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.
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, Universitŕ 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
