Talks will be at 4:00 Tuesdays with the exception of one. Room TBA. There will be refreshments before all talks in O'Connor 31 starting around 3:30pm.

- Apr. 8
Julie Glass, C.S.U. Hayward
*(We will be taking the speaker to dinner.)*Title: Lie algebras and beyond

Some of the basic notions involved in finite dimensional Lie algebra theory will be discussed. In particular, we will find a method of associating a matrix to each of a certain family of Lie algebras. With this correspondence it is possible to classify all such Lie algebras, no small feat! We will then see how a slight change in the matrix results in an infinite dimensional Lie algebra. In certain cases, there is a nice way of describing an infinite dimensional Lie algebra starting with a finite dimensional one; we will see how this works. Finally, as time permits, I will introduce the notion of a vertex operator algebra, and how to build one starting from an infinite dimensional Lie algebra. Most of this talk will be accessible to undergraduate math majors.

- Apr. 15
Tristan Needham, University of San Francisco
Title: Complex curvature

A new type of (complex) curvature can be associated with conformal mappings of the plane. This concept provides an elegant method of recognizing an harmonic function just by looking at a picture of its level curves. The talk should be accessible to any undergraduate who has taken complex analysis.

- Apr. 22
Alice Kelly, Santa Clara University
Title: Women of science and science on women

Who are they- what have they done - why did they do so much or so little. A look at the lives of some women who have made significant contributions to the sciences and how the sciences contributed to their lives.

- Apr. 24,
**Thursday**Peter Stevenhagen, Universiteit van AmsterdamTitle: . Squares and triangles in number theory

This elementary talk deals with natural numbers. I will discuss various methods for finding the most beautiful among these numbers. It is the task of the audience to find the most beautiful among the methods.

- Apr. 29
Bruce Sagan, M.S.R.I. and Michigan State University
Title: Going up and down with lattice paths

A sequence a_0,a_1, ... ,a_n is unimodal if there is an index m such that a_0 <= a_1 <= ... <= a_m >= a_{m+1} >= ... >= a_n. Such sequences arise in combinatorics, algebra, and geometry. For example, the nth row of Pascal's triangle (n choose 0), (n choose 1) , ... , (n choose n) is unimodal. We show how to give combinatorial proofs of various unimodality results using lattice paths, i.e, paths in the plane such that the endpoints of the edges of the path have integer coordinates. This talk will be quite elementary and accessible to undergraduates.

- May 6
Glenn Appleby, Santa Clara University
Title: d theta/dt, or, Raters of the Lost Arc

This talk will present some surprising mathematical developments that arise when studying the motion of a simple pendulum. We will encounter different points of view regarding just what a ``solution'' to this problem should comprise. The talk will be accessible to those who believe they know what a derivative is.

- May 13
Everett Howe, Center for Communications Research
Title: Real polynomials with all roots on the unit circle

Consider the set of real polynomials of degree 2n all of whose roots are non-real complex numbers on the unit circle. What can one say about the set S of coefficients of all such polynomials? I will present joint work with Stephen DiPippo in which we calculate the volume of the set S and find a very simple-to-describe subset of S. If time allows, I will indicate how real polynomials of this type arise in the study of abelian varieties over finite fields.

- May 20
Kevin Buzzard, U.C. Berkeley and Cambridge University
Title: The magic of modular forms

Modular forms are differentiable functions which satisfy certain ``periodicity properties'' (for example, f(z+1)=f(z)). They were initially in vogue early on this century, but the strangest and most interesting results about them were proved, not by analysts, but by the Indian number theorist Ramanujan, who made some surprising observations about the coefficients in the Taylor expansions of certain modular forms. For example, Ramanujan spotted that the coefficients in the Taylor expansion of the modular form called Delta(z) were all integers and satisfied some surprising congruences modulo the prime 691.

These kinds of observations were presumably thought of, at the time, as mere curiosities, and interest in modular forms waned, until the late 1960s when Serre, Shimura and several other number theorists proved theorems and made conjectures stating, broadly speaking, that a huge amount of number-theoretical patterns and data lay hidden in the Taylor expansions of these analytic functions, and that in fact Ramanujan had only scratched the surface! Suddenly there was a large amount of interest in modular forms again. In the 1980s, it was even proved that Fermat's Last Theorem was a consequence of certain conjectures about modular forms, and indeed in 1993 Andrew Wiles proved enough about modular forms to be able to deduce FLT as a corollary.

In my talk I shall give the definition of a modular form, and then give several examples. Then I shall point out the strange properties that the Taylor expansions of these examples have, and I shall go on to explain some of the things we can prove, and some of the things we conjecture, about Taylor expansions of modular forms in general.

* This page maintained by Dennis C. Smolarski, S.J.
Comments should be sent to
dsmolarski@scuacc.scu.edu.
Last updated April 1, 1997. *