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Melanie Wood: The Making of a Mathematician

by Deepti Scharf

Duke News & Communications, 05.08.2003

On a sunny, spring afternoon, with graduation barely three weeks away, Melanie Wood learned she’d won yet another scholarship to Cambridge -- bringing the total to three prestigious awards that would pay for her year there.

Settling her nearly six-foot frame into a chair, the Duke math senior took a moment to absorb this latest bit of news.

"I’ve just won the Fulbright to Cambridge," she said calmly. "And I’ve just decided I’m going to Princeton, though I’m going to defer for a year and go to Cambridge first."

With the Gates Cambridge scholarship and the National Science Foundation Graduate Fellowship already in hand, Wood said she would need to work out the details of who would pay for what.

Wood’s reaction -- understated, pragmatic -- may be a result of having achieved so much in her young, promising career. Profiled at age 17 as "The Girl Who Loved Math" by Discover magazine, Wood has a prodigious list of successes, including her recent naming as Putnam Fellow by the Mathematical Association of America -- making her the first American woman, and the second woman in the world, to be so honored.

But Wood is not simply a math prodigy. Last year, she won the Faculty Scholars award in Theatre Studies; assistant directed MacBeth, the Duke Players winter show; and produced a musical. "I think it gives me a fuller life that I can do both things," she said of her twin loves, math and theatre.

While at Duke, she also ventured into psycholinguistics, physics, and economics. "It may be interesting sociologically to see why I ended up that way, but you don’t find the typical male math student ending up that way," she mused.

The fact that grade-school boys who did math for fun were tolerated as nerds, whereas girls proficient at math or science were pressured into becoming well-rounded, might have something to do with it, she said. "I was already popular in middle school, and so it didn’t matter if I wanted to do some nerdy things."

But gender also concealed opportunities from girls. "If it hadn’t been for a teacher asking me, 'Hey, do you want to come to this math competition,' I would have never stumbled into it naturally because my friends weren’t in a math club after school or anything," said Wood, who went on to win math competitions from middle school on.

Still, it was hard for her "to separate the gender divide from the ability divide" through grade school, even as she found herself "drastically better" than anyone else at math. It was in high school, after she won the USA Math Olympiad, that the gender divide hit home.

Winning that Olympiad shocked her. "I thought, Why am I so surprised? And then I realized that it was just that I had this image of the people who won this competition -- and that image was of boys."

Female role models in math are scarce. "If I listed the top 20 faculty members in the country I might work with in graduate school, they are all male. And probably, if I listed the top 50, they are all male."

At Duke, Wood valued the opportunity to talk with Andrea Bertozzi, the only female, tenured faculty in the math department. "Having a woman in the department has been important to me. There was a point at which I needed to talk to a woman mathematician to deal with something, and I could, and that was great."

The fact that Bertozzi is leaving for another university is a blow for Duke because "there’s a big difference between one and zero," said Wood.

Gender became an out-and-out confrontation while Wood was shopping for colleges. At one university, a professor said dismissively, "I hear you’re supposed to be good, but I’ve never had a female student who really understood the mathematics I do."

Yet Wood is careful to distinguish the person from the institution. Women have cautioned her about specific departments and universities, citing serious, almost institutionalized discrimination. "I, personally, have never experienced that," Wood said. Instead, she has "experienced specific people who have had attitudes about women doing mathematics that were very negative."

As she prepares to leave what she says is a very nurturing environment at Duke, she’s learning about other issues confronting professional women mathematicians. At a recent lunch at another university, she learned that getting maternity leave is "a particular wall" for women in the tenure track process. And while daycare options are becoming standard at most academic conferences, math conferences are still not family friendly.

"It’s frustrating," she said, "but I really want to do math. I’m just hopeful that I’ll be able to deal with all of this because this is what I really love to do."

Because of her understanding of the gender divide in professional mathematics, Wood intends to lend help to other women. "I don’t think that sheer numbers have a chance of stopping me. I now believe that I can do this even if I’m the only woman.

"I think that it’s easier for a girl to see me and say, ‘Oh, I want to be like she is.’ I get a lot of emails, but I’m particularly concerned with younger women who are interested in math and want to know what I did and how I did it."

For more, see "A Conversation with Melanie Wood by Joseph Gallian, in MAA Math Horizons.

FindArticles > Discover > June, 2000 > Article > Print friendly

The Girl Who Loved Math - Melanie Wood is only female to represent US in International Mathematical Olympiad

Polly Shulman

EVEN IN THE REMOTE MOUNTAINS OF CARPATHIA, IN A SMALL TOWN CALLED SINAIA--WHICH very few Americans have ever heard of or are likely to visit--one is rarely far from a peculiar piece of hardware that seems to have captivated teenagers around the world: a basketball hoop. [paragraph] In Carpathia, the teens scrambling under the hoops tend to be Romanian, of course, but on this particular day in Sinaia there also happen to be some American teens who have been invited to play. But they have something they like to play a whole lot better than basketball. [paragraph] The Americans, six teens aged 15 to 17, are here to practice for a different sport. In 10 days, on July 16, they will gather with 427 other teens from around the world in Bucharest to face off against six mathematics problems so difficult that many college professors would find them taxing. This competition is called the International Mathematical Olympiad, and the American contestants have been selected from high schools across the United States.

Before those grueling days in Bucharest, though, the Americans have decided to practice--in exhibition games, if you will--against the Romanian math team. Now, after hours of work, the Romanian team is taking a much needed break playing basketball, but the Americans can't let go.

"The Americans are upstairs," says Titu Andreescu, their coach. "Our students are not very athletic. Well, they are athletes of the mind."

In a stuffy dorm room festooned with dirty laundry, Andreescu's mental athletes are sprawled on beds and linoleum. One is lost in thought, a place he finds comfortable for hours at a stretch. Another stares intently at handouts of problems from the last Olympiad. The other four have gathered around a notebook, brandishing the mathematician's weapon--number 2 pencils--to attack a particularly thorny brain boggler. Melanie Wood, tall and green-eyed with blond hair, is the only girl in the room. In fact, Melanie is the only female ever chosen to represent the United States. She turns to the five boys in the room and offers a way to solve the geometry problem at hand. She suggests inversion, a strategy for turning circles into lines to see if the simpler relationship of lines to lines will open up a solution to the problem posed by the circles. The other pencil-wielders nod their heads and join in with gusto.

What might seem ridiculous to many people--that mathematics can be more engaging to teens than basketball or video games or even dating--could stand for absolute truth in this room at this moment.

But mathematical fun does not come without stress. The U.S. team has an impressive history--three wins in 26 years of competition. And in 1994 all six Americans were awarded perfect scores. The competition is never easy: The Chinese and the Russian teams are always a threat, as are teams from countries with strong mathematical traditions such as Romania, Iran, and Hungary. Individual futures are also at stake. Although not one of these teens will have any difficulty getting into college, winning a gold medal at the Olympiad could earn a full scholarship.

Melanie may be under greater stress than anyone on the U.S. team. This is her second Olympiad. The previous year, in Taiwan, she won a silver medal. "Once you win," she says, "you have attention on you. Particularly because I'm the only American girl."

AFTER A PRACTICE EXAM THAT THE ROMANIANS finish with troubling speed, the Americans huddle. What does the Romanian victory mean? Perhaps only math can tell them. Melanie and her friends start with definitions used for solving problems involving inequalities. Do the Romanians "dominate us?" one asks. In math terms, dominate would mean the Romanians' worst player has beaten the best U.S. player. "Do they majorize us?" another asks. This term is more complicated: It would be true if the Romanians' top player got a higher score than the top U.S. player, and if the sum of their top two players' scores beats the sum of our top two, and the sum of their top three beats the sum of our top three, and so on. "What will it mean if there are teams that we beat or tie but don't majorize?" asks Melanie, who then answers her own question: "It'll mean we're clumpier than they are."

"What do you mean by clumpy?" asks Lawrence Detlor, from New York City.

"Our scores are closer together."

"That's the opposite of what I thought you meant," he says. "I thought clumpy meant `containing separate clumps.'"

Melanie's mind has already raced ahead to a consideration of how the size of a team's home country might affect clumpiness. "It would seem like big countries would be more clumpy than little countries, because if you take the top six people in a big country, they'll probably be good. But small countries will tend to be clumpy, too, because they might not have anybody good. So they're clumpy, but not in an interesting way."

Later, when Melanie asks if anyone has a pamphlet from a previous Olympiad, Lawrence answers: "I do. Wait! What do you mean by pamphlet? It's probably the opposite of what I mean by pamphlet."

WHEN MELANIE ARRIVED AT HER FIRST AMERICAN TRAINING program for the Math Olympiad, in the summer of 1996, she was thrilled to find a group of kids who were not only as smart as she but who also saw the world through the same prism: "I had finally found peers. For the first time in my life, I was an average student. That meant fast-paced classes, being bombarded with exciting math I'd never seen before, never being bored. Before, I'd always thought maybe I was just weird. Here nobody is ever, ever embarrassed to be doing math on Friday night, at six in the morning--whenever you get the urge."

The urge has been with her since childhood, a fact that brings up interesting questions about whether genes are more important than environment in a child's life. After Melanie's father, Archie Wood, a middle-school math teacher, died of cancer when his daughter was just 6 weeks old, Melanie's mother decided to keep him alive through the subject he loved most. She began teaching Melanie math at age 3. "By the time I was 4, when I got bored walking around the mall, my mother would give me linear equations to solve in my head," Melanie recalls. (For example, if 3x + 2y =12 and x = 2, then what is y?) Melanie's mother, Sherry Eggers, who was then a language teacher, also tried to teach Melanie French and Spanish, but the languages didn't stick. Eggers's genius was to let her daughter explore at her own pace. She never pushed, never made choices for her; instead, she asked Melanie to decide for herself what she wanted to do.

So Melanie did. The result was both amazing and, at times, sad. In seventh grade, for example, she entered a national middle-school competition called MathCounts and quickly realized, "I wasn't just the best student in my school, or as good as any other best student in her own school. I was probably in the top handful in the nation. That separated me from my school friends, who didn't understand the problems and what they meant to me."

Melanie's mother didn't understand the math her seventh-grader knocked off with ease either. Suddenly Melanie was alone in her world, and her sense of isolation was heightened by an awareness of what she had lost: the father with whom she might have shared these very triumphs. "That's when my dad became an important figure in my life, instead of just someone I'd never met. I have a picture of him standing in front of a blackboard; (mod 5) was written on it, a concept not a lot of people would understand. I thought about how amazing it could have been to come home and have someone I could talk to about this huge thing in my life."

What makes Melanie different from a lot of people is that she has accepted her loss, and accepted it in youth rather than in maturity. She achieved what her mother had hoped--Melanie's father is a part of her. "He is with me in the competitions, or even when I am just thinking about math," she says. "His spirit and memory are there in my mind."

Because she was able to forge a profound connection with her father, and because her mother gave her so much room to grow on her own, Melanie seems to be that rare teenage girl who is not stymied by self-doubt. When she encounters a problem that appears insoluble, she moves ahead deliberately and methodically, searching for the solution with unusual certainty that she will be able to find it, and a conviction that if she fails, her failure is not a reflection of her self-worth. "I try to understand all the mathematical structures involved in a problem, even when they're not necessary to solve it, because that helps me to understand the problem better."

Melanie also thinks of herself as cooperative, not competitive, which makes her an ideal study partner. "Mathematicians work together," she says, an attitude that has helped meld the team into a cohesive unit. The students enjoy one another's company keenly and, when not working on math, they often play made-up mind games. For example, they might plunge into a variation of chess in which each piece moves as if it were the piece to its left, so that kings slide around the board like bishops and bishops hop around like knights. They delight in a game called "No Fifth Symbol," in which players must speak without using words that contain the letter e; or "Silent Football," a game with unstated rules that newcomers must deduce by watching.

Such inventiveness isn't just about fun, says Melanie--it's also helpful for solving math problems. "My sense is that we have good intuition about how to win games, so it's useful to rephrase a problem as if it were a game," she says. A problem about polygons inscribed inside a circle, for example, might be thought of as a game between two players who take turns drawing lines that connect points on the circle.

The team is also united by a fierce belief in the purity of mathematics, which can make their world seem a lot more dependable and certain. Melanie defines it this way: "You start from nothing and deduce whole worlds just from logic. You don't have to take into account arbitrary facts about the world around you. If we were in a different world where atoms could combine in different ways and sulfur were a different color, math would still be math."

AFTER SEVEN DAYS OF PRACTICING, THE TEAM FORSAKES THE SYLVAN refuge of Sinaia for sweltering Bucharest and the competition. Finally, the moment of truth and proof has arrived. Teams from 82 countries begin the first of two 4 1/2-hour sessions over two days. Contestants will be expected to find a solution to each problem and also to prove that each answer is correct.

