ABSTRACTS

  ROSS MATHEMATICS PROGRAM
Reunion-Conference 2007

Abstracts for lectures on Friday, July 20, 2007

Right triangles and elliptic curves

  Karl Rubin
  Edward and Vivian Thorp Professor of Mathematics
  UC Irvine

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.

  William Dunham
  Koehler Professor of Mathematics
  Muhlenberg College

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

Labyrinths & Mazes

  Karan Singh
  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).

  Charles Misner
  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.

  Lynne Butler
  Department of Mathematics
  Haverford College

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 .

Games

  Jacob Lurie
  Department of Mathematics
  Harvard University

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

Arnold Ross's Influence on Teacher Education

  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:

1. 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?
2. Do experiences of immersion in mathematics have a productive role to play in teacher education?
3. Do all teachers benefit from such rich experience of mathematics? If not, which ones do? Why? or Why Not?
4. What are the essential features of such experience for mathematics education?
5. For those of us who teach (math or science), what do we choose to emphasize in our own classrooms? And why?
6. Can one generalize the Ross immersion experience so as to make it accessible to a broad range of teachers? Is such generalization desirable?

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.

Primes, Polynomials, and Patterns

  Paul Pollack
  Department of Mathematics
  Dartmouth College

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.

Patterns That May Not Last

  Keith Conrad
  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.