shi
arrangements, anti-invariant
forms and logarithmic forms"
Abstract: Let W be a finite
crystallographic group and A(W) be the arrangement of its reflecting
hyperplanes in the n-dimensional real vector space. The Shi arrangement
is A(W) together with the hypeperlanes defined by \alpha = 1 (\alpha is
moving on the set of positive roots of W). It has remarkable
combinatorial properties. For example, the number of chambers is equal
to (1+h)^n (h is the Coxeter number). The "principal part" of the Shi
arrangement is the double Coxeter arrangement. We prove that its
derivation module is a free moduole of rank n with a basis consisting
of derivations whose degrees are (h, h, ..., h). Explicit basis is
described using the flat ructure (the Forbenius system) of Coxeter
group. The relation between the anti-invariant differential forms and
the logarithmic differential forms is also discussed.
September 25, 2000
Title: "A Progress Report on Jacobian Conjecture"
Daya-Nand Verma
Abstract: The said unsolved
problem listed by Smale in his 1998 Math Intelligencer article as 16th
of 18 Mathematical Problems for the 21-st Century amounts to asking for
the ``Inverse Function Theorem of Algebraic Geometry,'' viz., to show
that any polynomial endomorphism of the n-space (say over the complex
numbers) with the Jacobian-determinant non-vanishing everywhere (IS
INJECTIVE and hence) admits a polynomial inverse! It may not be too
outrageous to say that the fact such a Statement is unproven (ever
since its formulation for the case when n is 2 in 1939 by Ott-Heinrich
Keller) is a `dark spot' for 20th Century Algebraic Geometry, which was
sometimes even taken as a matter of fact; lately the statement has even
been subjected to doubts, and some ``exotic'' consequences of its
falsity have been studied. After giving a quick survey of some of the
relevant developments (mostly dating back to the 70's) I shall dwell a
little on a recent new idea that could be called `A Jacobian-Wronskian
Approach to the Problem' <i.e. of the Proof of JC, for which I've
now more faith in its truth>. A bit more precisely, I ask the NEW
Question: ``How to give a GOOD Criterion to decide whether n given
homogeneous polynomials in m variables, all of the same degree <or
else, inhomogeneous polynomials in m-1variables>, are linearly
independent?'' The meaning of `good' should be guided by the answer
`classically available' when m is 2, in the form of (a slightly
revised, homogenized, version of) the celebrated Wronski Theorem, basic
to the Theory of Linear Ordinary Differential Equations. Thus one wants
<a la Wronski'> the answer to be (almost) independent of the
degree; however, for the actual application to the original Problem
it's enough to consider m=n and d=3. Hence our more general enquiry may
prove very valuable also to (an `Algebrization Programme' for) the
Theory of Linear Partial Differential Equation.
October 27, 2000
James T. Rogers, Tulane University
Title: "Boundaries of Siegel disks"
November 3, 2000
Title: "Random sets and the Loewner differential equation"
Steffen Rohde, University of Washington
Abstract: The Loewner differential equation, classically used by C.
Loewner to study questions from geometric function theory, has recently
been used by Lawler, Schramm and Werner to settle some outstanding
problems in stochastics (such as the determination of the Hausdorff
dimension of the Brownian frontier). In this talk I will explain the
Loewner equation, its relation to stochastic growth models (such as
loop erased random walk, percolation, Brownian motion) and will discuss
some recent developments. The talk will be aimed at the non-specialist,
and graduate students should be able to follow most of the talk.
November 13, 2000
Victor Nistor, Penn State University
Title: "Elliptic theory on singular and
non-compact manifolds"
Abstract: The analysis on
non-compact manifolds can sometimes be described by certain algebras of
vector fields on suitable compactifications. To generalize the
classical results on elliptic differential equations on compact
manifolds to non-compact manifolds, Melrose has asked for algebras of
pseudodifferential operators that quantize these vector fields and
shown how to achieve this quantization in several particular cases, the
best know being that of a manifold with cylindrical ends, which he
compactified to a manifold with boundary, thus obtaining the so called
"b-calculus." After reminding his results, as well as some related
results of Mazzeo, I will explain how one can obtain a solution to
Melrose's question in the spirit of Lie's third theorem (finite
dimensional Lie algebras are integrable), using an idea of Connes. This
then leads to the standard "elliptic package'' for operators in these
algebras including criteria for compactness, boundedness, or
Fredholmness. Using C^*-algebra methods, we also obtain the proof of
another conjecture of Melrose on the spectrum of the Laplace operator.