At the end of the first session, students emerge looking grim and shaken. The next day they are even more depressed. "It made difficult look easy," says one contestant. A television crew collars Melanie on her way out--the American girl has become a media magnet. With polish and politeness, she tells the camera that the test was challenging, points out that the Romanians have a tradition of choosing challenging problems, and says the team members won't know how they did until the awards ceremony. To her teammates she whispers: "Come on, we've got to get out of here." They hurry back to the safety and respite of the dorm.

With the problem-solving done, team coaches must fight for points, a process one coach describes as "horse trading." First, students meet with their coaches and go over copies of their exams, explaining what they had in mind. Sometimes they can find a partial solution to a problem hidden among false starts. The coaches try to convince the judges to grant partial credit. While math itself can be beautiful and universal, it turns out that a contest is, nonetheless, a contest.

When the dickering finally ends, Russia and China have tied for first. Romania is fourth, and the United States places ninth. Nevertheless, Reid Barton and Paul Valiant win gold medals; Gabriel Carroll, Po-Shen Loh, and Melanie get silver; and Lawrence Detlor is awarded a bronze. Characteristically, Melanie is not rocked by the loss. "As a team, we certainly didn't do better than average for the United States," she says. "Still, all the teams that placed above us are very good."

Coach Andreescu reassures his charges: "The coordinator congratulated me for the thoroughness of your solutions. Even though they were not the simplest, they showed mathematical maturity that impressed him." Later he adds, "That test would have been a challenge for a professional mathematician."

A PROFESSIONAL MATHEMATICIAN IS WHAT MELANIE PLANS TO be, although she just might "end up directing on Broadway," because she is majoring in theater as well as math. She has finished her freshman year at Duke University, which she chose over Harvard because she thought the math department there was "cold and competitive." She has been taking graduate-level classes in real analysis, complex analysis, and algebraic number theory, along with a drama class entitled Voice and Body Gesture.

This month Melanie is working with the U.S. math team's summer program as a grader, which gives her a chance to mentor younger mathematicians bound for next month's Olympiad in Seoul. All through high school she volunteered at MathCounts, the middle-school competition in which she had competed. "One of my jobs was emceeing the Cool Down Round, which follows the official competition. I run around with a mike among students who are furiously solving problems, and I jump on tables, yelling, `Hey, hey, we have an answer over here.' It's nothing like an actual math competition. Competitions are silent. Most of the math I do isn't competition math. It's openly working with others and full of laughter." Like her father, Melanie is passing along her love of mathematics to others in an exuberant, generous way that no doubt would have made him proud.

SATISFYING PAIRS

This was one of six problems contestants were asked to solve at the International Mathematical Olympiad in Bucharest last year. This year's Olympiad will be held in Seoul, South Korea, on July 18 and 19.

Find all pairs (n,p) of positive integers such that

* p is prime

* n [is less than or equal to] 2p

* [(p-1).sup.1] + 1 is divisible by [n.sup.p-1]

Hint: The cases P|n and p/n should be handled separately, In the latter case, consider the congruence [(p-1).sup.n] [equivalent] -1 modulo a suitable prime divisor of n.

MATH OLYMPICS SOLUTIONS

The pairs satisfying the given condition are (n,p) = (1,p) for any prime p, (2,2) and (3,3). The reader may easily check that these actually are solutions, and that there are no more solutions for p = 2, 3. So we may assume hereafter p [is greater than or equal to] 5 and n [is greater than or equal to] 2. In particular, since [(p - 1).sup.n] + 1 is odd and n divides this quantity, n must be odd.

First suppose p divides n; since n [is less than or equal to] 2p and n is odd, in fact n = p. Expanding [(p - 1).sup.p] by the binomial expansion reveals that

[(p - 1).sup.p] + 1 [equivalent] (p/1])p - (p/2)[p.sup.2] + (p/3)[p.sup.3] ... [equivalent] [p.sup.2] (mod [p.sup.3])

for p [is greater than or equal to] 3, which gives a contradiction for p [is greater than or equal to] 5. Alternatively (as noted by Lawrence Detlor), if [(p - 1).sup.p] + 1 = [kp.sup.p-1], then clearly k [is less than] p, but k must also be congruent to 1 modulo p - 1, so k = 1. On the other hand, [x.sup.y] [is greater than] [y.sup.x] for x [is greater than] y [is greater than or equal to] e (this reduces to the fact that (log x)/x is decreasing for x [is greater than or equal to] e, which may be shown by easy calculus), so [(p - 1).sup.p] [is greater than] [p.sup.p-1] for p [is greater than or equal to] 5, a contradiction.

Thus we may assume p does not divide n. We will give two proofs of the fact that [(p - 1).sup.n] [equivalent] 1 (mod n) has no solutions with p not dividing n.

* First Proof: Since n [is greater than] 1, it has a smallest prime divisor q. Since q - 1 has all its prime divisors less than q, n and q - 1 have no common prime divisor, that is, they are relatively prime. We now note that this implies [x.sup.n] [equivalent] [y.sup.n] (mod q) if and only if x [equivalent] y (mod q). This is obvious if either x or y is divisible by q; if not, we apply Fermat's little theorem in the form [x.sup.q-1] [equivalent] [y.sup.q-1] [equivalent] 1 (mod q). From Euclid's algorithm, we recall that there exist integers a, b with an + b (q - 1) = 1, and so

x [equivalent] [x.sup.an+b(q - 1) [equivalent] [x.sup.an] [equivalent] [y.sup.an] [equivalent y (mod q).

In the case of interest, this means p - 1 [equivalent] - 1 (mod q), a contradiction since p cannot equal q by our assumption that p does not divide n.

* Second Proof: Suppose the claim is false, and let n be the smallest integer greater than 1 such that [(p - 1).sup.n] [equivalent] - 1 (mod n). Let m be the smallest positive integer such that [(p - 1).sup.m] [equivalent] - 1 (mod n) and let d be the order of p - 1 modulo n. On one hand, m [is less than] d since otherwise e could have been replaced by e - d; on the other hand, d divides 2m since [(p - 1).sup.2m] [equivalent] 1 (mod n). Therefore d = 2e, and since [(p - 1).sup.2n] [equivalent] 1 (mod n), d divides 2n and so m divides n. From [(p - 1).sup.m] [equivalent] - 1 (mod n) and the fact that e divides n, we have [(p - 1).sup.m] [equivalent] - 1 (mod m). However, m [is not equal to] 1 (since n does not divide (p - 1) + 1 = p) and m [is less than] n (since m divides [Phi](n), which is less than n for n [is less than] 1), so m is a smaller counterexample to the assertion, contradicting the choice of n.

In either case, we conclude that the only solutions are those given above.

Bibliography for "The Girl Who Loved Math - Melanie Wood is only female to represent US in International Mathematical Olympiad"

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Polly Shulman "The Girl Who Loved Math - Melanie Wood is only female to represent US in International Mathematical Olympiad". Discover. June 2000. FindArticles.com. 21 Jun. 2008. http://findarticles.com/p/articles/mi_m1511/is_6_21/ai_62277745

Interview with Kiran Kedlaya, Mathematician and Puzzler

by Amy Hodson Thompson

Cogito, 11.27.2006

MIT professor Kiran Kedlaya is a mathematician, a juggler, a singer, a bicyclist, and a photographer – but he is first and foremost a puzzler.

Cogito members submitted questions to Dr. Kedlaya in November, 2006. (Jump to the Q&A)

MIT professor Kiran Kedlaya is a mathematician, a juggler, a singer, a bicyclist, and a photographer – but he is first and foremost a puzzler. He got hooked as a kid. A very rambunctious kid. “My parents, trying to figure out some way to keep me out of their hair, got me a subscription to Games magazine,” he says. Their scheme kept him from bouncing off the walls, and he’s been solving puzzles of one sort or another ever since.

He started competitive crossword puzzle-solving during graduate school and appears in a couple of scenes in the just-released documentary Wordplay about the New York Times puzzle editor Will Shortz and the American Crossword Puzzle Tournament, the oldest and largest crossword puzzle tournament in the nation. He finished fourth in the 2005 contest depicted in the movie, and in 2006, he won second place.

At the American Crossword Puzzle Tournament

Is there a connection between math and crosswords? Dr. Kedlaya thinks that math, music and computer science – popular professions among “solvers” – tap into a similar part of the brain. Wordplay, says Dr. Kedlaya, suggests that the link is using language in unique way. In a crossword, figuring out the word from the clue is not sufficient; decoding how the letters cross is vital, too.

For Dr. Kedlaya, solving and creating puzzles go hand in hand. One of his crosswords was published in the legendary New York Times crossword page, and he attended the National Puzzlers’ League conference in San Antonio in July of this year. In the NPL, where his nom de puzzle is “Kray” (Kiran means “ray” in Sanskrit, and he added a K because he had some to spare), some of the best amateur, semi-pro, and professional puzzle constructors and solvers in the country swap puzzles. He has also been a key player in the MIT Mystery Hunt. The winners of each year’s hunt construct the following year’s collection of interrelated puzzles whose answers must be combined (and sometimes recombined) in some fashion to provide clues to the location of a single coin hidden somewhere on the MIT campus. Dr. Kedlaya’s team won in 1999, 2001, and 2004.

Dr. Kedlaya solves and creates the puzzles of his profession as well. Winner of two gold medals (1990 and 1992) and one silver medal (1991) at the International Mathematical Olympiad, he has provided questions for the USA Mathematical Olympiad nearly every year since winning the prestigious international competition. During his college years at Harvard, he was a three-time Putnam Fellow (one of the top five finishers in the Putnam Competition). After earning his bachelor’s degree in Math and Physics in 1996 and his doctorate in Mathematics from MIT in 2000, Dr. Kedlaya spent the next three years at UC Berkeley holding a National Science Foundation Postdoctoral Fellowship. He then joined the Mathematics department at MIT in 2003 as an assistant professor, and thrives in the intense, exciting environment. This year Dr. Kedlaya was awarded a prestigious three-year Alfred P. Sloan Research Fellowship, intended for the very best young faculty members in particular areas of math and science.

Dr. Kedlaya, a number theorist, describes his work as being near the boundary of algebraic geometry, which he thinks of as a descendant of Euclidian geometry. Instead of studying simple polynomial equations which describe circles, straight lines, and parabolas, he studies much more general polynomial equations that don’t describe geometric or algebraic objects. While his primary focus is pure mathematics, he occasionally dabbles in areas of computer science. “Cryptography, coding theory – there are certain areas of computer science that relate nicely to polynomial equations,” he says.

His non-math interests vary widely. He speaks Spanish and Russian and a smattering of other languages as well, sings in the MIT Chamber Chorus, plays Ultimate Frisbee, and is working on improving his juggling skills. His work allows him to travel to conferences and seminars all over the world and gives him lot of opportunities to practice another one of his hobbies, photography.

“It’s important to tell kids who are interested in math as a career that there are many venues to do it, not just in the academic area within math departments,” says Dr. Kedlaya. People working in computer science, economics, finance, and other fields, whether in academia or industry, all can do interesting math. “Look for math anywhere you can find it,” counsels Dr. Kedlaya. Puzzles, after all, come in many types.

Questions and Answers

Here are Dr. Kedlaya's answers to your questions.

What are your primary fields of study?

My principal fields are number theory and algebraic geometry. I'm also quite interested in how those two areas relate to theoretical computer science.

Can you remember a particular event or experience that made you want to become a mathematician?

I think the pivotal sequence of events happened early in high school. First I started participating in math contests involving proofs, then I demonstrated an uncanny knack for such proofs, and finally I discovered that there were people in the world (research mathematicians) who actually made their living by constructing proofs!

How has your experience with puzzles helped you in your mathematical career?

I suppose it prepared me to think in unexpected ways about familiar objects.

Kiran Kedlaya, Ph.D.

Hometown: Washington, DC
Education:
A.B., Math and Physics, Harvard University
M.A., Math, Princeton University
Ph.D., Math, MIT
Interests: Bicycling, singing, juggling
Free-time Favorites: Puzzles

Can you say anything about the kinds of math questions you’re working on now?

I'm mostly interested in objects called zeta functions. The original such function, introduced by Riemann, can be used to analyze the distribution of prime numbers. One can use other zeta functions to study the ways a given polynomial factors when you consider its coefficients modulo different primes.

What do you think the compelling questions in mathematics will be in five years?

This is a tough thing to predict! Five years ago I might have suggested the Poincaré Conjecture, but that one is now settled. One cluster of compelling questions is the Langlands Program.

What kind of puzzle got you into solving?

When I was very young, I quite fancied arithmetic and logic puzzles, partly because they seemed much easier than word puzzles for which I didn't have the vocabulary yet. Somewhere along the way, I learned how to solve cryptic (British-style) crosswords, which was a good intermediate step: since every answer is clued both by a definition and by a bit of wordplay (i.e., as an anagram, or by putting two shorter words together, or adding one letter to another word), I could both solve for words I didn't know beforehand and along the way learn their meanings.

