ABSTRACTS OF LECTURES, Ross Reunion 1996

Thomas Banchoff, "Higher-dimensional geometry and the Internet"

New technology always challenges us to look at our favorite problems from fresh viewpoints. Interactive computer graphics on the internet is especially well-suited for the geometry of higher dimensions, offering new perspective on the way we teach, communicate, and do mathematics at all levels.

Terry Bisson, "Calculus with pebbles"

Disjoint union and Cartesian product of sets (of pebbles) underlie aritmetic. I will describe related constructions that underlie calculations with polynomials, including addition, multiplication, composition and differentiation. These constructions have topological versions that underlie natural operations in algebraic topology.

Keith Conrad, "Coefficients of cyclotomic polynmomials"

Why do the first 100 (well, 104) of the cyclotomic polynomials have coefficients 0, 1, -1 only, but the 105th has a coefficient of -2. This is a a nice example of how a pattern can last a while but then stop.

Charles Fefferman, "Turbulence"

The motion of fluids is often incredibly complicated. Although the equations governing fluid motion have been known for over a century, we are very far from any real understanding of the solutions of those equations. In this talk, I'll start by posing some of the main mathematical questions about fluids (blowup and scaling laws). Then, I'll explain some mechanical systems, much easier than fluids, for which one can ask analogous questions. The answers to these easier questions are still unknown.

Ronald Greenberg, "Some applications of sophisticated mathematics to randomized computing"

This talk will give an overview of a few powerful mathematical results and their application to the design of probabilistic computer algorithms.

The first result is a very general bound of Chernoff on the probability that a sum of n independent identically distributed random variables will exceed the expected value. The talk will touch upon applications to randomized algorithms for routing and sorting problems.

The second result, referred to as the Lovasz Local Lemma, gives a simple condition for there to be a nonzero probability that none of a set of interrelated events occurs. The condition is based only upon a bound on the probability of each individual event occurring and a bound on the number of other events upon which each event depends. The result is valuable to prove the existence of certain problem solutions, e.g., a short schedule for routing messages in a network. In addition, a recent version of the Lemma by Beck can be applied to the development of efficient algorithms for actually finding solutions to some problems. Time permitting, the talk will also sketch Bach's application of a theorem of Weil in algebraic geometry to the design of probabilistic algorithms. Most analyses of randomized algorithms assume that there is a source from which one may obtain many independent random values. Bach analyzes the approach more typically used in practice of attempting to start with one random seed and then using a pseudorandom number generator to get the other "random" values. He shows that certain randomized algorithms have very low failure probability even under the more reasonable assumption of just the initial seed value being random. For a particular example, we will look at an algorithm for computing square roots modulo a prime.

David Harbater, "Symmetries of fields and covers"

The study of field extensions has many parallels with the study of maps between curves. This can be viewed as providing a formal analogy between Galois theory and the theory of covering spaces. The analogy can be explained by considering the fields of functions of the spaces involved. In the case of curves, this approach yields the conclusion that every group is a Galois group over the field C(x) of complex rational functions. Extending this approach yields results over many other fields as well. But the most classical situation, for the field Q of rational numbers, remains mysterious.

Jeff Kahn, "Theory of finite sets, some linear algebra and a geometric application"

We will mention a few problems --- some solved, some not --- from the theory of finite sets, and will try to say a little about what linear algebra has to do with some of them, and about how some of them led to the solution of an old problem in elementary geometry.

Karl Rubin, "The arithmetic of elliptic curves"

Elliptic curves are polynomial equations with a particularly rich structure. Over the past twenty years tremendous advances have been made in our understanding of the arithmetic of elliptic curves, and many new questions have arisen. This talk will concentrate on a particular family of elliptic curves and attempt to describe what we know and what we don't know.

Alice Silverberg, "Points of finite order on abelian varieties"

Many important results in number theory, including Faltings' proof of the Mordell Conjecture and Wiles' proof of Fermat's Last Theorem, rely on understanding abelian varieties, their points of finite order, and their reductions modulo prime numbers. This talk will introduce the notions of abelian varieties and semistable reduction. Abelian varieties with sufficiently many rational torsion points have semistable reduction. This result, combined with the work of Wiles, Taylor, and Diamond, implies the modularity of the elliptic curves in 10 of the 15 infinite familes classified by torsion subgroup.

Glenn Stevens, "Continued fractions, SL2(Z), and modular elliptic curves"

The talk will explain what modular symbols are and why number theorists are interested in them. The main goal will be to describe an efficient method of computing modular symbols based on continued fractions and the "Magic Box". By looking at an example, I will try to motivate the view that modular symbols are concrete and readily computable objects that encode deep and subtle arithmetic information about elliptic curves.

Intended audience: A general mathematical audience, including participants of the Ross program.

David Sze, "The usefulness of rigorous math training in industry"

A strong and rigorous mathematical training program, together with the interest and motivation for applications, is a terrific combination for industrial scientists. Several examples of telecommunications research will be described, comparing the the techniques learned in mathematics with the skills needed to successfully solve industrial problems.

Colin Wright, "Juggling -- theory and practice"

Juggling has fascinated many for centuries. Seemingly oblivious to gravity, the skilled practitioner can keep several objects in the air at one time, and weave complex patterns that seem to defy analysis. In this talk the speaker demonstrates a selection of the patterns and skills of juggling while at the same time developing a simple method of describing and annotating a class of juggling patterns. By using elementary mathematics these patterns can be classified, leading to a simple way to describe those patterns that are known already, and a technique for discovering new ones.

The talk is suitable for all ages. Those with some mathematical background should find plenty to keep themselves occupied, while those less experienced can enjoy the juggling and the exposition of this ancient skill.