
Department of Mathematics 
University of Utah
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Graduate Colloquia
Spring 2001
Wednesdays at 4:30 in JWB 335
Math 6960
The goal of this colloquium is to encourage interaction among
graduate students, specifically between graduate students who are
actively researching a problem and those who have not yet started
their research. Speakers will discuss their research or a related
inrtoductory topic on a level which should be accessible to
nonspecialists. The discussions will be geared toward graduate
students in the beginning of their program, but all are invited to
attend. This invitation explicitly includes undergraduate students.
Talks will be held on Wednesdays at 4:30 in JWB 335, unless
otherwise noted.
Tentative speakers for this semester (not yet scheduled):
Aaron Fogelson
Jim Carlson
Brad
Frank
 Jan 16 Eric Cytrynbaum
 The Poincare Index Theorem and a swinging pendulum
Abstract:
The Poincare Index Theorem gives some concrete constraints on the
structure of vector fields, particularly, the nature of its critical
points. In this talk, I will sketch a proof for two dimensional
manifolds and use some examples from physics to illustrate the theorem
including a discussion of several bifurcations for a swinging pendulum.
 Jan 24 Angelo Vistoli
 Two open questions in the theory of finite dimensional
division algebras
Abstract:
A division algebra is a nonzero associative ring with identity, in which
every nonzero element has an inverse; of course commutative division
algebras are known as fields. The theory of division algebras is one of
the most attractive parts of algebra, and it has deep connections with
algebraic geometry and number theory.
The algebra of quaternions is the only noncommutative division algebra of
which the average mathematician is aware; in fact there are very many
examples. The basic problem in the theory can be stated as follows: is it
possible to describe all division algebras? Of course the answer is no,
because even describing all fields is a hopeless task. So we will
reformulate it: if we knew all fields, could we describe all division
algebras? This is a much more reasonable mathematical question.
I am going to explain some of the basic results on finite dimensional
division algebras, and two of the outstanding open questions. Most of the
talk should be accessible to anyone who is comfortable with the definition
of a field; to understand one of the two problems you will also need to
know what a Galois extension is.
 Jan 31 Misha Kapovich
 How to straighten a carpenter's ruler
Abstract:
Let L be an embedded finite polygonal chain in the the plane (the
"carpenter's ruler"). Can one deform L through embedded polygonal
chains (preserving the lengths of the edges) to a polygonal chain
contained in a straight line? This question known as the "Carpenter's
Ruler Problem" was first asked by Stephen Schanuel in the 1960s,
Kirby eventually included it in his list of problems in
lowdimensional topology. I will explain a recent solution of this
problem by Connelly, Demaine and Rote.
 Feb 7
 Title
Abstract:
 Feb 14 Emina Alibegovic
 The BanachTarski Paradox
Abstract:
Stated popularly, the BanachTarski paradox says that it is possible
to cut a pea into finitely many pieces which can then be rearranged to
form a ball of the size of the sun. We will look at this paradox from
a point of view of group theory. The talk will be accessible to all
first year graduate students.
 Feb 21 KaiUwe Bux
 Dead Languages, Killing Fish, and Exhausted Kangaroos
Problems and Metaphors in Combinatorial Optimization
Abstract:
Last millennium, we heard about genetic algorithms. However, it is
always good to have many weapons at hand, Thus, I will present two
other methods of combinatorial optimization based on more geometric
metaphors.
The optimization problem I will focus on arose in comparative
linguistics: given a list of cognates in two related languages (in out
case: avestan and vedic, which died long ago), find a measure for the
quality of this list. As typical for problems coming from outside
mathematics, it is awfully vague. I will discuss one way to turn it
into a well defined question to which discrete optimization can be
applied.
 Feb 28 Renzo Cavalieri
 Approximating Jacobians...maybe...
