Algebra & Discrete Math Seminar

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.

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