January 26, 2001
Title: "On the wild components of the family $\lambda \tan(z)$"
Janina Kotus, Warsaw Instutute of Technology
Abstract: It is natural to ask if
the hyperbolic components of the tangent family {\cal F}=\{\lambda
\tan(z): \lambda \in \C-\{0\}\}$ are the only open components of the
$J$-stable maps. The evidence of the computer pictures indicates an
affirmative answer to this so called {\it Densiy conjecture}. If the
density conjecture is not true, there are non-trivial components of the
$J$-stable set containing non-hyperbolic maps; we call them {\it wild
components}. We prove that there are no wild components in the family
$\cal F$ for which the orbits of the asymptotic values are unbounded,
what is a partial positive answer to the density conjecture.
February 16, 2001
Title: "A Problem in Differential Equations with Disappearing Solution"
Jim Serrin
1999-2000
September 14, 1999
Title: "Some aspects of the anticipating calculus for the Poisson
process"
Constantin Tudor, CIMAT and University of Bucharest
Abstract: We use the Poisson-Ito
chaos decomposition approach to define a variations derivative operator
and its adjoint, which is an anticipating integrali.e., it agrees with
the martingale Poisson-Ito integral for predictable integrands). Also
an integration by parts formula and characterizations of these
operators are given.
October 7, 1999
Title: "Multifractal structure associated with Lyapunov exponents for
dynamical systems"
Yakov Pesin, Pennsylvania State University
Abstract:
Lyapunov exponents are fundamental invariants of dynamical systems that
characterize stability of trajectories. They are often used in the
numerical study of dynamical systems and are well related to other
important invariants of dynamics such as entropy and fractal dimension.
The behavior of the Lyapunov exponents on the base point is known to be
extremely complicated and generates a highly non-trivial and remarkably
refined multifractal structure which will be described in the talk.
December 8, 1999
Title: "Unitary representations of Lie groups"
Peter Trapa, Institute for Advanced Study,
Princeton, NJ
Abstract:
Suppose $G_\R$ is a real Lie group. A classical and still open problem
is to describe all continuous, irreducible, length-preserving actions
of $G_\R$ on a Hilbert space. An impressive array of ideas has been
applied to this problem with varying degrees of partial success. The
purpose of this talk is to survey a handful of those ideas --- some
algebraic, some geometric, and some arithmetic --- with a view toward
optimistic applications of them.
December 17, 1999
Title: "Projections of self-similar sets"
Boris Solomyak, University of Washington
Abstract: Replace the unit square by the union of four corner squares
of side $1/4$. Iterating this construction yields a classical example
of a self-similar set $K$ in the plane, which has positive and finite
one-dimensional Hausdorff measure, but is purely unrectifiable. It
follows from Besicovitch's theory that $K$ projects into sets of zero
length on almost every line. I will present a self- contained proof of
this fact, using just the Lebesgue density theorem. There are several
related open problems which I am planning to discuss. (The talk is
based on joint work with Yuval Peres and K\'{a}roly Simon.)
January 28, 2000
Title: "Old and New Results on the Arc Structure of Singular
Algebraic Varieties"
Monique Lejeune-Jalabert, CNRS University
Versailles St-Quentin
Abstract:
An arc on an algebraic variety $ V $ defined over the complex numbers
is a mapping from a sufficiently small neighborhood of the origin in
the complex line into $ V $, given by convergent power series. By
Artin's approximation theorem, for any nonnegative integer $ k $, the
set of k-jets of arcs on $ V $ is a constructible set (i.e. defined by
polynomial equations and inequations). These constructible sets were
first studied by J. Nash in connection with Hironaka's resolution of
singularities. Further related developments will be reviewed.
