Graduate Student Seminar, Summer 2003 (June 11 - July 30)

Schedule of talks

Wednesday 2:30, room 302 Boyd Graduate Studies Research Center

Organizers: Carrie Wright and Kenyon Platt

Michael Guy, "Going up, going down: the ideal story"

Abstract: Rings are a fundamental topic of study in many algebraic areas (and ALWAYS occur on qualifying exams). One property people often ask as a first question about a ring is what are some basic properties of its ideals. One property that turns out to be quite useful in proving many basic results about a ring is contained in information concerning chains of ideals. Two seemingly dueling notions are the ascending chain condition (ACC) and the descending chain condition (DCC). What is this, and are these two conditions related? These two ideas will be discussed along with what may be a surprising implication of the DCC.

Kai Laemmle, "Cryptography with curves"

Abstract: What do you do if you want to send a message to your sweetheart, but nobody else should be able to read the message?

Encrypt your message, with elliptic curves.

How can you do encryption with elliptic curves? What are elliptic curves? Why do we use elliptic curves? Are elliptic cryptosystems more secure? And what does this have to do with the Discrete Logarithm problem (whatever that is)? How can we attack elliptic curve cryptosystems? Questions and questions.... Do you want answers? Do you know what abstract groups, modular arithmetic and cubic curves are? Come to my talk to find out.

Eric Pine, "(Almost) nothing but cubes"

Abstract: It's amazing how many interesting problems arise from simply looking at sums of cubes of integers. I'll discuss a couple of the problems I've worked on during my time at UGA, as well as give a survey of many other related results. Most of the talk will concern elementary number theory problems, so much so that "a = b mod n" is about as advanced mathematically as the talk will get. These problems are just the kind which have drawn many (including myself) into the study of number theory.

Phil Zeyliger (of Bob Rumely and Matt Baker's undergraduate research group), "Lost in 3-space: Electrical circuits and Polya's theorem on random walks"

Abstract: We will sketch the proof of Polya's theorem that a random walk on an n-dimensional lattice returns to the origin with probability 1 for n = 1 or 2, but escapes to infinity with non-zero probability for n greater than or equal to 3, by using intuition from classical circuit theory.

Carrie Wright, "Extensions of the Hill cipher "

Abstract: The Hill cipher is a cryptographic method that was developed in 1929 using matrices. In studying the Hill cipher, some questions come to mind: What happens when you compose two Hill ciphers -- is it any more secure? And there are many more questions to be asked and answered.

Joe Rusinko, "Lehmer's conjecture"

Abstract: Lehmer conjectured that the Mahler measure of a polynomial is bounded away from zero for all polynomials which are not a product of cyclotomic factors. In this talk, theorists will have an opportunity to learn what Mahler measure measures. Computer folks will learn where to look for polynomials with small Mahler measure. Finally, gamblers will learn whether or not to bet that the conjecture is true.

Kenyon Platt, "Pickin' up good vibrations: Molecular vibration and representation theory"

Abstract: Newton's Second Law of Motion allows us to write a second order differential equation which, if solved, would give the displacement of the n atoms in a molecule under internal forces. The problem is that this differential equation involves a 3n x 3n matrix A. It turns out that solving the differential equation amounts to finding the eigenvalues and eigenvectors of A. This, however, is an unwieldy task if there are very many atoms in the molecule. In this talk, I will illustrate how representation theory of the symmetric group can be used to solve the problem.

Emille Davie, "Characterizing continuity by preserving compactness and connectedness"

Abstract: Let us call a function from X to Y preserving if the image of every compact set is compact and the image of every connected set is connected. By elementary theorems, a continuous function is certainly preserving, but when does the converse hold? The aim of this talk will be to give a treatment of the case when X is a Frechet space with its coarsest topology, the cofinite topology.