Once, many moons ago, I attended a number theory colloquium and within the first five minutes I was lost. Even though this is a regular occurance for me, this particular time was due soley to my lack of understanding of inverse limits and profinite groups. In this colloquium, it is my intention to explain to all those who attend these relatively simple concepts so that they may understand at least the first ten minutes of a number theory colloquium! In short, I will explain the main differences between infinite and finite Galois Theory highlighting them in an example, and then I shall explain new methods for evaluating the Galois Group of an infinite extension using the afore mentioned inverse limit. A basic understanding of Galois Theory is all you will need.
Jeff has returned to us from D.C. and will tell us about his trip and how
to use the annual AMS meeting to find a job.
A number of you have probably seen the classical theta function defined
either on the real numbers or the complex upper half plane. What if you
wanted to define a theta function in a more general context, like in the
case of a function field of an algebraic curve for example. I'll tell you
a little bit about a more general theta function defined by Andre Weil and
how it can be used to solve this problem. I'll also show a few specific
examples that are relevant to my research. This talk will use some number
theory and representation theory. However, since I plan to basically omit
all the details, the talk will be introductory in nature.
The classical ecological theory of competitive exclusion states that
strong competition between 2 different species for the same resource will
inevitably lead to the extinction of one or the other. This theory assumes
that, left alone, each species will reach an equilibrium state. I will
derive the above conclusion mathematically, and then consider the
possibility of coexistence of 2 competing species which exhibit periodic
or chaotic dynamics.
In this talk, we will define the notions of both an elliptic function and
an elliptic curve (over the complex numbers), and show the interplay
between the two. We will look at a particular elliptic function, the
Weierstrass p-function, and show how it can be used to build an
isomorphism between an elliptic curve and a complex torus. Our results
will be even more surprising, as we will show a one-to-one correspondence
between the collection of elliptic curves and the set of complex torii.
Last semester when I was talking about factoring polynomials over Q, I used
the factorization over finite fields without justification.
Now I will go back and fill in this hole, focusing on Berlekamp's
algorithm which dates back to the late 1960's. I will also mention an
improvement to this algorithm developed in 1980 by Cantor and Zassenhaus.
In this talk some ideas and examples concerning a construction of a natural
symplectic structure on the space of representations of the fundamental group
of a surface into a matrix Lie group are reviewed. The general line of
reasoning follows the paper 'The Symplectic Nature of the Fundamental Groups
of Surfaces', Adv. in Math. 54, 100-225 (1984), by William M. Goldman.
I will briefly explain the method of Inverse Scattering for solving the
Korteweg de Vries equation. The solutions found by this method are not the
most general solutions of KdV. For periodic solutions to KdV Inverse
Scattering no longer directly works. To find periodic solutions we will need
to consider Schrodinger's Operator with a periodic potential, called Hill's
Operator. Roughly speaking the problem is to find potentials of this operator
from given information about the spectrum. I will review known results and
some terminology and then find solutions to the inverse problem in a specific
case.
I will talk about vector bundles and explain the classification of vector bundles over some schemes, e.g. curves, surfaces....
I will discuss the stanard construction of John Conway's sureal
numbers, and then discuss some of their basic properties and
operations. I will do some other stuff too, but Virgil's deadline
is coming to quickly for me too decide. In any case it will be nifty.
Given a ring \A, one can construct a topological space, called
Spec(A). This is a basic construction in Algebraic
Geometry. We'll see how to do it and look at lots of examples, focusing on
the interplay between the algebraic structure of the ring and the
resulting geometry of the topological space. The prerequisites for this
talk will be minimal: a basic knowledge of rings and topological spaces.
The first year graduate students are welcome and encouraged to attend.
PSL(2,C)+ is a semigroup sitting in the Lie group PSL(2,C). Its closure,
In this talk, I will discuss the research that I did with a fourth and fifth grade class from August 1999 - February 2000. By introducing the
students to mathematical games which emphasized different areas of
mathematics, they were able to design and create their own mathematical games.
From this, they broadened their scope on what constitutes mathematics as well
as developed their problem-solving skills. In addition, I will introduce you
to a game called SET, that I played with four students from the class. SET is
a game that requires skill at quick thinking and recognizing patterns. Using
this game, I tried to ascertain how these students are thinking logically. If
time permits, I will also discuss some ideas in which you can use mathematical
games in your classroom.
In this last colloquium of the semester we will use Java applets and
interactive software to explore some of the basic principles underlying
Hyperbolic Geometry. The talk is intended for a general audience.
Teachers and undergraduates at all levels are strongly encouraged to
attend.
January 19
Speaker - Aaron WOOTTON
Title - Infinite Galois Extensions and Profinite Groups
January 26
Speaker - Jeff EDMUNDS
Title - Annual AMS Meetings and How to Find a Job
Febuary 2
Speaker - Jeff CUNNINGHAM
Title - Defining a Generalized Theta Function
Febuary 9
Speaker - Jeff EDMUNDS
Title - Competitive Exclusion and Coexistence
Febuary 16
Speaker - Christopher RASMUSSEN
Title - Relationships Between Elliptic Curves and Elliptic Functions
Febuary 23
Speaker - Thomas HOFFMAN
Title - Factoring Polynomials Over Finite Fields
March 1
Speaker - Robert LAKATOS
Title - Symplectic Structure on the Space of Representations of the Fundamental Group of a Surface
March 8
Speaker - Virgil PIERCE
Title - An Inverse Spectral Problem
March 22
Speaker - Seog Young KIM
Title - Something About Vector Bundles
March 29
Speaker - Frederick LEITNER
Title - Happy Days for the Rest of Our Lives
April 5
Speaker - Susan HAMMOND-MARSHALL
Title - How to Turn a Ring into a Topological Space
April 12
Speaker - Jailing DAI
Title - Conjugacy Classes, Characters of PSL(2,C)+, and q-Invariant
April 19
Speaker - Laura KONDEK
Title - Playing Games in the Mathematics Classroom
May 3
Speaker - Guadalupe LOZANO and Jeff SELDEN
Title - Visual Hyperbolic Geometry - The Axiomatic Approach