Conference – Reunion – Party
In Honor of Dr. Ross’s Ninetieth Birthday and the 40th year of the ROSS MATHEMATICS PROGRAM
August 9 – 11, 1996
Abstracts for some lectures are available by clicking on the title.
Friday, August 9
- coffee and donuts in Math Tower lounge (room 724)
- David Harbater, University of Pennsylvania: “Symmetries of fields and covers“
- David Sze, Bellcore: “The usefulness of rigorous math training in industry“
- Chris Haase, Glynn Scientific, Inc.: “Trends in telecommunications”
- Ira Gessel, Brandeis University: “Combinatorial proofs of congruences”
- coffee, tea and snacks in Math Tower lounge
- Jeff Kahn, Rutgers University: “Theory of finite sets, some linear algebra and a geometric application“
- David Fried, Boston University: “Dynamical systems and continued fractions”
- Larry Taylor, University of Notre Dame: “The current state of affairs in the theory of 4-manifolds”
- Ultimate Frisbee game on field south of Stadium
Saturday, August 10
- coffee and donuts in Math Tower lounge
- Soccer game
- Colin Wright, “Juggling – theory and practice“
- 11:00-12:00, (An “Arnold Ross Lecture” sponsored by the American Mathematical Society)
- Thomas Banchoff, Brown University: “Higher-dimensional geometry and the Internet“
- Group photograph
- 14:00-15:00 (An “Arnold Ross Lecture” sponsored by the American Mathematical Society)
- Charles Fefferman, Princeton University: “Turbulence“
- coffee, tea and snacks in Math Tower lounge
- Terry Bisson, Canisius College: “Calculus with pebbles“
- Pierre Bouchard, Universite du Quebec a Montreal: “Introduction to Joyal’s theory of species”
- Banquet at Faculty Club
Sunday, August 11
- coffee and donuts in Math Tower lounge
- Karl Rubin, Ohio State University: “The arithmetic of elliptic curves“
- Alice Silverberg, Ohio State University: “Points of finite order on abelian varieties“
- Ronald Greenberg, Loyola University: “Some applications of sophisticated mathematics to randomized computing“
- Glenn Stevens, Boston University: “Continued fractions, SL2(Z), and modular elliptic curves“
- Keith Conrad, Harvard University: “Coefficients of cyclotomic polynomials“
- David Pollack, Harvard University: “The Hasse principle – relating local and global data”
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 arithmetic. 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.
Abstracts for lectures on Friday, July 20, 2007
Edward and Vivian Thorp Professor of Mathematics
For which positive integers D is there a right triangle with three rational sides and area D? This is a very old question, to which we still don’t have a complete answer. But over the last thirty years modern number theory has made a lot of progress on it, via the theory of elliptic curves. In this talk we will survey what is known about this problem, and discuss its connections with other important open questions in number theory.
Koehler Professor of Mathematics
We first sketch the life and work of Leonhard Euler (1707 – 1783), whose 300th birthday is being celebrated this year. We then address a specific problem from number theory: the construction of amicable pairs (recall that two positive integers are amicable if each is the sum of the proper divisors of the other). The Greeks knew the amicable pair 220 and 284, and two others were found prior to the 18th century, when Euler arrived on the scene. In an awesome display of mathematical power, he found 58 new ones!
We shall examine how he did it – i.e., how he single-handedly increased the world’s supply of amicable numbers twenty-fold. His argument is clever yet so easy to follow that we will generate a “new” amicable pair right before your eyes. And the topic seems fitting because, in the lives of both Leonhard Euler and Arnold Ross, number theory occupied a special place.
Abstracts for lectures on Saturday, July 21, 2007
Professor of Computer Science
University of Toronto
Labyrinths and mazes cradle millennia of legend and lore in their twisted articulations and are often considered mankind’s first creation borne purely of human imagination. This talk addresses the synthesis of labyrinthine and maze structures, represented as curves on 2D manifolds. The curves evolve subject to forces that capture properties of randomness, smoothness and van der Waals attraction. Artists interactively control the output patterns by spatially varying the simulation parameters and using text, sound or imagery to control the evolving curves. The resulting curves have wide range of applications from model construction and animation to data visualization.
This picture is a labyrinth interactively grown over a picture of Jim Morrison (lead singer of The Doors).
Professor Emeritus of Physics
University of Maryland
I’ll talk a bit about my career with an emphasis on Ross’s influence, perhaps describing some of my recent work on black holes. An abstract might be available later.
Raymond T. Pierrehumbert
Louis Block Professor in Geophysical Sciences
The University of Chicago
Though one doesn’t often have occasion to make use of number theory in the study of planetary climate, there are three things I learned from Dr. Ross which have served me in good stead throughout my attempts to understand the basic physics determining what kind of climate a planet has. The first is “Prove or disprove, and salvage if possible.” In science, there”s a big emphasis on the “salvage” part, since most theories are wrong in one way or another, and the key to progress is figuring out how to fix the failures while retaining the good bits. The second is “Generalize.” Theories get more interesting when they bring out the common features in a broad range of circumstances. The third, and most important, is of course, “Think Deeply of Simple Things.”
The physics of planetary climate is perhaps one of the clearest examples in science of the value of TDOST. The behavior of planets is complex, in the sense of exhibiting a broad variety of intricate and unexpected behavior. It is not complicated, though, in the sense of having a large number of parts each behaving according to different principles. Planetary atmospheres are more like the game of Go, with complex behavior emerging from pieces interacting via simple rules, than it is like a pinball machine.
