Fall 2004 GSS Schedule

Organizers: John Chrispell and Todd Michel

Date Speaker Topic Abstract
August 30 Christine Kraft ,
Nate Drake ,
Timothy Flowers ,
Shannon Purvis 		Life in the Department of Mathematical Sciences In this talk, graduate students with experience in the M.S. and Ph.D. programs in the department of Mathematical Sciences at Clemson will discuss some of their experiences at different levels and how they differed from their expectations. Christine Kraft is a second-year master's student here in the department. Nate Drake recently completed the M.S. degree and has now entered the Ph.D. program. Timothy Flowers is a Ph.D. student who has completed the qualifying examinations. Shannon Purvis is also a doctoral student.

The main purpose of this talk is to promote candid discussion among new and experienced graduate students about the realities of graduate school here.
September 6 Shannon Purvis Partition Functions Much research has been done in the area of partition functions. In this talk, we will discuss k-ary partition functions, a class of restricted partition functions. We will talk about previous work done in this area and many questions that arise from this work.
September 13 Robert Beeler The History of e While e is not as famous as its geometric cousin, pi, it has an interesting history of its own. The acceptance of the infinite which allowed for the development of calculus also allowed for the discovery of e, the first number defined by a limiting process. The importance of the constant was emphasized by Jakob Bernoulli, who showed why it should be considered as the "natural" base for the logarithms, and Leonard Euler, who showed the relationship between the trigonometric and the exponential. The impact of other mathematicians such as Napier, Briggs, DeMoivre, Hermite, and Cantor may also be discussed with regard to their contributions to the history of e.
September 20 Sarah Graham Error-Correcting Coding Theory: Affine Variety Codes with an Improved Bound In the classic communication model, when a message is sent from one location to another, interference can distort the message that is received. Error-correcting coding theory is a branch of mathematics that reduces the problem of interference in data transmission. Using mathematical techniques, the data can be encoded and decoded in a way that small errors can be detected and corrected. The size of the errors corrected depends upon the minimum distance of each code. This talk will give a brief introduction to coding theory and affine variety codes as well as give information a new bound on minimum distance and several families of codes that tie the best-known minimum distance for codes of fixed dimension.
October 4 Tim Flowers Factoring Large Integers: An Overview of the Quadratic Sieve In 1977, Martin Gardner's ``Mathematical Games" feature in Scientific American posed the challenge of factoring a 129-digit number (known as RSA-129). Using the resources available at the time, it was estimated that it would take 40 quadrillion years to factor this number. However, in 1994, Lenstra, et.al. announced a factorization of RSA-129. Using many computers, it took them eight months to complete this groundbreaking calculation. They used a modified implementation of the Quadratic Sieve which had been originally introduced by Carl Pomerance in 1982.

The goal of this talk will be to give a general overview of the Quadratic Sieve algorithm. Special attention will be given to the elementary number theory and linear algebra techniques used in the algorithm.
October 11 Todd Michel Thrust that you can trust. In my past life I was a mechanical engineer for a company called Pratt & Whitney which makes jet engines for commercial and military aircraft. I will discuss how a jet engine works and introduce all the major components of a jet engine. I will also detail some of the differences between a commercial engine versus a military jet engine.
October 18 Mark Liu Modern Economics and Mathematics In this talk, I will discuss the history of economic thought and its natural extension to mathematics. I will focus on the development of modern microeconomics, especially on some of the works by Gerard Debreu, the 1983 Nobel Laureate in Economics.
November 1 Fall Break
November 8 Bryan Faulkner Selmer Group Given any elliptic curve E, E(Q) is the ordered pairs on E with rational coordinates. It is known that E(Q) is a finitely generated abelian group. Currently there is no know algorithm to the find generators of E(Q). But, the closely related Selmer group is ``effectively computable" and can be used to bound the rank of E(Q). I will define the Selmer group and dicuss how it may be computed in the case E : y^2 = x^3 - n^2x.
November 15 Robert Beeler Problems Relating to the Chromatic Decompositions of Graphs Building on the recent work by Robert Jamison and Eric Mendelsohn, we will investigate the particular case of decomposing a graph, $H=(V,E)$, into a family of graphs $\mathcal{K}$. Such a decomposition exists if we are able to partition the edge set $E$ of $H$ so that for each part $P$ of the partition, the subgraph of $H$ induced by $P$ is isomorphic to a graph in $\mathcal{K}$. In particular, if $H=K_{1,N}$ and $\mathcal{K}= \{ K_{1,s_{1}}, K_{1,s_{2}} \}$ with $s_{1} \neq s_{2}$ then such a decomposition is possible if and only if there exist a non-negative integer solution $(x_{1}, x_{2})$ to the associated Frobenius Problem:

s_{1}x_{1} + s_{2}x_{2} = N.

We will also show that if $I(\mathcal{D})$ is the intersection graph generated by such a decomposition, then the chromatic index, $\chi'(\mathcal{D})$ of $I(\mathcal{D})$ is equal to the sum $x_{1} + x_{2}$. The range of all chromatic indices generated in this way is the chromatic spectrum. We will also discuss questions such as:

1. The structure of the decomposition of a tree into two trees.
2. When is a set of two positive integers the spectrum of a star when decomposing into two stars?
3. When is a set of three positive integers the spectrum of a caterpillar, when decomposing into two stars?
4. The spectrum of a fat cycle and a cube when decomposing into isomorphic copies of $P_{3}$
November 22 Jang-Woo Park Discrete Logarithm and Factorization Discrete logarithm and factorization are two most famous problems in public-key cryptography. This talk discusses the relation between them.