This lecture is also sponsored by the Charn Uswachoke Lecture Series
and the AGANT Lecture Series
March 6, 2000
Title: "From finite differences to finite elements--A short history
of numerical PDE"
Vidar Thomee, Chalmers University of
Technology, Sweden
Abstract:
We describe the historical development of the basic approaches to the
numerical solution of partial differential equations, namely finite
difference and finite element methods. The properties of these classes
of methods are compared for both stationary and evolution problems.
April 5, 2000
Title: "Complexity of Sequences and Dynamical Systems"
Sebastien Ferenczi, CNRS, Laboratoire de
Mathematiques et Physique Theorique, Tours, France
Abstract:
We study here the combinatorial notion of symbolic complexity: this is
the function counting the number of factors of length n for a sequence,
giving an indication of its degree of randomness. We give a survey of
an open question which is still very much in progress, namely: to
determine which functions can be the symbolic complexity function of a
sequence. Then, we investigate the links between the complexity of a
sequence and its associated dynamical system, and insist on the cases
where the knowledge of the complexity function allows us to know either
the sequence, or at least the system. This leads to another vast open
question, the S-adic conjecture, and to a conceptual (though still
conjectural) link with the notion of Kolmogorov-Chaitin complexity for
infinite sequences. Also, these links with dynamical systems have been
of considerable help to ergodic theory,and this prompted the
ergodicians to create their own notions of complexity, mimicking the
theory of symbolic complexity.
April 17, 2000
Title: "On Ostrowski's numeration system"
Valerie Berthe, CNRS Institut de
Mathematique de Luminy
Abstract:
The numeration scale of the Ostrowski
numeration system is given by the denominators of the convergents in
the continued fraction expansion of a given irrational number x. This
system is particularly suited to study discrepancy properties of the
sequence (nx). Our aim here is to explore the applications of this
system to combinatorics on words. In particular, Ostrowski's numeration
system is a very efficient tool for describing many arithmetic, ergodic
and combinatorial properties of sequences obtained as codings of
irrational rotations, the so-called Sturmian sequences. More generally,
we will evoke the generalizations of these properties to two-
dimensional sequences, three-interval exchange transformations and
sequences with sub-affine complexity.
May 1
Title: "A short history of matroid representation theory"
Geoff Whittle, Victoria University,
Wellington, New Zealand
Abstract:
Matroids were introduced by Whitney in 1935 to
axiomatise the combinatorial properties of a finite set of vectors in a
vector space over a field. A matroid is representable over a field F if
it is isomorphic to one that can be obtained from some set of vectors
over F. A fundamental problem, mentioned by Whitney, is to decide which
matroids are representable over which fields. This problem has turned
out to be deep and has attracted some of the best researchers in
combinatorics, eg Tutte and Seymour. Progress has been intermittent,
but punctuated by some impressive and exciting results. The talk will
survey the major results in the area. It is the speaker's intention and
earnest hope that the talk will be accessible to a very general
audience.
May 3
Title: "Forbidden Words in Symbolic Dynamics"
Filippo Mignosi, Universita di Palermo,
Brandeis University
Abstract:
We introduce an equivalence relation R on the
set of functions from
to N where N denotes the set of natural numbers. By describing a
symbolic dynamical system in terms of forbidden words, we prove that
the R-equivalence class of the function that counts the minimal
forbidden words of a system is a topological invariant of the system.
We show that this new invariant is independent from previous ones, but
it is not characteristic. In the case of sofic systems we prove that
the R-equivalence of the corresponding functions is a decidable
question. As a special application, we use this invariant to show that
two systems associated to Sturmian words having ``different slope'' are
not conjugate. We also exhibit some connections between this invariant
and a new data compression scheme.
