2003 MathFest - Boulder,Colorado

The Council on Undergraduate Research Award ($150) for the best research presentation at the Pi Mu Epsilon sessions went to Jennifer Novak.

"NICE SURFACE!" AND OTHER WAYS TO COMPLEMENT YOUR KNOT

**Jennifer Novak**

Texas A & M University - Texas Eta

The complements of hyperbolic knots often contain surfaces with
interesting geometric properties. However, identifying knots with
these totally geodesic surfaces has proven difficult. We will
explore extensions of manifolds, termed orbifolds, with unique
geometric structures that can generate these special knots.

The Pi Mu Epsilon awards ($150) for the best presentations went to the following seven students.

MATHEMATICAL FREEDOM

**Chris Jones**

Youngstown State University - Ohio Xi

In this talk we will begin with the definition of "freeness" as
it pertains to group theory and other objects in abstract algebra.
From there, we will investigate other times mathematical freedom
occurs, such as in category theory and topology.

A MATRICIAL ALGORITHM FOR POLYNOMIAL REFINEMENT

**Emily King**

Texas A & M University - Texas Eta

This project provides a methodology for obtaining sets of polynomial coefficients, refinement coefficients or integral displacements for finite polynomial refinement given the other two sets. Transforming the refinement equation of an *n*th degree polynomial into vectors in **R**^{n} utilizing a common basis yields a solvable system of matrices and vectors. Patterns that emerge in this system provide a foundation for many proofs involving refinement of polynomials.

KNOTS IN THE CUBIC LATTICE

**Marta Kobiela**

Texas A & M University - Texas Eta

Knots in a cubic lattice are placed in a 3D grid. I illustrate theorems concerning the composition of these knots, finding better upper bounds for the stick numbers of composed knots in relation to those of the original knots. Other theorems discuss composition of trefoil knots and figure-eight knots.

BIFURCATIONS OF THE HENON MAP

**Derek Pope**

Seton Hall University - New Jersey Delta

An overview of a classic example of a chaotic dynamical system, the Henon map, with special emphasis on creating a nearly complete bifurcation diagram for this map. Computer -generated images will be used to demonstrate the dynamics of this map for various parameters.

ALGEBRAIC STRUCTURES AND THE LONG-TERM BEHAVIOR OF DISCRETE DYNAMICAL SYSTEMS

**Brenda Russo**

Salisbury University - Maryland Zeta

Discrete dynamical systems known as cellular automata will be presented. We study the long-term behavior of these structures when generated over group and monoid alphabets. This research relates algebraic properties of these generating alphabets, such as those of automorphisms and subgroups, to the long-term dynamics of cellular automata.

CRYSTALLOGRAPHIC FRACTAL TILINGS

**Maria Salcedo**

Youngstown State University - Ohio Xi

In this talk I will discuss the seventeen crystallographic groups and show some representative patterns that I created which have fractal boundaries and tile the plane. Fractals were generated through an iterative process using Visual Basic.

This research was conducted while I was enrolled in the McNair scholarship program at Akron University.

MEANS OF COMPLEX NUMBERS

**Barbara Sexton**

Sam Houston State University - Texas Epsilon

A great deal is known about the arithmetic mean, harmonic mean, and geometric mean of real numbers. In this talk we will examine the arithmetic mean, harmonic mean and geometric mean of complex numbers. In particular we will examine properties that we expect such means to possess.