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.)

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 sub-triangulations 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 C² isometric (length preserving) immersion in R³. 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 {v₁,...,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 Kai-uwe 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.