May 24
Title: "Binary Self-dual Codes and Orthogonal Groups"
Akihiro Munemasa, Department of
Combinatorics and Optimization, University of Waterloo, Waterloo,
Ontario, Canada and Graduate School of Mathematics, Kyushu University,
Gakuoka, Japan
Abstract:
Tonchev (1989), harada and Kimura (1995) gave a method to construct new
doubly-even self-dual binary codes from a given one. Based on a joint
work with Masaaki Harada and Masaaki Kitazume, we give an
interpretation of their method in terms of transvections in an
apprpriate orthogonal group, and describe a generalization of this
method which leads to the enumeration of all doubly-even self-dual
binary codes. We then discuss an analogue of doubly-even self-dual
codes in arbitraty finite field of characteristic two. This new class
of codes is also characterized as maximal totally singular subspaces
with respect to a quadratic form.
1998-99
September 16, 1998
Title: "Spectral Concentration and the Sturm-Liouville Problem"
Malcolm Brown, University of Wales
October 23, 1998
Title: "The Theory of Optimal Stopping"
Ted Hill, Georgia Tech
October 30, 1998
Title: "Decay of correlations in hyperbolic dynamical systems"
Nikolai Chernov, University of Alabama, Birmingham
November 3, 1998
Title: "Picard's Theorem and its Generalization"
Min Ru, University of Houston
January 19, 1999
Title: "Polar Curves, polar varieties and their connections with
equisingularity and curvature of Milnor fibers"
Bernhard Teissier, CNRS Ecole Normale Superieure
January 20, 1999
Title: "Studying Fractals with Flows"
Albert Fisher, University of Sao Paulo, Brazil
February 11, 1999
Title: "Relatively Prime Numbers and Invariant Measures under the
Natural Action of SL (n,Z) on Rn"
Arnaldo Nogueira, Instituto de Matematica, Universidade Federal do Rio
de Janeiro, Brazil
February 19,2000
Title: "Effect of Aggregation on Population Recovery Modeled by a
Diffusion-Advection Equation"
Victor Padron, Universidad de los Andes/Univ. of Texas at San Antonio
February 25, 1999
Title: "What is the Fundamental Theorem of
Algebra"
Harold Edwards, New York University-Courant Institute
February 26, 1999
Title: "Fermat's Last Theorem: What Happened"
Harold Edwards, New York University-Courant Institute
March 22, 1999
Title: "Some Asymptotic Expansions for Parabolic SPDEs"
Samy Tindel, University of Paris 13 (Villetanneuse)
April 2, 1999
Title: "Sums in L1"
Joe Diestel, Kent State
April 16, 1999
Title: "Aperiodic Dynamical Systems"
Krystyna Kuperberg, Auburn
April 30, 1999
Title: "Turbulence for Polish group actions and its applications"
Alexander Kechris, Cal Tech
1997-98
October 21, 1997
Title: "Holomorphic Dynamics in Cn"
Stefan Heinemann, University of Gottingen
November 14, 1997
Title: "Semipositone Systems"
Ratnasingham Shivaji, Mississippi State University
November 24, 1997
Title: "A dual variational approach to a class of nonlocal semilinear
Tricomi
problems"
Kevin Payne, University of Miami
December 10, 1997
Title: "Recovery for an Aggregating Island Chain Model"
Victor Padron, University of Merida, Venezuela
February 6, 1998
Title: "Fermat's Last Theorem and Beal's Conjecture"
Henri Darmon, McGill University
February 20, 1998
Title: "Convective Stability of Solitary Waves on Lattices"
Robert L. Pego, University of Maryland
March 23, 1998
Title: "Sierpinski Gasket as a Martin Boundary"
Hiroshi Sato, Kyushu University
April 2, 1998
Title: "Edge K-Types and Restriction of Cohomology"
Mark Sepanski, Baylor
April 3, 1998
Title: "Combinatorial differential manifolds and matroid bundles"
Laura Anderson, Texas A&M
May 1, 1998
Title: "The Nonlinear Schroedinger Equation: Self-similar Solutions and
Scattering Theory"
Fred Weissler, University of Paris XIII
1996-97
September 16, 1996
Title: "Fredholm Alternative for Some Quasilinear Differential
Operators"