What is your favorite puzzle type, and do you do Sudoku puzzles?

Nowadays, I do mostly regular American crosswords and some cryptics; I find that they exercise a part of my brain that otherwise gets a bit restless. By contrast, at this point the math/logic puzzles like Sudoku feel a bit like work to me, without the benefit of a real discovery at the end!

How long does it take you to do the New York Times crossword?

The Times puzzles are graded by difficulty, Mondays being the easiest, Saturdays the hardest (and Sundays somewhere in between). I typically take about 3 minutes for the Monday puzzle, and anywhere between 5 and 15 minutes for the Saturday.

How long did it take you to create your New York Times-published puzzle?

I'd estimate a few hours, total, including coming up with the puzzle theme. Professional constructors tend to do it more efficiently.

How do you manage to balance work vs. play in your life?

I don't make any special effort to. My work feels enough like play (and occasionally vice versa) that the balance seems to take care of itself.

For more information

Dr. Kedlaya's homepage
Math Problems Directory, Dr. Kedlaya's project to catalog sources of information about mathematical problem-solving on the Web.

Math Whiz, 17, Hits the Big Time with Research

by Becky Bartindale

San Jose Mercury News, 03.07.2006

Yi Sun, of San Jose, California, won Second Place in the 2006 Intel Science Talent Search. This article was written before Sun went to Washington. In July, Yi Sun made news again when, as a member of the US team at the International Mathematical Olympiad, he won a Silver medal.

There's more than one way to become a teenage superstar.

For San Jose's Yi Sun, 17, a fascination with solving the puzzles of theoretical mathematics is the ticket to a meeting with President Bush, exchanging ideas with some of the nation's top mathematicians and scientists and explaining a relatively new branch of math to anyone who asks.

Is he nervous about proving his intellectual chops in the finals at the Intel Science Talent Search, the prestigious competition that begins this week in Washington, DC?

"Sort of," admits Yi, a senior at the Harker School in San Jose, but he's also excited.

Only 40 high school seniors in the United States, selected from among 300 semifinalists, make it to the Talent Search, the nation's oldest pre-college science competition. Yi is one of three California students -- and the only Northern Californian -- to move on to the finals.

Truth be told, demonstrating his scientific knowledge and critical-thinking skills to the judges is not what makes Yi most nervous. The real butterflies come from another part of the six-day trip. For two days, the Talent Search finalists set up shop in the rotunda of the National Academy of Sciences and answer questions about their research from all comers.

"Even though I'm sort of nervous, the public exhibition sounds really fun," said Yi, who hopes to become a college professor or research scientist after studying at Harvard University or the Massachusetts Institute of Technology. "I'm curious to see other people's projects and the opportunity to answer people's questions seems cool."

Yi already has won several gold medals at international olympiads in physics and math. These events are not about calculations but about proving propositions using mathematics. Training for olympiads has given him a much deeper understanding of math, Yi said.

Yi made it to the Talent Search on the basis of a research paper on the properties of random walks, a subject he had explored last summer at the Research Science Institute at MIT, working with graduate student David Pritchard.

Getting into that exclusive six-week summer program is harder than getting into college, said Yi's friend, Hailey Lam, 17, a Harker classmate. Yet despite all Yi's accomplishments, he said, his friend remains humble and is known for being just a very nice guy.

His friends and teachers already know that Yi is more than just a math geek. He's a varsity swimmer and Duke University basketball fan who drops in to play poker at lunch in the senior lounge. He speaks French and Chinese fluently and lists Joseph Heller, Kurt Vonnegut and John Steinbeck among his favorite authors.

When it comes to math, Yi just seems to have a certain intuition. "He can see how to break the problem down. He can take it apart really fast," Hailey said. And he keeps a cool head in competition.

Yi came to the United States as a toddler from China with his parents. His father, Lizhong Sun, is an engineer with Applied Materials, and his mother, Tianjing Shen, is an engineer with GSI Technology. His aptitude for science and math was apparent early and his parents encouraged it.

By third grade, Yi's principal was giving him a special math problem to solve every week in addition to his regular homework. When he was in middle school at Harker, Yi was invited to join the high school math olympiad team, said math teacher and olympiad coach Misael Fisico. As a freshman, Yi was chosen captain of that team.

Yi's research project for the Talent Search involves the mathematical field of combinatorics and looks at the winding number of a random walk, or path. At its core, Yi said, combinatorics is the study of counting groups of things. His research project, which produced towering stacks of paper filled with calculations, shows that the number of steps one can expect to take around an originating point on a coordinate plane is infinite.

Even Yi's dad admits, "I don't understand it well."

"It's definitely fun to know some things my dad doesn't know," Yi said.

Yi said he's drawn to math because "it doesn't have all the messy, real details of other subjects. You can see the underlying simplicity and beauty of things."

Plus, he added, "it's fun to find problems you can't solve."

If mathematicians and scientists were treated more like baseball stars, as they are in some countries, Yi said, probably more U.S. students would pursue those subjects.

That won't be a problem at the Talent Search finals, where the first-place winner receives a $100,000 college scholarship. Said Intel spokesman Mark Pettinger: "These students are treated like rock stars."

http://www.mercurynews.com/mld/mercurynews/news/local/14037129.htm
News & Views

Count Down: Six Kids Vie for Glory at the World’s Toughest Math Competition
by Steve Olson
Houghton Mifflin, 2004
Author Steve Olson followed six American teenagers as they prepared for and participated in the 2001 International Math Olympiad (IMO). These six students were chosen from almost half a million American young people who take the qualifying exams each year. In the international competition, the American students faced some of the most intelligent and mathematically talented students in the world. The IMO consists of six problems that utilize only basic math, but are extremely difficult to answer. Olson followed the kids from the time they attended training camp to the actual competition. The book includes biographical sketches of the kids, highlighting their other interests and hobbies, as well as their passion for mathematics. Olson also tackles interesting issues such as the lack of women in the field of mathematics. Additionally, he examines the roots of the kids’ mathematical genius in an effort to discover whether such talent is innate or if and how it can be learned. After evaluating data and observing the team, Olson proposes that mathematical talent can be cultivated, with the most important components being hard work and dedication. He also sheds light on the problems with mathematics education in the United States, citing that European students show much more enthusiasm for the subject.

Math's Architect of Beauty

by Jordan Ellenberg

Seed magazine, 09.22.2006

How Terence Tao's quest for elegance earned him a Fields Medal and a MacArthur Fellowship. An affectionate profile by a fellow mathematician who has known him since 1987, when they were high school students competing in the International Math Olympiad.
http://www.seedmagazine.com/news/2006/09/maths_architect_of_beauty.php

Cogito Interview: Terence Tao, Mathematician and Fields Medalist

Cogito, 05.11.2007

When UCLA math professor Terence Tao won a Fields Medal in August of 2006, it was the culmination of an amazing career of math achievements.

Dr. Tao took questions about his life and work through May 8, 2007. Read his answers.

In August of 2006, UCLA Professor Terence Tao won the Fields Medal, math's highest honor, "for his contributions to partial differential equations, combinatorics, harmonic analysis, and additive number theory."

The Fields Medal is awarded every four years to two to four mathematicians aged 40 or younger.

Dr. Tao's stunning mathematical ability was evident early in his life. When he was 10, 11, and 12 years old, he represented his native Australia at the International Math Olympiad, and won bronze, silver, and gold medals. He is still the only student to have ever won a gold medal before the age of 13.

He graduated from Flinders University at 15, got his PhD from Princeton at 21, and was a full professor at UCLA at 24. He was 31 last year when he won the Fields Medal. (One month later, he was awarded a MacArthur Fellowship, often nicknamed the "genius grant.")

Terence Tao, PhD

Hometown: Adelaide, Australia
Education:
PhD, Math, Princeton University, 1996
MSc and BSc, Math, Flinders University, 1992 and 1991
Research Interests: harmonic analysis, PDE, combinatorics, number theory
Last Math Book He Read: The Red Book of Varieties and Schemes by David Mumford
Last Non-Math Book He Read: Beyond Fear by Bruce Schneier
Favorite treat: Aussie meat pies

Whereas many mathematicians like to focus on one area of math, Dr. Tao likes to work in many areas in the field, learning as much as he can as he goes along. He works in non-linear partial differential equations, algebraic geometry, number theory, combinatorics, and harmonic analysis, an advanced form of calculus that uses equations from physics.

His work on prime numbers was considered by Discover magazine to be one of the 100 most important scientific discoveries of 2004. That year, Dr. Tao, along with Ben Green, a mathematician now at the University of Cambridge in England, solved a problem related to the Twin Prime Conjecture by looking at prime number progressions -- series of numbers equally spaced.

What kind of math were you doing when you were 8?

I think I was taking Year 11 or Year 12 maths classes in Australia at the time, which basically means high-school algebra (solving quadratic equations, etc.). Most of my other classes were at the Year 8 level.

What type of math questions intrigued you most when you were in high school and why?

I liked maths puzzles -- cute little questions which required some trick to them to unlock. Actually, I kind of thought this is what "real" maths was (and it was the type of maths which showed up in the high school maths competitions I was taking). It was only after I got into college that I began to realise that maths is not really about artificial puzzles at all, but about trying to understand patterns and phenomena in many kinds of situations, including some important "real life" problems (in physics, biology, finance, etc.).

Were you good at other fields than math, such as writing, reading, history, etc?

Ugh, not at all! I think creative writing was one of my worst. I liked situations in which there were very precise rules and procedures (such as maths or science), and so I had real trouble with assignments like "Write about something that happened to you." I did like history more, though, because there seemed to be more of a "story" that I could follow. I liked physics, but couldn't really deal with chemistry, especially organic chemistry -- far too many little facts to memorise. (There are facts to memorise too in maths, of course, but at least you can derive many of them from first principles. But I could never figure out how to use first principles to, say, describe the properties of butane.)

When you were in junior high or high school, what kind of summer programs did you participate in?

I went to training camps for the International Mathematics Olympiads. Other than that, I had a pretty free-form summer -- going to the beach, playing computer games, that kind of thing. I think it was more common back then to have a fairly unstructured life outside of school as a kid.

Do you feel that your achievements at the International Math Olympiad were largely based on extensive and arduous study or mainly because of your superior intellectual capacity?

That's an interesting question. I think back when I participated at the Olympiads, it was still a fairly informal affair -- many countries did not have formal training programs, or had very light ones consisting mainly of doing practice problems. So anyone who was bright could compete well after just doing practice problems for a few weeks. But now the training programs have gotten much more systematic (e.g. spending a single day focusing on one type of problem, or studying the history of what kind of problems appeared in previous competitions, etc.), and I would imagine that one would have to study intensively to be competitive nowadays.

In research-level mathematics, though, I would definitely say that hard work, experience, and an inquisitive approach count for much more in the long run than any innate ability. Solving a research problem is kind of like climbing a cliff; if you just try it with your bare hands, it doesn't really matter how strong or agile you are, it's unlikely that you will succeed. But if you have the right tools, and you've studied how other people were able to to get to lower places on the same cliff, what the hazards were, and which routes were likely to be easier, you have a much better chance.

You were obviously way ahead of everyone in your age group. How were you educated?

The short answer was that I was educated in the usual (public) school system, but at an accelerated pace, and also at a staggered pace (so some classes, e.g. maths and physics, I took at higher grade levels than others). For the longer answer, read this.

What advice would you give someone who isn't challenged in math class but has no real opportunity to move ahead because her school cannot accommodate her? What activities would recommend?

If for some reason an accelerated or special maths class is not an option, then there are other resources available, ranging from accessible maths books aimed at bright high-school or undergraduate students (the Mathematical Association of America has a nice selection) to the Internet, to competitions of various sorts, to maths clubs, to mentors. I myself was very lucky to be able to visit a retired maths professor each weekend as an undergraduate just to discuss maths in an informal manner, over cookies. Someone at the maths department at your local university might have some suggestions for what activities are available in your area.

What mathematical magazines or journals are good for someone in high school?

Hmm, there aren't many aimed at the high school level; the American Mathematical Monthly comes close, but is probably more at the undergraduate level instead. I found that there were many more maths books aimed at high school students (e.g. discussing the number systems, non-Euclidean geometry, elementary number theory, this kind of thing) than magazines.

Do you ever get to see applications of your discoveries in other fields of study? If so, is there any specific application which is your favorite?

I've only been doing research for about ten years, and in fairly pure areas of mathematics, so most of what I do is unlikely to lead to advances in the near future. But I do have one contribution which I am very happy to see have some application, which is that I helped work out the mathematics of "compressed sensing" -- which allows a measuring device (such as a camera) to take a medium-resolution picture (e.g. a 100KB image) using only a moderate number of measurements (e.g. using 300,000 pixels of measurements), as opposed to the traditional approach of taking a massive number of measurements (e.g. 5 million pixels) and then compressing all that data into a smaller file (e.g. a JPEG file). This type of approach to measurement may be useful for some future applications such as sensor networks, where for reasons of power consumption, one doesn't want to make too many measurements. There are already some prototype "single pixel cameras" based on this algorithm, and hopefully these sorts of devices will get deployed in the "real world" in a few years.

