Fall 2000
Algebra & Discrete Math Seminar
Time: Thursday 3:30pm
Room: Martin Hall M201

 Aug  30 None Sept   7 Robert T. Curtis University of Birmingham, U.K. Symmetric generation of finite groups I Sept 14 Robert T. Curtis University of Birmingham, U.K. Symmetric generation of finite groups II Sept 21 Kevin James Clemson University Nonvanishing theorems for elliptic curves Sept 28 Kevin James Clemson University Continued Oct    5 Anant Godbole East Tennessee State University Random Perfect Matchings, in Tandem, of the Complete Graph K_{2n} Oct  12 David Penniston Furman University Modular K3 surfaces Oct  19 Clemson Mini-Conference on Discrete Math. Oct  26 Neil J. Calkin Clemson University Counting sum-free and other sets of integers Nov   2 Morwen Thistlethwaite University of Tennessee Some applications of hyperbolic geometry in knot theory Nov   9 None Nov 16 Kelle Clark University of Virginia Bounds for the minumum weight of the dual codes of some classes of designs Nov 23 Thanksgiving Nov 30 George Fix Clemson University Minimal Homogeneous Bases for Ideals Dec   7 Robert T. Curtis University of Birmingham, U.K. Mathematics and the Art of M.C. Escher

September 7 &14, 2000

TITLE:                 Symmetric generation of finite groups  I, II

SPEAKER:              Professor Robert T. Curtis

AFFILIATION:     University of Birmingham, U.K.

ABSTRACT:

The remarkable fact that the symmetric group S_6 is capable of acting simultaneously, but non-permutation identically, on two sets of size 6 - and thus admits an outer automorphism - was used by J.A. Todd, Graham Higman and others to construct the Mathieu groups M_{12} and M_{24}.

As an easy example of the symmetric generation of groups we exhibit this automorphism in a novel and revealing manner. (In the language of Sylvester the two sets of size 6 were referred to as points and synthematic totals, the latter being rather unwieldy objects.)

We use this knowledge of S_6 to define the beautiful Hoffman-Singleton graph, and briefly explore its properties.

As a further example of symmetric generation we use the Hoffman-Singleton graph and its group of symmetries to define a new simple group, which will turn out to be the doubly-transitive group found by Donald Higman  and C.C. Sims. Its isomorphism with the group of the same order found by Graham Higman at around the same time will be demonstrated.

We conclude with a brief overview of other constructions of sporadic groups using the methods of symmetric generation.

September 21 & 28, 2000

TITLE:                 Nonvanishing theorems for elliptic curves

SPEAKER:              Professor Kevin James

AFFILIATION:     Clemson University

ABSTRACT:

In recent years, much attention has been focused on the theory of elliptic curves.  One can think of an elliptic curve as the set of rational solutions to a cubic equation in two variables.  The solutions to such an equation form a finitely generated abelian group. The torsion subgroup of such groups is well understood and therefore it remains to understand the rank of the group.  One surprising method of understanding the rank of elliptic curves is to study the behavior of their associated L-series near 1.

I will attempt to give an introduction to the theory of elliptic curves, their L-series and modular forms.  I will then present an example of a fairly elementry nonvanishing theorem.

Oct  5, 2000

TITLE:                 Random Perfect Matchings, in Tandem, of the Complete Graph K_{2n}

SPEAKER:              Professor Anant Godbole

AFFILIATION:     East Tennessee State University

ABSTRACT:

Let $K_{2n}$ be the complete graph on $2n$ vertices.  Form two complete matchings of $K_{2n}$ at random, and let $X$ denote the number of edges common to the two matchings.  Let $P_n=P(X=0)$ be the probability that the two matchings are disjoint.  We show that the distribution of $X$ can be closely approximated by a Poisson distribution with parameter 1/2, so that $P_n\approx e^{-1/2}$.  The problem is then generalized to dividing $K_{kn}$ into $n$ disjoint $k$-cliques.  Finally, for $k=2$, we explore the fine structure of the generalized cycle structure of the second matching, showing that the joint distribution of the cycle counts (up to size $b=o({\sqrt n})$) can be approximated by a Poisson process with independent components.  Comparisons are made with parallel results for derangements in random permutations, where the error bounds are
superexponentially small -- they are not in our case.

Oct  12, 2000

TITLE:                 Modular K3 surfaces

SPEAKER:              Professor David Penniston

AFFILIATION:     Furman University

ABSTRACT:

In his proof of Fermat's Last Theorem, Wiles showed that a large class of elliptic curves defined over the rational numbers is modular, and recently it has been proven that every elliptic curve over the rationals is modular (Taniyama-Shimura-Weil Conjecture).  Given this fantastic result, a natural question is what other objects are modular?"  In this talk we will focus on K3 surfaces, a natural 2-dimensional analog of elliptic curves.  In particular, we consider a one-parameter family of K3 surfaces which contains many modular examples.

