2002-2003
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.
September, 18, 2002
Title: "The Normalized Mean Curvature Flow
for a Small Bubble in a
Riemannian Manifold"
Nicholas Alikakos, University of North Texas
2001-2002
April 12, 2002
Title: "2000 Proof of the Double Bubble Theorem"
Frank Morgan, Williams College
Abstract: A round soap bubble is the most
efficient, least area shape for enclosing a given volume
of air. The Double Bubble Theorem, proved in 2000, says
that the double soap bubble, formed when two bubbles come
together, provides the least area shape for enclosing and
separating two given volumes of air. Undergraduate students
contributed to the proof of this theorem.
June 6, 2002
Title: "Squaring the Circle"
Miklós Laczkovich, Eötvös Loránd
University, Budapest, Hungary
Abstract: In 1925, A.Tarski asked whether or not
the disc can be decomposed into finitely many pieces and
the pieces can be moved to get a decomposition of a square.
We outline the history and the solution of Tarski’s
problem, and give a review of some recent developments and
of some open questions.
2000-2001
January 19, 2001
Title: "The spectral boundary of complemented
invariant subspaces in LP
(R)"
Zoltan Buczolich, Eotvos Lorand University, Budapest, Hungary
Abstract: This is a joint work with A.
Olevskii. It turns out that in the study of translation
invariant subspaces of LP(R) a certain, so called, "universal
coloring" property of the complementary intervals of
Cantor sets plays an interesting role. Related to this coloring
property a concept of arithmetic thickness is defined. Arithmetically
thick sets are large in the sense that they cannot be spectral
subboundaries. First we show that there exist arithmetically
thick sets which are of zero Hausdorff dimension then we
show that arithmetcially thick sets should contain arbitrarily
long arithmetic progressions.
February 28, 2001
Title: "Reflection groups and Semi-invariants"
Anne Shepler, University of California at Santa Cruz
Abstract: The symmetry group of the dodecahedron
consists of rotations and
reflections in R3 and is an example of a reflection group.
A reflection is a linear transformation of Rn or Cn which
has finite order and fixes a hyperplane (the mirror")
pointwise. A finite group which is generated by reflections
is called a reflection group. Reflection groups acting on
Rn include the Coxeter and Weyl groups. Reflection groups
acting on Cn include the symmetry groups of complex regular
polytopes.
The action of the group on the vector space induces an
action on the set of polynomials, vector fields (derivations),
and differential forms. We discuss objects that are invariant
under the group action and also objects that change by a
scalar when the group acts. Such "semi-invariants"
are invariant with respect to a linear (degree one) character
of the group.
March 2, 2001
Title: "Turing patterns in the CIMA reaction
diffusion system"
Moxun Tang, University of Minnesota
Abstract: In one of the most important
papers in theoretical biology of last century, Turing predicted
in 1952 that chemicals can react and diffuse in such way
as to destabilize a homogeneous stationary state, and result
in inhomogeneous spatial patterns. He suggested that this
mechanism could play a major role in biological pattern
formation. In spite of significant efforts devoted to studies
of his theory, the first experimental evidence for Turing
structures was only observed in 1990 by De Kepper and coworkers
in the chlorite-iodide-malonic acid-starch (CIMA) reaction.
In this talk I will describe some fundamental properties
of Turing patterns, through our mathematical analysis for
the Lengyel-Epstein model, a two-variable reaction diffusion
system which captures the crucial feature of the CIMA reaction.
Several analytic results concerning the global dynamics
of this system, especially our theorem showing existence
of various Turing patterns observed in the laboratory and
in computer simulations, will also be presented.
March 5, 2001
Title: "Abstract homomorphisms of algebraic
groups"
Lucy Lifschitz, Tufts University, Massachusetts,
Abstract: We study abstract homomorphisms
of simple isotropic algebraic groups (e.g. SL_n - the group
of n x n matrices with determinant 1) . It was conjectured
by A. Borel and J. Tits in 1973 that all such homomorphisms
come from a homomorphism of the underlying field and a rational
homomorphism of algebraic groups. They proved this conjecture
for the case when the image group is reductive. We take
up the case of the non-reductive image and prove the conjecture
for split and quasisplit groups. We also give an explicit
description of all field homomorphisms and rational maps
arizing in this situation.