What has motivated you the most? Who has supported you the most other than yourself?

I think one of my largest sources of motivation is simply an itch to find out how things work, especially in mathematics where everything is out in "plain sight." Take for instance, the twin prime conjecture: we believe that there are infinitely many pairs of prime numbers, such as 11 and 13, which are only a distance of 2 apart. This problem is easy to state, and we all know what a prime number is, there's no secret catch or anything to the problem -- so how come, after so many centuries of progress in number theory, we still can't answer this question? When there is something I feel I ought to know the answer to, but don't, then I get motivated to try to figure out what the problem is and whether I can improve my understanding of the issue.

As to who has supported me the most, that would certainly be my parents, who worked very hard to set up an education for me that fit my needs, and then followed by all my mentors (both formal, as in my undergraduate advisor and graduate advisor, and informal, such as the retired professor I mentioned earlier), who showed me what maths is really about, and how one should be a good mathematician.

What does your current research deal with?

I work in a number of different areas; right now, I guess I spend about a third of my research time on figuring out how to count patterns (such as arithmetic progressions) in prime numbers (and in other types of sets), and another third of my time in understanding the behaviour of various kinds of waves (water waves, sound waves, electromagnetic waves), which evolve according to a type of equation known as a partial differential equation. The other third is spent on all kinds of things, in particular I spend a lot of time learning new areas of mathematics (it's a vast subject, I only know a fraction of what's going on right now).

Do you plan to extend your work on prime numbers in the future to possibly find a way to find them?

Actually, in many ways, I'm trying to show the opposite, that beyond a certain point, it's very difficult to pin down where the prime numbers are exactly without expending a huge amount of effort, and it is in fact more profitable to think of the primes as being distributed somewhat randomly (though not completely randomly; for instance, the primes are almost entirely concentrated in the set of odd numbers).

In your spare time, what kind of hobbies do you have?

Well, back in college, I used to play some volleyball and foosball, bridge, computer games, and even a collectible card game, and also liked anime and tinkered around with dubbing music videos... but now that I work full time and have a wife and son, I've dropped all of these hobbies; I like to spend my free time with my family now.

If you weren't a mathematician, what else would you want to be?

Hmm. That's hard to say. I really like the creative freedom and flexibility that an academic job offers; I probably wouldn't function well in a 9-to-5 job where all your work is given to you by your boss, though if the work was enjoyable then it would probably work out. Being self-employed would of course offer freedom and flexibility, but I think I would find it stressful to take care of all the business side of things. I did work as a computer programmer during summers in college; I could imagine that I could have ended up in that profession instead.

Terence Tao, "Mozart of Math," Is Awarded the Fields Medal

by Stuart Wolpert

UCLA News, 08.22.2006

Terence Tao became the first mathematics professor in UCLA history to be awarded the prestigious Fields Medal, often described as the "Nobel Prize in mathematics," during the opening ceremony of the International Congress of Mathematicians in Madrid on Aug. 22. In the 70 years the prize has been awarded by the International Mathematical Union, only 48 researchers ever have won it.

"Terry is like Mozart; mathematics just flows out of him," said John Garnett, professor and former chair of mathematics at UCLA, "except without Mozart's personality problems; everyone likes him. Mathematicians with Terry's talent appear only once in a generation. He's an incredible talent, and probably the best mathematician in the world right now. Terry can unravel an enormously complicated mathematical problem and reduce it to something very simple."

"I'm not surprised," said Tony Chan, dean of the Division of Physical Sciences and professor of mathematics. "Someone like Terry comes along once every few decades. People all over the world say, 'UCLA's so lucky to have Terry Tao.' He has solved important problems in several areas of mathematics that have stumped others for a long time. The way he crosses areas would be like the best heart surgeon also being exceptional in brain surgery. What is also amazing is that Terry is still so young.

"The best students in the world in number theory all want to study with Terry," Chan added. "He's a magnet attracting the best students the same way John Wooden attracted outstanding basketball players." Chan said he is known as "the dean of the university where Terry Tao works." He described the International Congress of Mathematicians as "the World Cup or Olympics of mathematics."

Christoph Thiele, UCLA professor and chair of the mathematics department, said outstanding graduate students from as far as Romania and China, as well as throughout the United States, have come to UCLA for the chance to study with Tao.

Tao was awarded the Fields Medal "for his contributions to partial differential equations, combinatorics, harmonic analysis and additive number theory." In honoring Tao, the organization said, "Terence Tao is a supreme problem-solver whose spectacular work has had an impact across several mathematical areas. He combines sheer technical power, an other-worldly ingenuity for hitting upon new ideas, and a startlingly natural point of view that leaves other mathematicians wondering, 'Why didn't anyone see that before?'"

Like the summer Olympics and the World Cup, the Fields Medal is awarded every fourth year. Along with Tao, the Fields Medal also was presented to Andrei Okounkov, professor of mathematics at Princeton University; Grigori Perelman, formerly a Miller Fellow at University of California, Berkeley; and Wendelin Werner, professor of mathematics at the University of Paris-Sud in Orsay.

Tao's genius at mathematics began early in life. He started to learn calculus when he was 7, at which age he began high school; by 9 he was already very good at university-level calculus. By 11, he was thriving in international mathematics competitions. Tao, now 31, was 20 when he earned his Ph.D. from Princeton University, and he joined UCLA's faculty that year. UCLA promoted him to full professor at age 24.

One of the branches of mathematics on which Tao focuses is theoretical field called harmonic analysis, an advanced form of calculus that uses equations from physics. Some of this work involves, in Garnett's words, "geometrical constructions that almost no one understands." Tao also works in a related field, nonlinear partial differential equations, and in the entirely distinct fields of algebraic geometry, number theory and combinatorics — which involves counting. His research has been supported by the David and Lucille Packard Foundation and the Clay Mathematics Institute.

"Terry wrote 56 papers in two years, and they're all high-quality," Garnett said. "In a good year, I write three papers."

Discover magazine praised Tao's research on prime numbers, conducted with Ben Green, a professor of mathematics at the University of Bristol in England, as one of the 100 most important discoveries in all of science for 2004. A number is prime if it is larger than one and divisible by only itself and one. The primes begin with 2, 3, 5, 7, 11, 13 and 17.

Euclid proved that the number of primes is infinite. Tao and Green proved that the set of prime numbers contains infinitely many progressions of all finite lengths. An example of an equally spaced progression of primes, of length three and space four, is 3, 7, 11; the largest known progression of prime numbers is length 24, with each of the numbers containing more than two dozen digits. Green and Tao's discovery reveals that somewhere in the prime numbers, there is a progression of length 100, one of length 1,000, and one of every other finite length, and that there are an infinite number of such progressions in the primes.

To prove this, Tao and Green spent two years analyzing all four proofs of a theorem named for Hungarian mathematician Endre Szemerédi. Very few mathematicians understand all four proofs, and Szemerédi's theorem does not apply to prime numbers.

"We took Szemerédi's theorem and goosed it so that it handles primes," Tao said. "To do that, we borrowed from each of the four proofs to build an extended version of Szemerédi's theorem. Every time Ben and I got stuck, there was always an idea from one of the four proofs that we could somehow shoehorn into our argument."

Tao is also well-known for his work on the "Kakeya conjecture," a perplexing set of five problems in harmonic analysis. One of Tao's proofs extends more than 50 pages, in which he and two colleagues obtained the most precise known estimate of the size of a particular geometric dimension in Euclidean space. The issue involves the most space-efficient way to fully rotate an object in three dimensions, a question of interest to theoretical mathematicians.

"Terry is the world's expert on this set of five problems, and has been since he finished graduate school," Garnett said. "When Terry made a new estimate of how big the dimension must be, he also produced the solutions, or partial solutions, to many other problems."

Tao and colleagues Allen Knutson at UC Berkeley and Chris Woodward at Rutgers solved an old problem (proving a conjecture proposed by former UCLA professor Alfred Horn) for which they developed a method that also solved longstanding problems in algebraic geometry and representation theory.

Speaking of this work, Tao said, "Other mathematicians gave the impression that the puzzle required so much effort that it was not worth making the attempt, that first you have to understand this 100-page paper and that 100-page paper before even starting. We used a different approach to solve a key missing gap."

Tao found a surprising result to an applied mathematics problem involving image processing with California Institute of Technology mathematician Emmanuel Candès; their collaboration was forged while they were taking their children to UCLA's Fernald Child Care Center. Chan said that Tao and Candès work is providing important insights into how to compress images, which has applications for medical imaging.

"A lot of our work came in the preschool while we were dropping off our kids," Tao said.

"Outstanding mathematicians love working with Terry," Garnett said. "You could build the best mathematics department in the world by hiring his co-authors."

What are Tao's secrets for success?

Tao, who was raised in Australia, offered some insight. "I don't have any magical ability," he said. "I look at a problem, and it looks something like one I've done before; I think maybe the idea that worked before will work here. Nothing's working out; then you think of a small trick that makes it a little better but still is not quite right. I play with the problem, and after a while, I figure out what's going on.

"Most people, faced with a math problem, will try to solve the problem directly," he said. "Even if they get it, they might not understand exactly what they did. Before I work out any details, I work on the strategy. Once you have a strategy, a very complicated problem can split up into a lot of mini-problems. I've never really been satisfied with just solving the problem. I want to see what happens if I make some changes; will it still work? If you experiment enough, you get a deeper understanding. After a while, when something similar comes along, you get an idea of what works and what doesn't work.

"It's not about being smart or even fast," Tao added. "It's like climbing a cliff: If you're very strong and quick and have a lot of rope, it helps, but you need to devise a good route to get up there. Doing calculations quickly and knowing a lot of facts are like a rock climber with strength, quickness and good tools. You still need a plan — that's the hard part — and you have to see the bigger picture."

His views about mathematics have changed over the years.

"When I was a kid, I had a romanticized notion of mathematics, that hard problems were solved in 'Eureka' moments of inspiration," he said. "With me, it's always, 'Let's try this. That gets me part of the way, or that doesn't work. Now let's try this. Oh, there's a little shortcut here.' You work on it long enough and you happen to make progress towards a hard problem by a back door at some point. At the end, it's usually, 'Oh, I've solved the problem.'"

Tao concentrates on one math problem at a time, but keeps a couple dozen others in the back of his mind, "hoping one day I'll figure out a way to solve them."

"If there's a problem that looks like I should be able to solve it but I can't," he said, "that gnaws at me."

Most of Tao's work is pure theoretical mathematics. Of what use is that to society?

"Mathematicians often work on pure problems that do not have any applications for 20 years, and then a physicist or computer scientist or engineer has a real-life problem that requires the solution of a mathematical problem and finds that someone already solved it 20 years ago," Tao said. "When Einstein developed his theory of relativity, he needed a theory of curved space. Einstein found that a mathematician devised exactly the theory he needed more than 30 years earlier."

Will Tao become an even better mathematician in another decade or so?

"Experience helps a lot," he said. "I may get a little slower, but I'll have access to a larger database of tricks. I'll know better what will work and what won't. I'll get déjà vu more often, seeing a problem that reminds me of something."

What does Tao think of his success?

"I'm very happy," he said. "Maybe when I'm in my 60s, I'll look back at what I've done, but now I would rather work on the problems."

Terence Tao Named a MacArthur "Genius" Fellow

The MacArthur Foundation, 09.19.2006

Tao, 31, was recently awarded the Fields Medal, the "Nobel Prize in mathematics." Now his genius has been honored beyond the mathematics community.

Popular Science has a very enjoyable introduction to Tao's work, Math's Great Uniter.

UCLA produced an engaging video profile of Tao, which includes photos of him as a child. (Duration: ~ 20 min.)

And here is the MacArthur Foundation's less engaging description: "Terence Tao is a mathematician who has developed profound insights into a host of difficult areas, including partial differential equations, harmonic analysis, combinatorics, and number theory. He has made significant advances in problems such as Horn’s Conjecture, which he showed can be reduced to a geometric combinatorial configuration known as a “honeycomb”; this problem holds deep implications for more abstract mathematical relationships in algebraic combinatorics. His analysis of the Schroedinger equation, a central element of quantum mechanics, has provided new avenues for solving nonlinear partial differential equations. Recently, with Ben Green, Tao offered a proof of the longstanding conjecture that there exist arbitrarily long arithmetic progressions consisting only of prime numbers. (For millennia, mathematicians have studied the properties of prime numbers, which find important applications in cryptography among other things.) In addition to his research, Tao has taken a leadership role in educating mathematics students through his web site, commentaries, books, and lectures. His work is characterized by breadth and depth, technical brilliance and profound insight, placing him as one of the outstanding mathematicians of his time."