Oct  26, 2000

TITLE:                 Counting sum-free and other sets of integers

SPEAKER:              Professor Neil J. Calkin

AFFILIATION:     Clemson University

ABSTRACT:

A set $S$ of positive integers is said to be {\em sum-free} if the equation $x+y=z$ has no solutions in $S$.  Cameron conjectured that the number
of sum-free sets contained in $\{1,2,\dots,n\}$ is $O(2^{n/2})$.  We will prove a slightly weaker theorem, showing that the exponent $n/2$ is correct (Cameron's original conjecture is still open).  We will then show how the techniques used here can be applied to other types of sets defined by forbidden equations, and will conclude with some open problems.

Nov 2, 2000

TITLE:                 Some applications of hyperbolic geometry in knot theory

SPEAKER:              Professor Morwen Thistlethwaite

AFFILIATION:     University of Tennessee

ABSTRACT:

The power of hyperbolic geometry as a tool for 3-dimensional topologists has only come to light within the last 20 years or so.  In this talk we'll look at some applications of hyperbolic geometry to the problem of classifying knots. In particular, the canonical cell decomposition will be  described; this has rightly been described in the literature as a magic wand" for knot theorists, and was vitally important in the classification of knots and their symmetries up to 16 crossings.

Nov 16, 2000

TITLE:                 Bounds for the minumum weight of the dual codes of some classes of designs

SPEAKER:               Kelle Clark

AFFILIATION:     University of Virginia

ABSTRACT:

The dual codes of the codes from the designs of points and subspaces of finite projective and affine geometries, and those of the codes of finite planes, were of interest initially in the 1960's and early 1970's following the work of Massey, Rudolf and others, in which an efficient decoding algorithm for these codes was described.  Effective use of the codes in practice requires knowledge of the minimum weight of the code, which had only been determined for these duals in a few cases. In particular, for the desarguesian planes of even order $q = 2^m,$ the minimum weight is
$q + 2$ and the minimum-weight vectors are the incidence vectors of the hyperovals in the plane.

In this talk we discuss new results for the minimum weight of the duals of some codes of $2$-designs that significantly improve the known upper and lower bounds for the minimum weight in the case of geometry designs and projective planes.   In particular we prove that the minimum weight of the dual code of any known non-desarguesian projective plane of order $25$ is either $42$ or $45$ and that it is 45 in the desarguesian case.
For  translation planes, we describe the construction of a set of points that could lead to either a word of small weight in the dual code or a hyperoval in the even case, the former giving improved bounds for the minimum weight of the dual code of these planes. Our construction leads to a possible formula for the minimum weight in the case of desarguesian planes that would apply for any order $p^m.$

Nov 30, 2000

TITLE:                 Minimal Homogeneous Bases for Ideals

SPEAKER:              Professor George Fix

AFFILIATION:     Clemson University

ABSTRACT:

A homogeneous basis for an ideal has the key property that homogenization of the elements of this basis generates the ideal of the projectivization of  the associated algebraic variety. Groebner bases have this property, however in important problems in computer vision these bases may be excessively large. A notion of a minimal homogeneous basis is introduced, and necessary and sufficient conditions are developed for determining when a given set of generators forms a homogeneous basis. In addition, it is shown that all minimal bases have the same number of elements. Finally, a modified version of the Buchberger algorithm is introduced for computing minimal homogeneous bases.

This is joint work with T.Luo and E. Yilmaz of UTA.

Dec 7, 2000

TITLE:                 Mathematics and the Art of M.C. Escher

SPEAKER:              Professor Robert T. Curtis

AFFILIATION:     University of Birmingham, U.K.

ABSTRACT:

The impossible' posters of the Dutch wood-cut artist M.C.Escher are well-known to most people: men trudge wearily up a never-ending
staircase; water flows down a channel and over a waterfall, mysteriously arriving back at the top in defiance of gravity. Indeed, at one time, it seemed that almost every student's room contained one of these prints.

The works which brought Escher to the attention of the mathematical world, and led to his becoming personal friends with several leading mathematicians of the day, were those which concerned themselves with symmetry in the plane.

The interaction really took off in 1954 when the International Mathematics Congress was held in Amsterdam and coincided with an exhibition of Escher's work. The fruitful collaboration between Mathematics and Art which ensued owes a great deal to the mathematician H.S.M. Coxeter who made a number of suggestions of mathematical concepts which Escher might like to illustrate.  These include the M\"{o}bius strip, with ants crawling along it, and hyperbolic geometric, exploring ideas of infinity.  The results often convey a mathematical idea more eloquently than
pages of symbolism.

In this talk we trace Escher's interest (he himself refers to his obsession') in Mathematics from his encounters with arab geometric designs in the Alhambra, to his later work in the 1960s.