March 7, 2001
Title: "Random Probability Distributions and Average-Optimal
Stopping"
Pieter Allaart, University of North Texas
Abstract: In a typical no-information optimal stopping problem,
one observes a finite sequence of random variables with
unknown distributions, and the objective is to find a stop
rule that maximizes the expected return on the average,
over all possible distributions. To make the term "average"
precise, it is assumed that the distributions are generated
at random according to some measure on the space of all
possible distributions. Thus, to analyze this optimal stopping
problem it is essential to study natural constructions of
random probability distributions. Several such constructions
will be described, with special emphasis on a method by
Dubins and Freedman (1967). An important aspect of a random
probability is the distribution of its mean M. I will show
how to calculate the moments of M for the Dubins-Freedman
construction. The distribution of M can then be approximated
using a classical inversion formula.
March 12, 2001
Title: "The Status of Mathematics Reform"
Bowen Brawner, University of Texas at Austin
Abstract: The presentation will offer
evidence of the current state of mathematics education,
comparing the performance of U.S. students with our international
economic competitors and reporting recent trends in nationwide
and statewide assessments. These results, coupled with the
need for mathematics literacy in today's world and the necessity
for equity with excellence, provide the impetus for the
current reform initiative. Algebra is recognized as the
gateway to higher mathematics and is the core course in
the secondary curriculum. The centrality of the function
concept, the role of problem solving, and the use of technology
are the focus for the algebra reform debate. The effects
of a function-based approach to algebra will be discussed.
March 16, 2001
Title: "The Effects of Teaching Self-questioning
Skills and Using Activity-based Instruction to Improve Mathematics
Instruction"
Marvin Harrell, Emporia State University
Abstract: In the 1830s, Warren Colburn
stated: By the old system, the learner was presented with
a rule which told him how to perform certain operations
on numbers and when this was done he would have the correct
answer. But no reason was given for a single step. When
he was through and had the answer, he neither understood
what it was nor what use it was. As he began in the dark,
so he continued and the results of his computations seemed
to be obtained by some magical operation rather than reason.
Similar sentiments have been echoed over time and serve
as the basis for many recent national reports. These reports
urge educators to teach students how to think about and
solve nonroutinue problems. This talk will summarize the
use of self-questioning skills to help regulate one’s
thinking to improve the mathematical performance of undergraduate
students. In addition, activity-based instruction will be
discussed as a way to improve mathematics instruction. Activities
appropriate for students in grades 5-9 will be examined.
1999-2000
Dec. 10, 1999
Title: "Combinatoric and Topological Toolsin
1-D Dynamics"
Karen Brucks, University of Wisconsin-Milwaukee
Abstract: In the 1980's F. Haufbauer and
G. Keller defined Kneading Maps and Hofbauer Towers to investigate
'Quadractic maps without asymptotic measure'. These tools
have since been used to obtain many results in 1-D dynamics.
This talk will be a survey talk focusing on recent results
in 1-D dynamics where these tools are used.
Dec. 14, 1999
Title: "Mixing Properties in Dynamical Systems
and Localised Return Times"
Nicolai Haydn, University of Southern California
Abstract: Uniformly hyperbolic dynamical systems are known
to have very strong mixing properties with respect to a
natural class of invariant measures. This is expressed by
the fact that correlation functions decay exponentially
fast. Similar results apply to rational maps which, due
to conformality, behave in may respects like expanding maps.
We link strong mixing properties to statistical distribution
of return times on small neighbourhoods of points. We have
developed a general scheme that allows to prove that normalised
return times to small neighbourhoods are Poisson distributed
in the limit when the neighbourhoods shrink to single points.
1998-1999
March 29, 1999
Title: "Something Old and Something New and
Something to Do"
Paul Humke, Saint Olaf College, Northfield, Minnesota