About the MacArthur Fellows Program:

"The MacArthur Fellows Program awards unrestricted fellowships to talented individuals who have shown extraordinary originality and dedication in their creative pursuits and a marked capacity for self-direction. There are three criteria for selection of Fellows: exceptional creativity, promise for important future advances based on a track record of significant accomplishment, and potential for the fellowship to facilitate subsequent creative work.

Fellowships are awarded to women and men of all ages and at all career stages; the extraordinary creativity of MacArthur Fellows knows neither boundaries nor the constraints of age, place and endeavor.

The MacArthur Fellows Program is intended to encourage people of outstanding talent to pursue their own creative, intellectual, and professional inclinations. In keeping with this purpose, the Foundation awards fellowships directly to individuals rather than through institutions. Recipients may be writers, scientists, artists, social scientists, humanists, teachers, entrepreneurs, or those in other fields, with or without institutional affiliations. They may use their fellowship to advance their expertise, engage in bold new work, or, if they wish, to change fields or alter the direction of their careers.

Although nominees are reviewed for their achievements, the fellowship is not a reward for past accomplishment, but rather an investment in a person's originality, insight, and potential. Indeed, the purpose of the MacArthur Fellows Program is to enable recipients to exercise their own creative instincts for the benefit of human society."

http://www.macfound.org/site/c.lkLXJ8MQKrH/b.2070789/apps/nl/content2.asp?content_id={6DBB4260-1605-496C-8311-36328C702E50}&notoc=1

Cogito Interview: Keith Devlin, The Math Guy

Cogito, 04.18.2007

Keith Devlin is a math guy. He’s a mathematician at Stanford, and he’s literally the Math Guy on NPR’s Weekend Edition, where he talks about topics in mathematics and computing. He’s also a frequent guest on other radio programs, both in the United States and Britain. That’s pretty good for a guy who didn’t particularly like math when he was in grade school.

Questions were submitted to Keith Devlin through April 15. Scroll down to read his answers.

Photo by Richard Ressman

When Devlin was a teenager, he read a couple of books on math by W. W. Sawyer -- Prelude to Mathematics and Mathematician's Delight. Those books helped him develop a love for math and ultimately influenced his decision to become a mathematician.

Today, he tries to do the same thing for other people. He’s made a career not only out of doing math, but also out of making math understandable, even enjoyable, to non-mathematicians. He's also the author of 25 books and over 70 research articles.

And Devlin has argued some interesting things. For example, he says that we all have an innate mathematical ability, the most basic of which is “number sense,” or the ability to recognize small quantities without counting. Humans also are born with spatial reasoning abilities. And we aren’t the only animal that can do math. Chimps have symbols for numbers. Dogs can figure out the fastest path to a stick thrown in the ocean.

"Do any other animals have a concept of 1, 2, and 3? Can they, like humans, learn math? The answer is a definite yes. And we’re not just talking about apes and chimpanzees here, our nearest neighbors on the evolutionary tree. Small brained creatures such as rats and birds also have numerical abilities, which can be improved with training."

from The Math Instinct: Why You're a Mathematical Genius (Along with Lobsters, Birds, Cats, and Dogs) by Keith Devlin

This ability, Devlin says, is very similar to our ability to learn language. For Devlin, our ability to understand the relationship between numbers is very similar to our ability to understand the relationship between words. That’s one reason why Devlin is also executive director of the Center for the Study of Language and Information at Stanford. He researches using different media to teach and communicate mathematics to diverse audiences, and on designing information/reasoning systems for intelligence analysis.

You can submit questions to Keith Devlin through April 15. Ask him about the life of the mathematician, about how he translates math to the layperson, about the relationship between math and language, about the innate ability of all of us (and our pets) to do math, and how he thinks the digital age will affect the work of the mathematician.

Keith Devlin's Website

When did you begin to like math? Did you like math in high school?

I went to high school (in England) in 1958, and that was the year the Russians launched Sputnik, the first human-made spacecraft. Like many other 11-year-old schoolboys, that blew me away and I decided right away I wanted to be a space scientist. I didn’t know what that really meant, but I knew I’d have to be good at math. Trouble was, I wasn’t. Not that I was that bad, but I didn’t like it and it didn’t interest me – just a bunch of techniques to solve problems, or so I thought. But with the strong motivation to become a space scientist, I worked hard on the math, and over the years got pretty good at it. Then, when I was sixteen or seventeen, and starting to think about what to study at university (in England back then you had to choose your major before you went to university), it suddenly hit me. I was enjoying the math much more than the science (the physics and the chemistry). And the reason was, it had all started to fall into place and make sense. It was no longer a collection of disjoint techniques. I saw that it was a vast and beautiful mental landscape that generations of human beings had created over three thousand years.

Without the motivation of wanting to be a space scientist (and of course I never became one), I would never have discovered what mathematics really is. I suspect many other young people are less fortunate than I was. It seems that many people need to learn quite a lot of math before it makes sense (I did), and if you don’t enjoy the early stuff, as I didn’t, chances are you won’t get far enough to see what you’re missing.

You've worked significantly to bring many issues in math to the general public, from NPR to books. This might seem general, but why should math matter to non-mathematicians? Beyond the obvious skills for, say, taxes and groceries, so much of math often seems too abstract, too "out-there" for it to apply to actual reality; so where does it come into play?

Well, if you view education as being learning “useful skills”, then few people need to learn much math. You should drop it immediately. And while you are at it, I suggest you stop spending time learning anything else that is not really “useful” to you – reading (apart from the sufficient basic skill to read a bus timetable or the contents list on a packet of breakfast cereal), music (apart from being able to sing happy birthday to your friend), art, drama, science, etc. Hey, when you start to think about it, just how much of what you learn at school do you ever really NEED or USE (to pay taxes and buy groceries)? Almost nothing. We could scrap school altogether.

Of course, your life is likely to end up pretty dull. What if you wondered how that little iPOD managed to store so much music? (The answer is that it uses some of that abstract math that is just “out-there”, so if you maintain that it is doesn’t apply to actual reality then your iPOD can’t be real, can it?) What if you wondered how your mobile phone works, how modern movies and videogames are made, why airplanes can fly, how the ATM at your bank works, how music synthesizers work, or how Google manages to find just the article you need to finish that essay. All of these use masses of that “out-there,” abstract math. In fact, I’ll let you into a secret. Almost EVERYTHING you use every day depends on abstract math. Are you really not interested? Not even a little?

Sure, you can use all those things without understanding how they are made or how they work. But isn’t life more interesting, and more fun, when you know something about how it works. That’s why it’s really important for everyone to learn some mathematics: so you can understand the world you live in. More generally, that’s what K-12 education is about. It’s not to train you to do “useful” things; it’s to equip you to live a rewarding and enjoyable life as a full member of society.

What do you mean when you say that humans are born with "spatial reasoning abilities"? Wouldn't that be contradicted by the fact that some people (like me) are incapable of determining what 3D shape a given 2D cutout would fold into?

Visualizing 3D shapes from 2D cutouts is a highly specialized ability that many people don’t seem to have, though it does appear that anyone can get better at it with practice. But you use huge amounts of spatial reasoning ability when you catch a ball, return a tennis serve, or judge whether you have time to make a left turn before that car heading towards you reaches the junction. And we all do those things. Those are the aspects of spatial reasoning ability I am referring to in my book The Math Instinct.

The concept of zero seems so obvious to us today, but it took many centuries for it to take its place beside the whole numbers. What stopped mathematicians from making the leap, and what historical or cultural forces eventually led to zero? How did the lack of zero impact mathematicians, how did they work around this limitation for so long, and how did the eventual introduction of zero influence mathematics?

The difficulty people had in accepting 0 as a number was that for hundreds of years, numbers were viewed essentially as adjectives, that tell you how many objects there are in a collection or how long a line is, etc. And in that context, 0 has no meaning. You can’t have a collection with NO members – you don't have a collection! Likewise, you can’t have a line of NO length – you don’t have a line! The place-value system for writing numbers required some form of marker to show that a column did not have an entry, but that was viewed as just that: a place marker. It was relatively recently that mathematicians started to think of numbers as abstract entities (nouns rather than adjectives), and then it made sense to include 0. So the introduction of 0 did not really influence mathematics, rather it was a consequence of a shift in the way people thought about numbers.

I know how to find the values of hyperbolic functions with complex arguments, but what about with regular trigonometric functions? The output appears to be the complex conjugate of the corresponding hyperbolic function, but I don't know how they came up with this answer, except by using the exponential definitions for the functions. Is there another way?

The theory of functions of complex variables is one of the most beautiful and elegant parts of mathematics, that shines a high intensity floodlight on many parts of mathematics dealing with real numbers, where there are results that seem curious or surprising. It would take way too much of a digression to answer this question properly, but I urge the questioner to seek the answer by reading a book on complex analysis. There are few experiences in mathematics more rewarding. Go for it!

I know Pi is a Greek letter, but do you know how and why was it selected for the mathematical formula Pi?

The ancient Greeks knew that when you divide the circumference of any circle by its diameter, you always get the same number, and they knew that this number is just bigger than 31/7, but they did not denote it by the Greek letter “pi.” The first person who did that was a little known English mathematician called William Jones, in a book published in 1706. Most likely it stood for “periphery.” It became the standard symbol when the famous Swiss mathematician Leonhard Euler used it in a book he published in 1737. Incidentally, this year is the 300th anniversary of Euler’s birth.

You’ve said that animals have some mathematical ability. What symbols do chimps have for numbers? Can my cat do math? What kind and how?

Scientists have taught some chimps to recognize the same symbols for numbers that we use, up to around 20 or so. But I would not call that mathematical ability, it’s just recognizing certain patterns. Talk about using symbols misses the point, however. One of the main things I was trying to show in my book The Math Instinct, where I give many examples of mathematical abilities exhibited by animals, is that a lot of math has nothing to do with using symbols. Using symbols is just one way of doing certain kinds of math. When birds or salmon or whales or monarch butterflies migrate over thousands of miles, they MUST be doing trigonometry, but they don’t do it using symbols the way students are taught to do it in the math class. In fact I am pretty sure they are not in any sense aware that they are doing math. But so what? Neither is your calculator, but it can do arithmetic, trigonometry, and all sorts of other mathematics.

As a teacher of high school mathematics, how can I connect with my students to make them interested enough to stay with the algebra or geometry long enough to realize the benefits of their perseverance? How do you reach those students who struggle?

Why not show them (notice I said “show”, not “tell”) some of the benefits? Few of us enjoy a long journey, particularly if it is difficult, if we don’t know where we are going and why we are heading there. My own view is that all of mathematics up to calculus is within anyone’s grasp, and 95% of success is motivation.

How you motivate depends on your interests and to a greater extent those of your students. As my first answer above showed, what worked for me as a school pupil was the importance of algebra and geometry in understanding physics. That was my motivator. Algebra and geometry are after all languages, and by and large people will put in the effort to learn a language only if they want to go to a country when it will be useful to understand what is going on or make themselves understood.

The world we live in is full of applications of algebra and geometry. A few years ago I worked on a six-part PBS television series called “Life by the Numbers” where we interviewed dozens of people from all walks of life to see how they made use of mathematics in their careers and lives. We filled six hours of program that way. We could have filled sixty hours. You can still buy the entire series on DVD from Monterey Media. So one easy way to motivate your students is show them some clips from the video; each segment is about 7 to 10 minutes long.

I am very curious as to what the life of a mathematician is like - do you try to create new theorems or teach math?

Mathematicians do all kinds of things, depending on their preferences and their jobs. As a university mathematician, I sometimes try to solve problems (most recently for the US government, to try to find better ways to prevent terrorist attacks, a task that involves a lot of mathematics); I sometimes try to prove a theorem (at least I used to, I haven’t done that kind of thing for many years now); and I teach mathematics to students. I also run a research center at Stanford that studies issues of language and communication, I travel a lot across the country and internationally as part of my job, I write books (I just finished a companion book to the CBS television crime series NUMB3RS), I do my radio appearances as “the Math Guy;” I ride my bike (usually between 20 and 100 miles a day), watch movies occasionally, and have a level 63 warrior in “World of Warcraft” that needs my help from time to time. (The more math you use the better you succeed in World of Warcraft, which is built entirely on mathematical principles.)

I am under the impression that all languages have a tie to math in some manner. Proper grammar is composed of relatively rigid rules, and consequently, one could say that the calculations involved are somewhat mathematical. And I think math is related to how our brains function. Don't our brains use numbers to perform processes? I know that ultimately biology is the key to our brains' functions, but are numbers not used to help us in many ways? In addition to standard calculations, aren't numbers used to make decisions, etc.?