Abstract:
Jacobians are a very classical and fascinating topic in the study of
algebraic curves. You can associate to any Riemann Surface its
Jacobian, which is a complex torus; the amazing thing is that given a
Jacobian and a little extra information you can pin down exactly the
curve you started with. In the first part of the talk we'll get
acquainted with this construction.
Then, I'd like to present an idea by G.B. Shabat and V. Voevodsky
(~1990), on how to use the combinatorial data of a dessin d'enfant on
an algebraic curve over Qbar to create a sequence of abelian varieties
supposedly converging to the Jacobian variety of the curve. I'd also
like to point out some aspects of this construction that to me are
still quite obscure.
 March 7 Paul Bressloff
 Synchronization in networks of coupled oscillators
Abstract:
The Kuramoto model describes a large population of coupled limitcycle
oscillators with a prescribed distribution of natural frequencies. It
has been used to study the phenomenon of collective synchronization in
a variety of biological systems (neural oscillators, flashing
fireflies, circadian pacemaker cells, cardiac pacemaker cells) and
physical systems (laser arrays, superconducting Josephson junction
arrays). We survey some of the mathematical approaches to
understanding the onset of synchronization, which occurs when the
coupling strength exceeds a critical threshold. This draws from a
broad spectrum of ideas in mathematical biology, statistical physics,
kinetic theory, bifurcation theory and plasma physics.
 March 14 No colloquium (spring break)
 Title
Abstract:
 March 21 Annual Graduate Student Meeting
 Title
Abstract:
 March 28 Martin Deraux
 Hyperbolic Tessellations
Abstract:
I will give a basic introduction to the hyperbolic plane, and consider
some questions related to its tessellations by triangles.
For instance, it is an easy exercise to determine which Euclidean
triangles tessellate the Euclidean plane; I will discuss the
corresponding question in hyperbolic geometry.
 April 4 Nick Cogan
 The BelousovZhabotinskii reaction
Abstract:
 April 11 Andrew Oster
 Population Projection Via Computer Simulation
Abstract:
Over the last two years while attending Cal Poly, I worked on a
project for the reintroduction of the California Condors back into the
wild. We developed a computer simulation program to help accomplish
this task. I will discuss the biology behind our model and the
mechanics of the program. Also, I will give some insight on the
implications of the output.
 April 18 Cord Erdenberg
 Evolutionary Algorithms
Abstract:
Evolutionary Algorithms are a class of search algorithms which is inspired
by evolutionary change in populations of individuals. Although the first
origins can be found in the early 1960s, the area of Evolutionary
Algorithms is still quite young. They have been applied successfully to a
broad variety of problems from many different fields. This led to a
strongly practical oriented interest. Thus, today, theory is far behind
"experimental knowledge". A solid theory has to be built...
We will analyze the behavior of a simple Evolutionary Algorithm, the so
called (1+1)EA, on linear and on unimodal functions. Several examples
using plateaus of constant fitness will illustrate how small changes to
this algorithm can influence its behavior significantly.
 April 25 Tom Robbins
 How in the hell can you say anything about that!
Abstract:
There have been numerous developments in the modeling of systems that
seem to obey the age old phenomenon in which "What goes up, must come
down". In comparison, very little is known about systems in which
stochastic factors often govern the evolution of the system. In this
talk, I will outline several deterministic models and recent advances
stochastic models that have been used to model the spread of
biological systems. In addition, I will outline a hybrid
mathematicalnumerical model to describe spread of terrestrial plant
species. To test this model, I will propose several ways in which the
model can be used to describe the reintroduction of vegetation at
Mount St. Helens in Washington state. At the end of the talk, I will
"Drop my pants" and ask the question, "What the hell can you say about
that!". [I believe this is supposed to be read as "I will expose my
ignorance to the careful proding of well thought out questions".]
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Department of Mathematics
University of Utah
155 South 1400 East, JWB 233
Salt Lake City, Utah
841120090
Tel: 801 581 6851, Fax: 801 581 4148
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