Northeastern Section of the Mathematical Association of America

Fall 2004 Meeting - WPI

Abstracts of Invited Presentations



Quantum mechanics and combinatorial designs
- P. K. Aravind

Recent proofs of Bell's theorem in the foundations of quantum mechanics have led into some unexpected areas of discrete mathematics such as regular polytopes, projective configurations and combinatorial designs. In particular, this work has led to the emergence of the concept of a "quantum block design" that is a generalization of the balanced incomplete block design known to experts in combinatorics and statistical design. The purpose of this talk is to expose this connection between quantum mechanics and combinatorial mathematics for the benefit of mathematicians who know next to nothing about quantum mechanics. This will be done by focusing on a particular example that illustrates this connection in detail rather than by dwelling on abstract generalizations. Having done this, I will discuss some open questions connected with quantum block designs that mathematicians may feel inclined to address. The possible physical applications of quantum block designs will also be touched upon.

Some things I should have known when I first started teaching differential equations (and didn't!)
- Joseph McKenna

I taught my first course in differential equations thirty years ago. It was a basic cookbook course and I just followed the textbook slavishly, often barely grasping what I was teaching. Here, I will describe some of the things I've learned since I started.

Elliptic curves, the silver bullets of modern mathematics
- Ezra 'Bud' Brown

Elliptic curves made their first appearance seventeen centuries ago and are among the most beautiful objects in mathematics --- and the most useful. This talk will be about elliptic curves and their connections to such things as: A talk on all of these would run for days, so the audience will choose the topics to be presented.

Alternating Sign Matrices
- David Bressoud

This will be an overview of what is known and what is conjectured about Alternating Sign Matrices, a combinatorial structure with ties to partition theory, representation theory, and statistical mechanics. The talk will include an overview of the story of the Alternating Sign Matrix Conjecture, a tale that begins with a Lewis Carroll algorithm for evaluating determinants and ends with Kuperberg's realization that the 6-vertex model of Izergin and Korepin held the key to the solution.

The CUPM Curriculum Guide 2004
- David Bressoud

The MAA Committee on the Undergraduate Program in Mathematics (CUPM) updates its recommendation for the undergraduate curriculum in mathematics roughly every ten years. The most recent and the most extensive set of recommendations ever produced by the CUPM was published in February: CUPM Curriculum Guide 2004. This is the first CUPM curriculum guide to look at all mathematics courses and the needs of all students taking mathematics rather than dealing solely with the preparation of majors in the mathematical sciences. This workshop will explain what can be found in this guide and how it can be used.

Counting on Determinants
- Arthur Benjamin

We demonstrate how determinants solve many interesting combinatorial problems. Determinants count nonintersecting lattice paths, spanning trees, and permutations with specified descent points. Elegant proofs of these results are based on the definition of the determinant and occasionally the principle of inclusion-exclusion. Applications to Pascal's Triangle, Fibonacci numbers and Catalan numbers will also be given.

This talk is based on joint work with Naomi Cameron of Occidental College.

Folding Robot Arms, Proteins, Origami: a Combinatorial Approach
- Ileana Streinu

Robot arms can be modeled as simple polygonal chains with rigid bars and rotating joints. Planning non-colliding motions between two configurations of a robot arm is a notoriously hard problem, for which the currently known best algorithms run in exponential time. An efficient solution in dimension 3 could have an impact in understanding apparently unrelated questions, such as how proteins fold. In this talk I will present a suprisingly simple combinatorial solution to the 2-dimensional version. The Carpenter's Rule Problem, "Can every planar polygonal chain be convexified with non-colliding planar motions?" was open since the '70. It was answered in the affirmative in the early 2000, and my contribution - the subject of this talk - was to give an efficient algorithmic solution. I will present it with a lot of graphical props: animations, games, even some 3d-graphics. Along the way, we'll use tools ranging from a 19th century theorem of J. Clerk Maxwell to graph embeddings, oriented matroids, combinatorial rigidity theory and visibility computations in Computational Geometry. I will conclude with some algorithmic insights into origami folding induced by this approach.

Industrial Mathematics: Real Analysis of a Different Kind
- Arthur Heinricher

Students and faculty at WPI have been working on industrial mathematics projects for almost 10 years. The Center for Industrial Mathematics and Statistics at WPI was founded in 1997 to coordinate these efforts. This talk will give three snapshots of mathematics at work in business and industry. The examples include: The work described comes from projects completed by graduate and undergraduate students at WPI as well as work done during the summer in WPI's Research Experience for Undergraduates in Industrial Mathematics and Statistics. Two main messages: First, there are good mathematics problems everywhere, but the math is often hidden. Second, mathematics students have valuable skills and can contribute to the solution of real problems of immediate interest to business and industry.

Spherical Codes, Fullerenes and the Structures of Solutions to the Problems of Thomson and Tammes
- Jack Graver

Spherical Codes (maximum families of non overlapping identical spherical caps on a sphere), Fullerenes (large carbon molecules) and the structures of solutions to the problems of Thomson (minimize the potential of n unit charged particles on the sphere) and Tammes (distribute n points on the sphere to maximize the minimum distance between them) all give rise to large planar graphs. This talk is about the structures of these graphs and the relationships between the graphs that arise from different problems.