Well, as the linguist Noam Chomsky showed, you can use mathematics to study certain aspects of language. But linguists also tell us that there is no such thing as “proper grammar” and that the “rules” that people write down are not at all rigid. Now we are getting into an area that has interested me for about twenty-five years, and is why I ended up directing a university center called the Center for the Study of Language and Information here at Stanford. The fact is, the structures of language, particularly the grammatical structure (what we call “syntax”) does exhibit features that I would describe as “mathematical,” and that is why Chomsky’s approach in the 1950s was as successful as it was. There are still mysteries surrounding language, particularly how the human brain acquired the capacity for language, but the best introductory account I know of the mathematical study of language is Steven Pinker’s book The Language Instinct.

As to using numbers to make decisions, it is certainly possibly to make deliberate, conscious decisions using numbers, and often you get much more reliable answers if you do that, but there is some pretty overpowering evidence that most of our decision making does not involve numbers at all. The evidence is this. Humans have had numbers a mere 10,000 years, whereas our decision making abilities go back to much much earlier than that. We (and some other creatures) do use counting, and we (and many other creatures) make judgments about sizes of collections, heights of trees, etc., but neither of those require numbers.

There is quite a bit of evidence today that the human brain is not at all numerical, rather numbers are things the brain constructs using language. I describe some of that evidence in my book The Math Gene.

Being more of a language person, I'm not really the most interested in math. I like the concept behind it, but doing it doesn't really engage me. Are there any ways to change this? Do you know of any books that could help me become more at ease with mathematical operations?

Since mathematics is just another language, I guess I don’t really understand the question, unless you mean you’re more of an ENGLISH language person. Russian is the language of Russia, Chinese is the language of China, musical notation is the language of music, and, as Galileo famously observed, mathematics is the language of the universe. So I can paraphrase your problem (if you think it’s a problem, I’m not sure I do) as you not being interested in the language that describes the universe we live in.

Well, I have never been interested in Russian or Chinese. Most likely that is because I never had the opportunity or the strong desire to go and live in Russia or China. I did live in Germany for many years, and learned German, and I love to spend time in Italy so I have learned some Italian. So I imagine I COULD learn Russian or Chinese, I just never tried. It’s hard learning any foreign language, and there’s only so much time to do the things we’d like to. But I can’t say I’ve ever thought that I’m more “an English language person” than “a Russian language person.” That’s why I find your question strange, at least as you formulated it.

Now, if we dig deeper, I think there is a lot more that can be said. There are certainly differences between the language “mathematics” and languages like English and Russian, but the differences are really about the worlds the languages they describe, not the languages themselves. But maybe this is what your question is really about. If so, then this is what I address in my book The Math Gene. Since we generally feel more at ease with something when we understand why we start out feeling UNEASY, you might want to read that book. Not only is it now available in cheap paperback format, it’s written in ENGLISH (i.e. not in mathematics), the language you like!

37 Under 36: America's Young Innovators

Smithsonian, October/November 2007

This issue of Smithsonian magazine profiles 37 amazing people under the age of 36 who are helping shape the world. Innovators include:

Elizabeth Catlos, geologist, takes a new look at where the world's highest mountains come from.

Philippe Cousteau, taking up the family business, campaigns to save our oceans and rivers.

Matt Flannery, software engineer, pioneers Internet microloans to the world's poor.

Christina Galitsky, engineer, designed an energy-efficient cookstove to make life a little easier for Darfur's refugees.

Brian Hare, primatologist, investigates the social behavior of chimpanzees and bonobos in Africa. But dogs and foxes showed him the way.

Lisa Kaltenegger, astrophysicist, analyzes light from distant stars for evidence we're not alone.

Jon Kleinberg, computer scientist, helps us see the invisible networks that pervade our lives.

Aaron O'Dea, paleobiologist, studies underwater extinctions.

Jennifer Richeson, psychologist, explores how prejudice affects people.

Joshua Schachter, del.icio.us developer, invented a deceptively simple tool that helps us all de-clutter the Internet.

Beth Shapiro, biologist, has figured out a recipe for success in the field of ancient DNA research.

Terence Tao, mathematician, is regarded as first among equals among young mathematicians, but who's counting?

Amber VanDerwarker, anthropologist, is unraveling the mysteries of the ancient Olmec by figuring out what they ate.

Luis von Ahn, computer scientist, develops "games with a purpose" that accomplish all sorts of useful tasks.

John Wherry, immunologist, is racing to develop a vaccine that provides lifelong immunity against influenza.

Michael Wong, chemical engineer, is cleaning highly polluted groundwater with a detergent based on gold.
http://images.smithsonianmag.com/content/innovators/index.html

Cogito Interview: Harold Reiter, Problem Spinner

by Amy Hodson Thompson

Cogito, 05.02.2007

Preparing for a math competition? Like solving math problems? Harold Reiter is the guy to know. In the UNC Charlotte math professor's nine years of composing problems for the MATHCOUNTS national math enrichment, coaching, and competition program for middle-schoolers, he's composed more than 2,000. More about Dr. Reiter...

Discuss MATHCOUNTS in Cogito’s forums.
More related forums...

Q & A with Dr. Reiter

What math class did you take as a 9th grader?

I took Geometry in ninth grade, just like everyone else. It was a good class at a good school (in Lafayette, Louisiana) and I learned what it means to prove a theorem. Today we don't teach the course in this way. We didn't have acceleration, but the courses were what today would be called enriched.

What is the best way to study for a math competition in general? For MathCounts?

Do problems. Put yourself in the same position as a competition participant and take an old contest using the same time allowed during the contest. Then learn about the problems you've missed.

Harold Reiter, PhD

Hometown: Shreveport, Louisiana
Education:
PhD, Mathematics, Clemson University
Interests: travel, spending time with my one- and three-year-old grandsons
Free-time Favorites: reading light novels (like John Grisham), running, racquet sports, watching or playing bridge at Bridge Base Online
Favorite treat: Dark Chocolate

What is your favorite "principle" in mathematical problem solving? Is it the invariant principle?

My favorite idea for problem solving is to 'trade in' a hard problem for an easier problem. Then make the easier one a little harder as I get a better understanding.

What are your favorite math questions?

I have two favorite MATHCOUNTS questions, both from MATHCOUNTS Handbooks. The first is one about the difference between number and numeral. Here it is.

Consider the following list.

Are these numbers getting larger or smaller?

A few years later, I was asked to write a handbook Stretch about combinatorics. What appeared was a sequence of 10 questions each of whose answers was the following binomial coefficient.

I picked six different ways to model the problem, but the solution was the same.

As for higher level problems, check out my column in Mathematics and Informatics Quarterly and my problems at the London Sunday Times.

What is the most interesting MATHCOUNTS question you've written?

Although it has been a few years since I've written MATHCOUNTS problems, one that comes to mind is a shortest path problem on a many sided polyhedron. The net of faces is given and the endpoints of the path are given. I think some students actually built the polyhedron to solve the problem.

What thought processes do you use when creating particularly enjoyable math problems?

I like to try creating problems in bunches of 8 to 12 using a combination of two or three ideas. I'm particularly fond of counting and existence problems that use (a) the inclusions/exclusion principle, (b) polygons in the plane all of whose vertices are integer lattice points, and (c) the Pythagorean Identity. Here's an example. How many squares in the plane have two or more vertices in the set {(0,0), (0,1), (1,0), (1,1)}?

What kind of topics and thinking do you try to cover with your questions?

I enjoy problems that combine two or three ideas. I play around with number theory problems that involve number of divisors, sum of digits, and product of digits. I'm especially fond of problems that hinge on a repeated process that has an easily discovered invariant. See the M&IQ problems mentioned in my first answer.

How do you think of all of the interesting properties of certain groups of numbers that make the problems work? Do you start with a group of numbers and look at properties, or do you think of an interesting property and try to find numbers that work with it?

Usually, the latter. For example, suppose I'm thinking about problems involving remainders. I would like to test students' understanding that you can compute the remainder of a sum by first taking the sum of the remainders. Now what set of numbers could we use for such a problem? Of course, Fibonacci (or Lucas) numbers! The problem I write is: What is the units digit of the 2007th Fibonacci number?

How is mathematical problem solving related to mathematical research, and what is the relationship between the two? How does one combine these sometimes seemingly different creative endeavors in a harmonious and fruitful way?

These are hard questions. For some mathematicians, problem solving IS research. For me, that is sometimes the case. For others, problem solving might come into play, but the focus might be on theory building. In that case the problem solving is peripheral. The excitement and satisfaction we all get from reaching inside ourselves to find ideas that we didn't know were there is certainly part of both research mathematics and creative problem solving.

More about Harold Reiter…

Reiter's daughter Ashley drew him into the world of composing when, as an inquisitive sixth grader, she'd ask him to make up problems for her as they were driving in the car. "We began a family problem solving adventure," says Reiter. Reiter was composing puzzles for Ashley regularly, even on family vacations. "She liked it, so I began to like it too. I realized that I could write good problems and it was fun." Reiter's plate

Ashley was the first girl to make it to the top 10 in the MATHCOUNTS national competition (1987), and she grew up to earn a PhD in mathematics herself (See the article The Unity of Mathematics that she wrote for Imagine magazine in 1998). Hooked on composing, Reiter began writing problems for the American Mathematics Competitions, then volunteered to write for MATHCOUNTS in 1990, starting his first four-year term on the seven-member committee in 1991.

Although Reiter no longer composes for MATHCOUNTS, he is still passionate about the program and problem solving. When I caught up with him, he was just back from a jam-packed weekend of coaching eight North and South Carolina middle-schoolers in preparation for the MATHCOUNTS national finals on May 11, 2007. Although weekends like these admittedly wear out the soon-to-be 65-year-old Reiter, he has no intention of stopping. "I have no plans to retire," he says, "my work has never been more satisfying and enjoyable."

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Discuss Math, or MATHCOUNTS, share some Math Humor, or solve some problems.

Related Content

The Unity of Mathematics

by Ashley Reiter

Imagine, March/April 1998

If linguists study languages, and chemists study chemicals, what do mathematicians study? Children might identify numbers as belonging to the realm of mathematics. Students of algebra might observe that mathematicians study equations and solutions to equations. More advanced students might expand this view, identifying sets and functions as major objects of mathematical study. As students continue to study mathematics, they may encounter an even greater number of mathematical objects—groups, rings, fields, vector spaces, operators, metric spaces, manifolds, and so on. But what do all of these objects have in common?

As a math major in college, I became acquainted with a wide array of mathematical objects much in the manner of a child visiting a zoo—intrigued by each species in succession, but rarely stopping to consider what features distinguish amphibians from reptiles, or what characteristics all mammals share. But in graduate school, as my mathematical vision widened and deepened, I began to appreciate the unity in the menagerie of mathematical structures.

Learning by Abstracting

One of the most important unifying ideas of mathematics is abstraction. You use abstraction when you first understand 5 as a concept, independent of any particular five objects, such as five fingers or five apples. You then have the ability to abstract further, to perform arithmetic operations, such as 5+5, without referring to a particular set of five things. Furthermore, once you learn that 5+5=10, you never need to repeat the process of counting the fingers on each of your hands to see that the total number of fingers is 10. Your discovery—that 5+5=10—is independent of all objects and can apply to all objects.

As you progress through elementary and high school mathematics, you continuously abstract from what you've learned earlier to understand new mathematical structures. In algebra, for example, you need not test all possible values for x to determine a solution to the equation x+5=10; rather, you can solve the equation by using your knowledge about the properties of addition that hold for all real numbers. Next, you can learn to solve all equations of the form x+5=? by considering a new mathematical structure, the function f(x)=x+5. At each step, the new structure you encounter is not an entirely new idea but a generalization of those studied earlier.

Abstracting makes the original mathematical concepts easier to understand by showing how both old and new pieces fit together into one big picture. For example, when you learn to solve both linear and quadratic equations, you might notice differences between them: a linear equation has at most one solution, while a quadratic equation has two. But as you generalize from linear (first-degree) and quadratic (second-degree) functions to study solutions of higher-degree polynomial equations, you see that the degree of the polynomial reveals the number of solutions. That quadratic equations have exactly two solutions is just a specific case of this more general truth.

Expanding Circles of Mathematical Study

In one way or another, nearly every branch of mathematics is a generalization of either arithmetic, the study of numbers and their interactions, or geometry, the study of shape. In fact, students may be surprised to see where generalizations of the math they're learning now can lead them.

Polynomial functions are an important abstraction of arithmetic. After learning to find solutions to these functions in high school, calculus students delve deeper, detecting maxima and minima of polynomials and other functions, and noticing the ways in which the functions increase and decrease. Students can use polynomials to approximate many different functions, from the exponential growth of a bacteria culture to the periodic behavior of the ocean tides.

