Fall 2002

[ 12/3 | 11/18 | 11/7 | 10/28 | 10/7 ]

Tues. 12/3/02 (12:30) - The "Coin-Exchange Problem" of Frobenius
Prof. Matthias Beck
Visiting Mathematics Professor
SUNY in Binghamton

How many ways are there to change 42 cents? How many ways would there be if we did not have pennies? How about if nickels were worth four cents? More generally, suppose we have coins of denominations a_1, ..., a_d. Can one find a formula for the number c(n) of ways to change n cents? A seemingly easier question is: can you change n cents at all, using only our coins?

We will see that if a_1, ..., a_d are relatively prime then we can be certain that we can change n, provided n is large enough. A natural task then is to find the largest integer that cannot be changed. This problem, often called the linear Diophantine problem of Frobenius, is solved for d=2 but wide open for d>2. We will use the above counting function c(n) to recover and extend some well-known results on this classical problem. En route we will discuss some basic Number Theory and Discrete Geometry connected to c(n).

11/18/02 - When topology controls geometry: a generalization of the 180 degree theorem
Dr. David Richeson
Department of Mathematics and Computer Science
Dickinson College

Every schoolschild knows the fact that the sum of the angles of any triangle is 180 degrees. A stronger and more useful version of this theorem is that the sume of the exterior angles of any polygon is 360 degrees. Not coincidentally there is a smooth version of this theorem: there are 360 degrees in a circle. In this talk we show that there is a beautiful generalization to three dimensional shapes, the so-called Gauss-Bonnet theorem. We begin by looking at polyhedra, then we look at smooth topological shapes such as the sphere, the torus, etc. We see that the total angle excess of a polyhedron or the total curvature of a surface is related to the Euler characteristic.

Thurs. 11/7/02 (12:30) - A Defense Application of Math & Computer Science
1st Lt. John Gulick
United States Air Force
Air Force Laboratory

Mathematicians and computer scienctists play a key role in addressing computational challenges in the prediction of electromagnetic wave scattering. This talk presents the challenges from the perspective of physics, mathematics, and computer science along with specific areas where innovative solutions can dramatically improve current computational electromagnetic technology. As a related issue, fidelity of the geometry representation of the scattering target affects thre correlation between the prediction and measurement results. Hence, geometry representation presents another technical challenge. Both topics require creative math and computer science solutions to overcome current challenges.

10/28/02 - Symmetric Polynomials: From Babylon to Newton to Now
Dr. Marc Renault
Department of Mathematics
Shippensburg University

A polynomial in several variables is said to be "symmetric" if any permutation of its variables leaves the polynomial unchanged (e.g. 2x3 + 2y3 + 2z3). In other words, a polynomial in n variables is symmetric if it remains fixed under the permutation action of the symmetric group Sn. It is a classical fact that any symmetric polynomial can be written as a polynomial in "elementary symmetric polynomials", and the proof of this is given by a constructive algorithm. I will discuss the origins of symmetric polynomials, the algorithm above, and some generalizations that people are considering presently.

10/7/02 - Introduction to Game Theory
Dr. Richard Forrester
Department of Mathematics and Computer Science
Dickinson College

Game theory is a formal method of dealing with the general features of competitive situations in an abstract way. Founded by the great mathematician, John von Neumann, game theory is based upon mathematics, economics, and other social and behavioral sciences. In this talk we discuss the basic elements of game theory and consider various applications.