In this lecture, I will provide a number of examples of how a range of different planetary climates emerge from simple physical principles. I will give some examples from the Solar System, focusing on Earth, Mars, Venus and Titan, and how these planets could have had radically different climates in the past. In these cases, I’ll show how the interaction of three simple physical laws account for much of the range of behavior of these planets in the present and past: the Clausius-Clapeyron law determining the amount of a vapor (e.g. water or methane) an atmosphere can hold in saturation, the Stefan-Boltzmann law for emission of infrared radiation, and the albedo law determining reflection of solar energy by a partially ice-covered planet. From these we get phenomena as diverse as Snowball Earth, the runaway greenhouse, Methane rain on Titan, the Dry-Iced greenhouse effect warming Early Mars, ice ages, and global warming. In particular, we’ll see why Titan, with a temperature of about 173 Celsius below zero, is dynamically like a hotter all-tropical Earth.
I will also give some flavor of the wealth of exotic extrasolar planets that have been discovered, and describe the factors that determine whether such planets could be habitable to life as we know it. Exotic possibilities include long-lived biospheres around red M-dwarf stars, and tide-locked large Earths with a too-hot sunside and a glaciated farside, with a habitable zone near the terminator. Thinking of this sort gives us an idea of how things might play out differently in other solar systems.
Department of Mathematics
Stochastic models used in speech recognition, cryptanalysis and hedge fund management include a simple class called Hidden Markov Models. I’ll talk about the senior paper of a Haverford math major on HMMs, a topic she learned about in my cryptography class. You will see how to use HMMs to model the string of letters in English text and the sequence of percentage changes in the Standard & Poor’s 500 Index. Though their use in cryptanalysis is classified, researchers like Jim Simons and Nick Patterson who mastered their use in cryptography and language modeling have profitably employed them in finance and genomics.
See the 12 December 2006 New York Times article “A Cryptologist Takes a Crack at Deciphering DNA’s Deep Secrets”. Here is a link to that article .
Department of Mathematics
In this talk I’ll describe what has come to be known as “combinatorial game theory”: that is, the theory of games with no chance and no hidden information. It turns out that the collection of such games possesses a surprisingly rich algebraic structure, reminiscent of (and containing) the structure of the real numbers, but with some subtle and surprising new features.
Abstracts for lectures on Sunday, July 22, 2007
Glenn Stevens, Max Warshauer, and Daniel Shapiro
Departments of Mathematics
Boston University, Texas State University, and Ohio State University
As we celebrate the fiftieth anniversary of Arnold Ross’s program, we hope also to use our time together to reflect on some of the ideas Arnold contributed to the American mathematical scene and, in particular, to consider the ongoing influence of those ideas on mathematics and science education.
It is perhaps not entirely well known that Arnold Ross’s famous program began not as a program for secondary school students, but rather as a program for secondary teachers. Indeed, Arnold’s commitment to teacher education continued to be strong throughout the Ross Program years. Many of us still remember the clusters of teachers who sat quietly in the back of the room diligently taking notes of Arnold’s lectures during our time in the program.
The panel presentation will describe three teacher education programs that were designed by alumni of Arnold’s program. All three programs are deeply influenced by the principles of the Ross program, but our respective programs express these principles in different ways. Our collective experience raises important questions, which we hope will stimulate discussion among the reunion participants. For example:
- Maybe our energies would be best spent in programs aimed directly at students, not teachers. Wouldn’t focusing on students have just as great if not a greater impact in the long run?
- Do experiences of immersion in mathematics have a productive role to play in teacher education?
- Do all teachers benefit from such rich experience of mathematics? If not, which ones do? Why? or Why Not?
- What are the essential features of such experience for mathematics education?
- For those of us who teach (math or science), what do we choose to emphasize in our own classrooms? And why?
- Can one generalize the Ross immersion experience so as to make it accessible to a broad range of teachers? Is such generalization desirable?
Mathematicians and educators are increasingly beginning to ask questions like these. What do Ross alumni have to add to this discussion?
To loosen tongues around the room, we may pull out some old samples of the Ross problem sets and consider specific ways in which these problem sets might be relevant (or irrelevant) to teacher education today.
As alumni of the Ross Program we have vivid recollections of the program’s effect on the ways we approach life intellectually and in the spirit of exploration. We all remember the many bright young people we interacted with as young people. We know these interactions made a difference in our lives and how the experience of immersion in mathematics changed us in the way we approached learning and living.
Department of Mathematics
Questions about prime numbers constitute some of the most celebrated problems in all of mathematics. At the popular level, one frequently hears asked: “Is there a pattern in the set of primes?” This is often allied with the question of whether or not the primes are “random.” These queries are somewhat ill-formed, but in various guises they reappear even at the upper echelons of research in number theory.
In this talk we will look at some attempts to answer these questions, ranging from recreational to research-level. Towards the end of the talk we discuss how some of these questions look from the vantage point of generalization: suppose that, in the spirit of the Ross program, one replaces the ordinary ring of integers by the ring of polynomials modulo p. This is the subject of the author’s doctoral dissertation, currently in progress at Dartmouth College.
Department of Mathematics
University of Connecticut
Observing patterns in numerical data can be a source of important (and unimportant) mathematical developments. However, while numerical experimentation has a useful role in suggesting new ideas or conjectures, it is no substitute for a proof. There really are patterns that last for 100, 1000, or even billions of terms but which eventually break down. We will look at patterns (largely drawn from number theory) which persist for a while. Students will have a chance to decide whether they think the trends continue forever or stop working. Then the truth will be revealed.