Further abstraction of these same algebraic pursuits can lead to other fascinating areas, such as the study of fractals. When students study linear algebra in college, they generalize from the linear equations they learned about in high school, which have a straight line for a graph, to functions in many dimensions, whose graphs are flat lines, planes, or spaces. In linear algebra, students learn how to classify objects as one-dimensional, two-dimensional, and so on. If students generalize these ideas to sets with fractional dimension, they'll find themselves studying fractals.

College math students might also use their understanding of linear equations in many dimensions to study other functions in several dimensions. Solutions to these functions look quite exotic, like a whole sphere or the surface of a doughnut, and are studied by topologists and algebraic geometers. Again, by progressively generalizing from the knowledge gained in high school mathematics courses, students can eventually end up in some of the most interesting and current areas of mathematical inquiry.

It's tempting to perceive different areas of mathematics as separate from one another, especially when they're taught in different courses. But by studying the history of mathematics, students can start to see relationships between these different areas. Just as modern mathematicians build on the discoveries of those who came before them, students can understand mathematical ideas in an order that prepares them for more advanced study and, perhaps, to make further discoveries of their own.

About the author: While on hiatus from graduate school in math, Ashley Reiter taught at the Maine School of Science and Mathematics. She is looking forward to spending the summer traveling, reading, and thinking more about mathematics.

For Further Reading:

Newman, James R., The World of Mathematics. New York: Simon and Schuster, 1956.

Steen, Lynn Arthur, ed. On the Shoulders of Giants: New Approaches to Numeracy. Washington, D.C.: National Academy Press, 1990.

Cogito Interview: Jason Bardi, author of The Calculus Wars

Cogito, 03.02.2007

Science writer Jason Bardi has just published his first book, The Calculus Wars, about the dispute between Newton and Leibniz over the invention of calculus. Cogito interviewed Bardi January 29-February 20, 2007.

Who is Jason S. Bardi?...

Mr. Bardi, I know studying Calculus is worthwhile if you are pursuing a higher education in math/engineering. But how has Calculus benefited you in your everyday life decisions? (from a Cogito member turning 15 today)

Happy birthday, and thank you for your interest in my book The Calculus Wars.

If you are 15 today (or rather a few weeks ago when you originally wrote this), then you are probably already staring down the dizzying array of big choices you must face in your last few years of high school and first few years of college. Probably you and your classmates are already asking each other “where do you want to go to college?” Soon enough you will make your choice, and in college the question will become “what is your major” and then “what are you going to do after graduation?” etc.

Want to talk about Math? Do problems for fun? Members can discuss and solve in Cogito's forums.

All along, it will seem like these major choices engender a thousand lesser (but often no less difficult) choices. For instance, what classes to take can become quite complicated as you balance requirements, recommendations, and your own interests. Often the easiest thing may be to self-limit your choices and take only those classes that are required of you. So if you are getting a degree in math, science, or engineering, you don’t really have to consider the worth of calculus because it’s a required course. And if you are in the humanities, you don’t even consider taking calculus because you are not getting a degree in math, science, or engineering.

Well consider this: studying calculus and mathematics in general may be more worthwhile if you are NOT pursuing a higher degree in math, physics, or engineering because the classes you take will expose you to analytical abilities you never knew you had. You learn how to solve problems logically. You are exposed to mathematics that gives you a better understanding about physics and the world around you. It may not help you throw a perfect curveball on the baseball field, but you will certainly understand why the ball curves.

I am a big believer that math education at all levels is useful regardless of where you wind up going in your career. I know I benefit every day from having learned calculus, and I wish I could clone myself and study it even more.

How did you get interested in this subject? Sometimes writers stumble upon a subject by researching something related. Is that what happened to you?

In my case, this is not really what happened. The fight between Newton and Leibniz is a famous story, and anyone who has ever taken a single calculus course in their life has probably heard about it. My own college introductory calculus textbook was where I first read about Newton and Leibniz, in fact. I can still picture the sidebar—one of those spot-color-filled boxes with a picture or two and a few paragraphs about the dispute. Even after all these years, I can still remember what it was that stuck with me. I remember being fascinated by the picture of Leibniz because he had a peculiar smirk on his face. Was he angry? Was he amused? I couldn’t tell.

This was my first exposure to the subject matter of my book, and when I was first starting to write it, I searched desperately for that old textbook, but I must have discarded it years before. It’s funny the books you throw away, thinking you will never look at them again.

Find out more about the book: read Cogito's Walking with Giants: Synopsis of The Calculus Wars.

But none of this is to say that you are incorrect. Writers do often stumble upon their subject matter by chance, or they find inspiration from unusual corners. Take the example one of the masterpieces of German literature, the tragic play Faust by Goethe. Faust is the story of the man who makes a bargain with the devil in order to enjoy worldly power, and it derives from the play Doctor Faustus, which was written by Shakespeare’s contemporary Christopher Marlowe more than a century before Goethe was born. My understanding is that Goethe’s first exposure to the story was in the form of a silly puppet show, and it was his genius to recognize that it could be turned into great art.

Did you have to have a good understanding of mathematics to be able to write a book like this?

At first glance, I thought you were asking whether you need a good understanding of calculus to read my book, and I reflexively started answering that question. I have told so many people I know who have no background in mathematics that indeed they do not need to know calculus in order to read and enjoy my book. Certainly it helps, but I make no assumptions as to my readers’ level of education.

Now your actual question is a very good one. While my own background in mathematics is not so solid or so recent that I would consider myself a mathematician, there is no question that the numerous calculus courses I had in college helped me out a lot while researching and writing this book. For instance, in my book I discuss the brachistochrone problem, which was devised to challenge Newton’s mathematical ability in 1696. I can actually recall having this same problem assigned to me in one of my physics classes. I remember puzzling over it long and hard one weekend and hitting the wall a number of times.

There is no question that this sort of experience helped me write this book. I am not a mathematician, but I drank from the cup as it were—albeit just a sip and probably choked even on that. A real mathematician like Newton is someone who would drink deeply from the cup of mathematics. Incidentally, he solved the brachistochrone problem in just a few hours late one night after working all day.

But to finally answer your question, I guess I would say that some people might have been able to write this book with no background in mathematics, but I doubt whether I could have.

How did you do the research for this book? How do you determine what is a valid source?

Newton and Leibniz were obviously both extremely famous, and pretty much everything they ever wrote or did has been recorded, reported, analyzed, debated, and reanalyzed. These figures sit atop two mountains of scholarship built so high, climbing up seemed almost insurmountable. For me the most relevant question in some ways was not what to read but what to ignore.

But to answer your question, I did read many, many books and journal articles, which are described at the end of my book. I spoke to a few top scholars in the field. I traveled to England and Germany and examined some of the original source documents and walked some of Newton and Leibniz’s old haunts. I spent a considerable amount of time in a handful of different libraries across the U.S. and Europe reading rare books in their collections, and I cobbled together quite an extensive library of my own by buying used books from bookstores all across the United States and Europe.

Reading became a big part of my life for a long time. For a year and a half I never went anywhere, it seems, without carrying along one or more books related to this subject. I stopped reading about anything else. I got married while working on this book, and I brought a dozen books with me on my honeymoon (and I am happy to add that I did not find the time to read them).

What were most surprised to learn about Newton and Leibniz?

Newton was into a lot of things that we would consider strange and even cultish today, such as his livelong love of alchemy, his fascination with interpreting biblical revelations, and his obsession with history and chronology (something for which he was highly regarded in his day). But the most interesting thing for me was that he had a whole second career as a public servant in charge of the British mint. The last few volumes of his collected correspondence are filled with detailed administrative letters pertaining to these duties, and some of the more interesting bits I discuss in the book.

Leibniz was just simply a fascinating character, and there was not any one thing about him that amazed me so much as there were hundreds of things. All the mind-blowing schemes and plans he had, for instance, which I also briefly discuss. I suppose if I had to choose one thing that surprised me the most, it would be the story that I used to open Chapter Eight in my book. Leibniz is on a ship battered about in high storms and about to be killed by the crew, who think he has visited some ill fate upon them because he does not pray in the same way that they pray. But Leibniz knows enough of their language to deduce what’s coming, and he torpedoes their plan by pretending to pray as they would pray.

Why did you decide to become a science writer as opposed to a scientist? How was JHU's science writing major?

Personally I was always more interested in writing about science than I was in doing it, but looking back, I don’t think I really realized that until I had spent some time in the laboratory working on actual research projects (and not the challenging but neatly designed sort of projects you might do while taking a class in science). My graduate school advisor was very understanding and supportive when I decided to switch, and this was a great help.

In some respects I really do miss biology. The intensity. The informality. The great ambitions. The strange and diverse personalities. The ceaseless learning. Many laboratories resemble the U.N. in the sense that the students, postdocs, and even professors hail from all over the world. The human interactions alone can be richly rewarding. And most of all, biology and science in general offers the rare opportunity to contribute to research that may eventually lead to some direct benefit for humankind.

Still, the grass is always greener. Some scientists I talk to are very envious of what I do because to them I have the ability to learn about a wide variety of the most cutting edge science in such a wide variety of areas. This is very appealing to them.

The JHU science writing program was top notch. I loved it… especially the flexibility to choose electives that suited my interests and the opportunity to interact closely with the students in my own program as well as the fiction and poetry programs. The advisors were great, and looking back, that was one of the most intense years of my life. If there is one thing I regret, it is that I did not pursue an internship. I believe these can be valuable not only for getting experience and clips but for really discovering what it means to be a science writer. But as Frank Sinatra said, that’s life.

You were a biophysics major before, right? That's an interesting combination! I haven't heard much about that major. What's it like?

I suppose a technical and expansive definition would define biophysics in terms of its techniques, its community of scientists, its departments and programs, its vast literature of past discoveries, and its leading theories. If that’s what you want, stop reading this and go enroll in a biophysics course. What might be more interesting for you to hear here is what biophysics is not. Biophysics is not really a discipline distinct from the rest of the areas of biology. It is really as much a part of biology as biochemistry, molecular biology, or any of the other biological disciplines.

When I was in graduate school, more than ten years ago now, the traditional divisions between these disciplines were breaking down. Johns Hopkins in general and my biophysics program in general were on the cutting edge of encouraging this breakdown. My program brought in people from different backgrounds—computer programmers, physicists, engineers, etc.—and we worked together with chemists, biologists, and many others on topics like the structure and function of various biological macromolecules, computer modeling of protein-protein interactions, and protein design, just to name a few.

Nowadays I think that the distinctions between disciplines has blurred even further. This is probably a good thing because in our post-genomic, so many of the burning questions in biology can probably only be approached with a multifarious approach that employs all the latest tools of biophysics, biochemistry, chemistry, and molecular and cellular biology combined.

By the way, when I was an undergraduate, I majored in physics and English. A lot of people I knew thought that was a particularly interesting combination.

Bardi signing off: OK that’s all for now. Thank you for the questions and be sure to visit my own web site The Calculus Wars.

Who is Jason S. Bardi?

Jason Bardi earned graduate degrees in molecular biophysics (M.A., 1998) and science writing (M.A., 2001) from Johns Hopkins University, and has since worked as a professional science writer. He spent a year as a writer at NASA's Goddard Space Flight Center, five years as the senior science writer at the Scripps Research Institute in La Jolla, CA, and is currently a writer and editor specializing in the sciences. He lives in College Park, MD. He's currently working on his second book, about the development of non-Euclidean geometry. (information from The Calculus Wars book jacket)

Your Brain on Music, Magnets, and Meth

01.01.2008

No one has seen oddities of the mind quite like Oliver Sacks has.

by Susan Kruglinski

Tucked away in the cabinets of Oliver Sacks’s Greenwich Village office are hundreds of small black notebooks, each filled with jottings and sketches, newspaper clippings, and photos. These are the accumulated reflections from a lifetime spent observing the extraordinary ways the human brain can misfire and misbehave: a man who believes his own leg does not belong to him, an autistic woman with a gift for understanding animals, and the man who mistook his wife for a hat—the case that inspired one of Sacks’s most famous books.

What people may not know about Sacks, however, is that the 74-year-old neurologist has spent much of his career regularly treating patients in mental-health facilities around New York City. Those patients have more commonplace problems such as dementia, sciatica, gait disorders, and seizures. He does love the challenge of an unusual case, of course, and those kinds of cases keep finding him. After his book Awakenings was adapted into an Oscar-nominated movie starring Robin Williams, the letters started pouring in, and they continue to today. Many are from people who are experiencing an interesting neurological phenomenon, or know someone who is. “My assistant Kate removes about nine-tenths of them,” Sacks says. “That leaves me about a thousand per year to read.”

In his latest book, Musicophilia: Tales of Music and the Brain, Sacks focuses on unusual cases having to do with music’s effects on the mind, such as a man who found relief from Tourette’s syndrome by playing the drums, and another who was driven to the edge by an unwelcome and unending tune that cycled uncontrollably through his head.

