Princeton University - Council of Science & Technology

POPE PRIZE

Essay 1
The Smartest and the Fastest: Investigating the limits of nature
Charles Graham Wells

Essay 2
Oldfield Mice, Theoretical Biology and the Search for True Love
Sarah Helen Labun

Essay 3
John Horton Conway, John von Neumann Distinguished Professor of Mathematics
Lillian Beatrix Pierce

John Horton Conway, John von Neumann Distinguished Professor of Mathematics

It is an early morning at Princeton University-early for the students who are shuffling into Room 401 of the Princeton mathematics department. The twelve story building is nearly silent. Only the whir of the elevator sliding up and down the core of the building with an hourly freight of students indicates that these floors are indeed inhabited. Somehow, though, it seems that all twelve floors are glowing with mental activity, with mathematicians' conjectures about an abstract world. The windows lining the back of this triangular classroom on the fourth floor open only a few inches, and then are stopped by large iron bars. "Suicide," one of the students explains. "They think we are too stressed to be trusted around here." This building, Fine Hall, towers quite literally over the rest of the Princeton campus. Named after one of the first mathematicians at Princeton, it currently houses some of the most famous mathematical brains in the world: Andrew Wiles, John Nash, Elias Stein, Peter Sarnak, John Conway.

It is John Conway whom these students are here to see this morning. At first it appears that there is to be embarrassingly poor attendance: in a classroom made to seat twenty, four students, three girls and a boy, are waiting silently with notebooks open to clean pages. But then it becomes clear that these are all the students there are to be-the students say they are a more surprising collection for the large percentage of girls than for the small total number. The elevator whirs again, its bell sounds, and John Conway appears around the door. He throws on the lights, tosses the New York Times on a table, drops his parka and woolly ear-flapped cap in the corner, and methodically bows at the waist to place a tall cup of coffee on the front desk, his grizzled beard pointing forward. Without a glance to the room he faces the board. "Theorem," he says. "The graph is a tree."

John Conway is a genius, an eccentric, a mathematician, and a teacher. His mathematical discoveries are many and famous. Unlike many mathematicians who never think of presenting their mathematics to the public, Conway wants to blazon his favorite number tricks to the whole world. Not only does he publish mathematical papers, he writes playful books such as "The Book of Numbers," which can be read with awe and pleasure by a practiced mathematician or a fourteen year old. To his students, Conway's genius is in finding the connections between different subjects, reducing matters to the very simplest terms, and defining a new vocabulary that makes the material vivid. Read his books or watch him lecture and the material he presents will seem simple-deceitfully simple.

Conway's personal oddities and his dedication to teaching come clear after a visit to his Princeton office. He spends his days here thinking about numbers and talking with everyone around him about more numbers. Over the years his office has become crammed with multicolored geometrical puzzles, a bright geode in the twelve floors of mathematicians' cubicles. The rumor is that he once let a homeless artist live in it for two years and that the man repaid him by painting portraits of the most famous mathematicians around the ceiling of the room. When Conway tells the story he ends mournfully, "But he left before he finished coloring in the face of Euler."

A sign on the door of the office reads "Let he who has no knowledge of geometry enter here," a hint at Conway's fondness for teaching. Students never quite know what to expect from Conway. One day in the middle of a lecture he started yelling "Green elephants! Green elephants!" Another day he put his shoes on first and then his socks, so that students would understand the importance of performing certain complicated mathematical procedures in the correct order. Sometimes he throws chalk. He hands out new business cards to the classroom with a sort of innocent pride: John Horton Conway, John von Neumann Distinguished Professor of Mathematics. Once he had a class of students stand up and practice screaming. Perhaps students don't know exactly what Conway will do in class, but they do know that they will learn a lot of beautiful math. He once said, "If it is alive and is sitting somewhere near me, I'll teach it."

That is what he is doing this morning in Room 401. The course is Math 403, "Advanced Topics in Algebra." The topic, quadratic forms. This is a very different algebra from the 2x + y = 3 of junior high days, and Conway is presenting his own signature version of the material. "We have a well, a simple or a double well, when the quadratic form changes signs," says Conway. "Wetness or water is where a sign change occurs. Here we see river edges form a number of doubly infinite chains." The students nod and note this new geographic entity in the mathematical landscape. "And here we have a finite river running between two lakes." If the blackboard weren't covered with numbers it would seem as if this were some type of ecology lecture. "And here we have the flood!" Conway fills the board with zeroes, gleeful.

Once again Conway has demonstrated his special knack for making mathematics intuitive. Perhaps it isn't quite intuitive why a certain bit of mathematics "changes signs," but it is intuitive that a river can lead to a lake, and this playful and apt vocabulary gives the mathematics a special beauty and accessibility. It is a rare mathematician who enters the mathematical realm of abstraction as utterly and successfully as Conway, and yet retains a human, linguistic playfulness. This is where the genius shows, as well as a Puckish sense of humor. For example, Conway's treatise on quadratic forms, upon which the lecture this morning is based, is titled "The Sensual^^(quadratic) Form." One of the students in the class tells the story of trying to convince the university bookstore to order the risqué title. They wouldn't believe it was a text for a mathematics course.

