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, PSL(2,C)⁺, is Olshansky's semigroup. We will start by describing how PSL⁺ and q-invariant are related. Then, we will classify conjugacy classes of PSL(2,C)⁺: it turns out the conjugacy relation is the same as the q-invariant on PSL⁺. We will then derive character formulas of PSL⁺ which, again, are closely related to the q-invariant. If time allows, we will extend the characters of PSL⁺ to PSU(1,1) by analytic continuation. The extended characters are exactly Harish-Chandra character of PSU(1,1).

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.