Pavel Drabek, Czech Republic
November 8, 1996
Title: "Semisimplicity of Representations"
George McNinch, University of Notre Dame
November 15, 1996
Title: "Mathematician, Sculptor: Recent Works"
Helaman Ferguson, Supercomputing Research Center, Maryland
November 22, 1996
Title: "Action as a Function of Period for Ground State Solutions of
Semilinear Elliptic Equations"
Y .S. Il'yasov, Steklov Mathematical Institute, Russian Academy of
Sciences
January 24, 1997
Title: "The Prime Ideal Spectrum of a Noetherian Ring"
Chandni Shah, Colgate University
March 12, 1997
Title: "Rellich Inequalities"
Andreas Hinz, München University
March 28, 1997
Title: "Puig's Conjecture on Blocks of Finite Groups"
Radha Kessar, Yale
March 31, 1997
"On Matroids Representable over Both GF(4) and GF(5)"
Dirk Vertigan, LSU
April 4, 1997
Title: "Global Solutions and Self-similar Solutions of Semi-linear
Evolution Equations"
Fred Weissler, University of Paris
April 24, 1997
Title: "Linearly Recurrent Systems and S-adic Systems"
Fabien Durand, Université de la Méditerranée
Aix-Marseille II
May 2, 1997
Title: "Evolution Semigroups and Stability of Time-varying Systems"
Tim Randolph, University of Missouri
May 14, 1997
Title: "Semilinear Elliptic Equations"
Djairo De Figueiredo, University of Campinas, Brazil
1995-96
September 8, 1995
Title: "On Projections of High-Dimensional Measures"
Heinrich v. Weizsacker, Kaiserlautern
September 29, 1995
Title: "Topological Graph Theory and Venn Diagrams"
Peter Hamburger, Indiana-Purdue University at Fort Wayne
October 12, 1995
Title: "The Two Squares Theorem of Fermat"
W. N. Everitt, University of Birmingham (UK)
October 19, 1995
Title: "Oscillatory Radial Solutions of Semilinear Elliptic Equations"
William R. Derrick, University of Montana
November 17, 1995
Title: "Fair-Division Problems: Cake-cutting and Convexity"
Ted Hill, Georgia Institute of Technology
December 14, 1995
Title: "Equilibrium States and Families of Multipliers"
Nicolai Haydn, USC
January 26, 1996
Title: "A Survey of Dowling Lattices"
Joseph Bonin, George Washington University
February 16, 1996
Title: "Knots and Matrices"
Gail S. Nelson, Carleton College
March 15, 1996
Title: "Convective Stability and Instability of Solitary Waves"
Robert L. Pego, University of Maryland
April 3, 1996
Title: "Implicity Theorems for Quasi Self-Similar Multifractal
Measures"
Toby O'Neil, University of St. Andrews
April 26, 1996
Title: "Subintegrality and Weak Subintegrality"
Leslie G. Roberts, Queen's University, Kingston, Ontario
1994-95
November 18, 1994
Title: "Diophantine Approximation: So Close, Yet So Far"
Edward B. Burger, Williams College
December 9, 1994
Title: "Elasticity of Factorization in Integral Domains"
Scott Chapman, Trinity University
January 27, 1995
Title: "K Theory and Cyclic Homology: What Are They and Why Should You
Care"
Sue Geller, Texas A&M University
February 3, 1995
Title: "Semigroups with Differentiable Operations"
J. P. Holmes, Auburn University
February 10, 1995
Title: "Graphs and Groups"
Margaret Morton, University of Auckland
February 17, 1995
Title: "Modular Representations of Simple Groups"
Peter Sin, University of Florida
February 23, 1995
Title: "Harmonic Measure on Fractals"
Alexander Volberg, Michigan State University
February 24, 1995
Title: "Dynamical Systems"
James Yorke, Institute for Physical Science and Technology, University
of Maryland
March 20, 1995
Title: "Fractal Geometry of Self-Avoiding Processes"
Kumiko Hatori, University of Tokyo
March 31, 1995
Title: "Flag Varieties and Exterior Powers of the Reflection
Representation"
Mark Reeder, University of Oklahoma
April 7, 1995
Title: "A Survey of Uniform Homeomorphisms between Banach Spaces"
Bill Johnson, Texas A&M University
April 12, 1995
Title: "Stability and Instability in Gases and Plasmas"
Walter Strauss, Brown University
