
Department of Mathematics 
University of Utah
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Graduate Colloquia
Fall 2000
Wednesdays at 4:30 in LS 101
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 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 PM in LS 101, unless
otherwise noted.
 Aug 30 Klaus Schmitt
 Critical point theorems and PDE's.
Abstract:
Solutions to many PDE problems may be found as critical points of
certain associated functionals. In the lecture I shall discuss
some critical point theorems (a minimization theorem, a mountain
pass theorem, and a saddle point theorem) and give applications to
nonlinear PDE's.
 Sep 6 David Eyre
 A geometric method for calculating traveling waves in coupled
systems of diffusive equations
Abstract:
Geometric equations to find dispersive traveling waves of coupled
reaction diffusion equations are derived. The waves considered
here consist of a state space connection between critical points of
the equation's nonlinearity, and a physical space traveling wave
that takes values on the state space connection as a function of
physical space and time. This talk will describe a method that
decouples the process of calculating these two features. This
decoupling is accomplished with a natural change of variables that
utilizes a curve defining the state space connection. The change
of variables results in a scalar reaction diffusion equation
describing the traveling wave on the curve, and a two point
boundary value problem that describes the curve.
 Sep 13
Abstract:
 Sep 20 Peter Alfeld

Multivariate Splines And The Four Color Map Problem
Abstract:
This talk describes an approach to solving a particular
multivariate spline problem using the same techniques that were
used to solve the four color map problem. Thus we construct an
unavoidable set of subtriangulations using a discharging
technique. The ideas are illustrated by proving a simpler result by
the four color map techniques. That result, however, can also be
obtained by simpler means. The work described here is very
tentative but it does illustrate a perhaps unexpected connection
between multivariate splines and the four color map
problem.
 Sep 27 Andrejs Treibergs
 Isometric embedding of negatively curved surfaces
in Euclidean Space
Abstract:
In 1901, Hilbert proved that the hyperbolic plane doesn't admit a
C^{2} isometric (length preserving) immersion in
R^{3}. This led geometers to ask which manifolds can
be isometrically embedded. A complete surface with curvature
bounded above by a negative constant admits no such immersion
(Efimov, 1963) whereas a complete surface whose negative curvature
is allowed to decay has such immersions (Hong, 1993.) These
geometric problems are unusual in that a hyperbolic PDE plays an
essential role. After developing some geometric background, we
shall sketch the proof of Hilbert's theorem and comment on the
proofs of the others.
 Oct 4 Bob Guy
 Modeling HIV Infection Dynamics Using Delay Equations
Abstract:
I will present various models for the spread of HIV infection in a
population of cells. One interesting model involves delay
differential equations. I will provide a short introduction to delay
equations, discuss why they are needed in the HIV model, and briefly
discuss numerical methods for solving dde's.
 Oct 11 Robert Hanson
 Reconstructing a graph from a collection of subgraphs
Abstract:
F. Harary delivered a lecture in 1963, presenting the very
interesting idea of reconstructing a graph from a collection of
subgraphs. He lets G be a graph with p vertices labeled
{v_{1},...,v_{p}}. He defines the subgraph
G_{i} as G  v_{i} for each i = 1,...,p. Harary's
problem is to reconstruct G using the set {G_{i}  i =
1,...,p}.
 Oct 18 David Kebler
 Perturbation and Multiple Scale Techniques applied to the
diffusion equation with a sink
Abstract:
Perturbation methods are used to determine under what boundary
conditions on the sink a system of coupled diffusion equations may
treated independently. Thus so the method of multiple scales is
applied to determine under what conditions fast/slow, small/large
scales are seperated near and away from the sink.
 Oct 25 Renzo Cavalieri
 Algebraic curves defined over \bar{Q}. An introduction to
Grothendieck's theory of dessins d'enfants.
Abstract:
The first part of the talk will give a basic introduction to
Grothendieck's theory of "dessins d'enfants", a geometric construction
that puts in a somewhat bijective correspondence algebraic curves
defined over /bar{Q} and some elementary graphs defined over
topological compact oriented surfaces; such graphs were named by
Grothendieck dessins d'enfants, children drawings, "as a scribble of a
child on a piece of paper is a perfect explicit example of them
(Esquisse) ".
In the second part of the talk we will discuss a possible application,
originarily conceived by G.B Shabat and V.Voevodsky, of this
theory. We'll use the combinatorial data of the dessin to construct a
sequence of abelian varieties that converge to the Jacobian variety to
the curve.
 Nov 1 Kaiuwe Bux
 Ouch, my ruler is too short  topics in projective geometry .
 Nov 8 Brynja Kohler
 A mathematical model of the stretch reflex and the muscular
tremor clonus.
Abstract:
The stretch reflex is most dramatically demonstrated with a tap to the
tendon below the knee cap which causes a quick jerk of the lower leg,
however, its significance is more subtle  it is important for
maintaining muscle tone, posture, and dynamic control of muscle
length. Certain types of neurological injury can result in a
disruption of this reflex in such a way that an involuntary muscle
oscillation or tremor occurs known as clonus. I will present a
mathematical model of the each of the components of the reflex: the
motorneurons, the muscle dynamics, and the muscle spindle which is the
sensory apparatus of muscle, and show some numerical solutions of the
healthy stretch reflex and some cases of clonus.
 Nov 15 Wieslawa Niziol
 Galois representations and geometry
Abstract:
Galois representations coming from cohomology of algebraic varieties
have special properties. I will try to explain what we know about them
and how this knowledge can be used to solve arithmetic geometric problems.
 Nov 22 No colloquium (Thanksgiving)
 Nov 29 Chris Staskewicz
 Self Avoiding Walks
Abstract:
We will discuss the problem of how many random walks of size n there are
which begin at the origin and do not intersect themselves. There are
applications in the physical sciences including polymers and percolation.
I will present some recent advances in the theory and discuss current
research and open problems pertaining to the theory.
 Dec 6 Andrew Oster
 A Brief Introduction To Genetic Algorithms
(Plus A Concrete Example On A Graph Coloring Problem)
Abstract:
The concept of how a genetic algorithm works shall be discussed. A survey
of some of the types of problems that genetic algorithms are used
to solve will also be given. We will then examine the primitive genetic
algorithm approach my team and I took on a graph coloring problem during
the Mathematical Competition in Modeling 2000. During the talk, I will
attempt to keep the coding jargon to a minimum. So the talk should
be quite accessible to all.
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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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