Conway is the inventor of the Game of Life. It is characteristic of Conway both that he invented a stunningly simple mathematical model for the complex process of the birth, life and death of an organism, and that he called it a game. For Conway, most things are games, games to be pulled and tugged at by his immense brain, games to be discussed in arcane terms by the mathematical community, but also games to be enjoyed by all people. He himself enjoys the role of mathemagician, provider of number puzzles, tricks, and surprises. Recently he of stunned his students with one of his games. Walking into a classroom, he wrote a string of fourteen different fractions, 17/91, 78/85, 19/51, and more, across a chalkboard. "This is PRIME GAME. Give me a number," he said, "and I will calculate the prime numbers for you."

This is as bold a statement as that of Archimedes when he said "Give me a lever long enough and a fulcrum on which to place it, and I shall move the world." The prime numbers are 2, 3, 5, 7, 11, 13, and so on: every whole number divisible only by itself and by 1. These elusive and strangely alluring numbers are still giving mathematicians and supercomputers alike quite a chase. How many primes are there? How big is the next prime? Most importantly, how to find the next prime? Conway's invention, PRIME GAME, looks at first sight like a miracle: it can show you one prime after another, after just a few manipulations of fractions on a chalkboard. On one level it is just an amusing little game to play. But behind it lies an entire new theory of computer programming, a computer language Conway invented called FRACTRAN. This language is based on simple mathematical units, everyday fractions like 1/7 and 5/13, but it can do the remarkable calculation of finding the primes. Mathematicians know there are infinitely many of these prime numbers: no matter how big a prime number you find, there is another prime number even bigger. You may not know what it is, but you know it exists. But mathematicians want to know more. What kind of an infinity of prime numbers is there? A big infinity? A small infinity? What is the difference between a big infinity and a small infinity, anyway?

"Curses!" The lecture has ground to a halt. Conway mutters and his students stare at the board. They are studying Conway's "Little Methuselah Form." Conway explains the name: "No other positive-definite three-dimensional form lives so long." But something has gone wrong in the list of numbers on the board and he stares at it, running a hand through his pale brown hair, his pointed, gray-streaked beard tilted upward at the board. "Really, I'm sorry I'm not seeing this," he says in his British accent to the four students. It is hard to believe the John von Neumann Distinguished Professor of Mathematics is apologizing for a momentary pause in a complicated manipulation that only a few people in the world understand. For all his quirks and surprises, Conway is a gentleman. There may be an eerie feeling in the room, a suspicion that rather than lecturing to the people in the room, at whom he never looks, he is conversing directly with intellects hovering in the air. However, the prevailing atmosphere is one of benevolence, of intimate mathematical camaraderie. He may not look at his students, but he knows exactly who they are. If he needs a starting point, a number to test in a computation, he'll ask without even turning from the board, "What is your favorite prime today, Lillian?"

This morning, still puzzled, Conway paces in front of the board with his hands behind his back, something rattling in his pocket. He is wearing his usual soft black leather shoes, khaki pants, and a bright orange and black shirt marked with wisps of chalk dust. He stands up very straight, his posture itself expressing eagerness, curiosity, interest in the world around him. He takes a few swallows of coffee. Each morning he comes in to class with a fresh cup, sometimes large, sometimes small, a pastel paper frustum of the brew he reaches for whenever his thoughts slow to mortal speed. Today again it does the trick. He finds the bug on the board and the lecture is off and running.

So what about those prime numbers, those infinities, big and small? Conway is one to know about infinities. Years ago he himself discovered a new type of number, the surreal numbers, which fill in the cracks between big and small infinities. This new kind of number may sound impossible, confusing, perhaps unnecessary, but to mathematicians, it is a new continent discovered, a new territory to colonize with mathematical ideas. These days what is Conway doing besides teaching about rivers, lakes, and quadratic forms? He is busy most every day, writing a new book. And he has the Princeton math department agog over his latest excitement, a new connection he has made between something he calls the "Monster group" and a type of math called "modular forms." And what has he named his new idea? Monstrous Moonshine.

It is difficult not to be interested in seeing Conway's complicated, abstract, freshly discovered math, couched in whimsy and scented with genius. Like a magic show, his lecture ends as suddenly as it began. "Problem," Conway writes on the board. "When is diag(a,b,c,…) = diag(A,B,C,…) mod Q?" Conway turns from his work at the board and looks silently over his students' heads, out the window and into the wintry morning. He turns back to write "Answer: When and only when they have the same determinant mod squares and the same p-signatures for all primes." And he walks out of the room.