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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 Thursdays at 3:05 PM in 104 Stewart, unless otherwise noted.Abstract. The goal of this talk is to introduce the idea of excitability through common examples as well as examples in biology. Ordinary differential equations can be used to describe excitability, and I will use phase plane analysis to understand these ODE's. The biology of the nerve and the mathematical description, again using ODE's, developed by Hodgkin and Huxley will be described. Spatial considerations, in the form of diffusion, will be added into the ODE's. I will briefly discuss concerns in cardiology such as defibrillation and atrial arrhythmias. A short video will allow visualization of numerics describing some cardiac arrhythmias.
ABSTRACT: A group G can be described by means means of generators and
relations as follows: Fix a set X:={g_1, ..., g_i, ...} of elements
in G, called "generators", closed with respect to taking inverses;
and fix a set R:={R_1=1, ..., R_j=1, ...} of identities where each R_j
is a product of generators that evaluates to the trivial element
1 in G . The pair of these two sets determines the group G up
to isomorphism provided that
Not every group admits a presentations where X is finite, and among
those that do, only a few admit a presentation where R is finite as well.
To find those finite presentations, one can make use of geometric
spaces upon which groups act nicely.
a) every element of G can be written as a product of generators, and
b) whenever a product of generators evaluates to 1 in G , this is a
formal consequence of the identities collected in R.
In this case, the pair
ABSTRACT: The Riemann sphere (the one-point compactificaton of C) is the natural geometric "model" for the field of rational functions C(t). Similarly, any finite extension K of C(t) has a geometric model, which is a compact (Riemann) surface. The genus (number of holes) of the surface is therefore a number we can assign to K. We will use these surfaces and a little topology to prove L\"uroth's theorem, that any subfield S of finite index in C(t) is of the form C(f(t)). If there is time, we will also discuss some of the considerable difficulties which arise when we try to push this circle of ideas out to fields C(t_1,...,t_d) of rational functions of several variables.
ABSTRACT: Some binary alloys exhibit a property called phase transition, which describes the spatial and temporal distribution of components of the alloy. In particular, for certain parameters, striking patterns result in short time. These patterns are dynamic on a longer time scale. The Cahn-Hilliard model of phase separation will be derived using energy methods, and several physical assumptions. Some numerical simulations will be presented as well as some typical analytic results.