Seated at an aging wooden desk in front of a wall of tacked-up photos that include snapshots of past patients, Sacks spoke to DISCOVER about his recent musical investigations, his experiments with unlocking his own secret wells of creativity, and why he accentuates the positive with his patients whenever he can.

You are so famous for your books that most people don’t realize you have a day job. What is your regular work like?
Well, I see a few patients. Some of them are in nursing homes or in chronic disease institutions, like Beth Abraham Hospital, where the events in the book Awakenings happened 40 years ago, or the Little Sisters of the Poor, whom I’ve also been with for almost 40 years. I also go to a clinic, and I do a few house calls, which I’m fond of. For example, I personally called on a lady with amusia [the inability to perceive musical tones and rhythms] in the Bronx.

A world without music: What is her life like?
This is a delightful, intelligent lady, a former schoolteacher, who from her earliest years has been unable to recognize any piece of music, or indeed to hear it as music. She herself said to me, “You want to know what I experience when you play music? Go into the kitchen and throw the pots and pans around. That’s what I hear.” So although there are people who are considerably tone deaf, this is nothing compared to the absolute inability to perceive or conceive of music, which this otherwise gifted and articulate woman has. She was extremely relieved to find out that this so-called congenital amusia has a clear neurological basis. It’s not just in her mind; other people have it, although it’s pretty uncommon. She used to go to concerts with her husband. She says she wished she had been diagnosed 70 years earlier. She might have been spared a lifetime of being polite but bored, bewildered, and sometimes excruciated while listening to music.

I found myself with this useless, floppy leg on a mountain. Then I found the Volga Boatmen song going through my mind. I was being “music-ed” down the mountain.

In Musicophilia, you describe an intense personal experience with music 33 years ago, after you badly injured your leg while mountain climbing. How did music help you?
I found myself with this useless, floppy leg, and I was up at five or six thousand feet on a mountain. No one knew where I was—this was before the era of cell phones—and I had to try and save my life. I happened to have an umbrella with me, and I snapped off the top and splinted my leg. I tried to move myself down the side of the mountain, pushing myself along with my elbows, which was quite a cumbersome movement. Then I found the Volga Boatmen song going through my mind. I would make a big heave and a ho on each beat in the song. In this way, it seemed to me that I was being “music-ed” down the mountain. It became fun and easy and efficient. Everything was sort of coordinated and synchronized by the beat.

And music helped with your recovery, too?
Yes, after I was safe and my leg had been put together, I was still out of action neurally. In an injury like this, the body image changes. There have been functional MRIs that show this sort of thing. If a limb has been inactive for a while, it starts to lose its representation as part of the body image that is mapped in the cerebral cortex, and becomes difficult to use. Sheer will alone may not be enough to get it back; you almost have to be tricked back into action. Something spontaneous has to happen. For me, one form of this was when music suddenly came to me—this tape I’d been listening to again and again [Mendelssohn’s violin concerto]. It sort of came to me like a hallucination and got me going, triggering the ability to walk again. I once saw an older woman who had had a broken hip; no one knew why her leg didn’t move. She told me that it had moved once, when she was listening to an Irish jig.

You’ve also found that people with aphasia—the inability to speak because of neurological damage—can sometimes sing.
A good proportion can. So automatically when I see aphasia patients, I engage them in “Happy Birthday.” Beyond propositional speech and making sentences, there seems to be language embedded in song and in automatism of various sorts. I don’t like the word “automatism,” but when you recite a poem, it’s there, and it’s used as a different form of memory [than the memory of regular speech]. It’s used as procedural memory. But this doesn’t mean that it’s just mechanical.

+++

Music seems to be involved with so many functions of the brain: It can aid memory, assist movement, and trigger emotions. Why is that?
However music started—and it may be that the evolution of rhythmic sense is quite different from that of tonal sense—it has now taken up residence and demands many, many different parts of the brain, certainly more than language. And by the same token, music is very robust neurally. There are people with a huge amount of cerebral disease who are still responsive to music.

Does that suggest that music is somehow essential to human survival, or at least to social survival?
This is a big question. I can only say that there is no culture without music. There are almost no individuals without music. The lady in the Bronx is a one-in-a-million sort of exception. And in every culture, music forms a social cement for dancing, for singing. It’s invariably part of ritual and religion, and then there are things like work songs and martial music. Steven Pinker said, “Music could vanish from our species and the rest of our lifestyle would be virtually unchanged.” I strongly disagree with that and I think no anthropologist in the world would agree with that.

You’ve been fascinated with music for so long—why are you only writing about it now?
Going back 40 years, I was very struck by the therapeutic power of music with many of the patients I saw: Parkinson’s patients, patients with aphasia, patients with dementia. But just in the last 20 years, there has grown up an ability to examine the living brain when people are listening to music or imagining music or composing music and to define—in a way which would have been unimaginable 30 years ago—what goes on in many different parts of the brain when one listens to music, imagines music, composes music, et cetera. Although I was experiencing both the power of music and the varieties of musical experience 20 or 30 years ago, I couldn’t have given it the scientific backing which is possible today.

In Musicophilia, you argue that emotional responses to music may be distinct from other emotional reactions. What do you see as the difference?
I think the emotional responses to music can be unbelievably complex and mysterious and deep. You can be sort of agonized, sort of ecstatic, and you don’t know what’s happening. You can’t even say what the feeling is. The usual feelings just can’t begin to match the musical experience. On the clinical side, in some cases, people—maybe after a head injury or a stroke—suddenly cease to enjoy music, while still enjoying everything else, and while perceiving music perfectly well. And then there’s the opposite of this, which gives the title to my book: people who develop an oddly specific need for music—they must have it.

It seems strange that music can become a hunger, like the need for food or sleep or sex.
I agree. And it can be very, very specific—because often, you don’t just want music; you’ve gotta have Brahms, or you have to have a particular pianist. That exact music will speak to your condition and will fill a particular void—and nothing else can.

There’s a notion that savant abilities may be universal or latent in all of us and could be released.

And other people are musical savants, with musical abilities far beyond the norm. What have you learned about them?
Savants are people with extraordinary capacities of calculation or music or drawing, mixed with generally low intelligence—a very startling anomaly. I first saw savant syndromes in an institutionalized autistic population at Bronx Psychiatric Center. The savant I have seen in most detail is Steven Wiltshire. One really does have the feeling with him of something autonomous. There will be a brief sidelong glance at a landscape, and then he’s drawing it. He may be looking around as he’s drawing it; he’s whistling. There doesn’t seem to be concentrated attention. And it may be done in a very odd way. He doesn’t do a sketch first; he doesn’t do salient features. He will start at one edge of the paper and go over like that [Sacks makes a trilling noise and motions from one side of the page to the other]. He’s also a musical savant. Not only does he have absolute pitch, he is able to get the structure of a fugue. It’s very, very startling to see someone who is dazzling in one way and grossly defective in others. This disparity tends to be increased by practice and possibly obsession—because, of course, this may be the one highly pleasurable and rewarding thing in their lives.

What is happening in the brain of a person like that?
Some neurologists think that what may go on in the savant may be a relative preservation and heightening of primitive perceptual and computational powers in the right hemisphere—powers of a sort that are normally inhibited with the development of abstract intelligence and language. If abstract intelligence and language don’t develop, it could be possible that they may be, in a word, freer. Something which might support this idea may be the late appearance of savant-like powers in people, say, with frontal temporal dementia; it is precisely with the decline of verbal and abstract intelligence that we sometimes see this emergence of artistic powers. There are much-discussed and somewhat disputed experiments in Australia, where a researcher named Allan Snyder is using TMS—transcranial magnetic stimulation—to try and damp down the left dominant temporal lobe. I tried this myself, but it just gave me a headache after 15 minutes. And I was actually slightly afraid of the effects of TMS on my own nervous system.

+++

You stimulated your own brain in an attempt to unlock your creativity? What happened?
I was asked to draw a dog. I’m very bad at drawing. And my dog—well, it doesn’t quite look like an amoeba, but it’s a diagrammatic quadruped. It could equally well be an elephant or a mouse, pretty much. The experimenters wanted to see whether I might lose some of this abstract formulaic quality and draw an appealing dog in profile. Maybe if I’d stayed with it longer . . . but I was getting this peculiar sort of face ache. I don’t know why. It may be an idiosyncratic reaction. But certainly, there’s a tantalizing notion that such savant abilities may be universal or latent in all of us, and could be released in certain circumstances. But if the release entails a loss of enunciation—of our higher powers—it may not be such a good bargain.

Have you ever tried anything else to alter your brain function?
Well, I mention a rather autobiographic thing with amphetamines in a footnote in my new book—this is more than 40 years ago. [Sacks had experimented with massive doses of amphetamines.] I got into a very strange state for two weeks, a state in which I, who cannot draw, found myself able to do the most accurate anatomical drawings. I have a notebook from that time full of anatomical drawings of a sort I had never done before and have never done since. This also affected things like musical reproduction and sense of smell. I could recognize most people and most places by smell. And so I did have an experience myself of having various perceptual powers released. When it all disappeared, I had mixed feelings. It was a great relief, and also some regret. However, I think the amphetamines are terribly dangerous, and I’m glad I survived that time.

Some critics suggest that you romanticize your subjects—that you write sentimentally, even joyously, about some very sad cases. Why do you focus so much on the positive?
Well, I want to draw attention to it, but it’s there with the negative. In the old-fashioned medical notes, one would write about the HPC, or “history of the present complaint.” The patient comes to a doctor because something is the matter; they have a complaint. And one goes through it with the patient, but one also wants at the same time to remind them of the powers which are preserved and which they can perhaps use and which can mitigate life. My interest is very much in rehabilitation. I won’t say “recovery.” And maybe rehabilitation sounds rather technical, but it’s making the fullest possible life under the circumstances.

One of the most dramatic examples of music and rehabilitation comes in the story of your patient Clive. Can you describe this case?
Clive was an eminent musician and musicologist in England. In the mid-1980s he had a rare form of encephalitis caused by a virus, a herpes encephalitis. This caused massive destruction to various parts of his brain, but especially the temporal lobes and the hippocampal system, which is crucial for personal memories. As a result, when Clive recovered from the high fever—an acute symptom of the illness—he was profoundly amnesic; that’s to say he could not remember anything said to him or anything which happened in front of him for more than a few seconds, and there was also a retrospective deletion of memory. So what had happened for many years preceding this illness, and to some extent throughout his whole life, was gone. In a sense, he was a man without a memory. As such, he seemed utterly devastated and not there, very terrified and disorganized. But it was discovered by his wife, Deborah, who is also a musician and had been in his choir, that his musical sense—his ability to recognize and perform music at the highest professional level—was completely intact. He was able to sing, to play the piano, to conduct an orchestra or conduct a choir beautifully. So here is a paradox: The memory of events had been practically wiped out, whereas the memory of how to perform was completely intact, and in particular, how to perform music. He remained at a virtuoso level—it’s still the situation more than 20 years later.

Your chapter on Clive is strangely uplifting because he seems so happy when he conducts. That paradox really comes out in your book.
Clive was—I was going to use the F-word. Clive was tragic. One can pick out the high points, but he’s been wiped out in so many ways. I mean, mercifully, there is the music. . . . I hope I don’t romanticize unrealistically.

I think my business and the business of the physician is, one way or another, to try and help someone live—live realistically.

Walking with Giants: Synopsis of The Calculus Wars by Jason S.Bardi

by Amy Hodson Thompson

Cogito, 02.10.2007

Jason Bardi’s account of Gottfried Wilhelm Leibniz (1646–1716) and Sir Isaac Newton's (1642–1726) 10-year, down-and-dirty intellectual property rights war over the invention of calculus will dispell any idealized images readers might have of these two great thinkers. The truth is that Leibniz and Newton were (as we all are) human. Bardi’s account of “the calculus wars” puts events in historical context, and gives us a rounder picture of the two intellectual giants.

Painting Giants

Newton invented calculus in two years (1665-1666) spent at his family’s country estate avoiding a plague outbreak. By the time Leibniz discovered calculus independently in Paris between 1672 and 1676; Newton had written several articles and two books (one unfinished) on the subject, all merely shared with friends. Leibniz published his work on calculus in the mid-1680’s. But Newton published nothing about calculus until 1693.

Click here to go to our online interview with author Jason Bardi. Photo by Pam Hazen

Newton’s delay in publishing his ideas doesn’t make sense from our publish-or-perish 21st century standpoint – until Bardi explains. As a 28-year old Cambridge professor, Newton had published his highly original (and correct) idea that light is not a wave, but a particle, and even more surprisingly that white light, instead of being an absence of color, is actually a combination of all the colors. Expecting praise, Newton instead was attacked by the Robert Hooke, one of Britain’s most revered optics authorities. “Hooke,” explains Bardi in the Calculus Wars, “was also infamous as one of the most outspoken and intellectually cutthroat of the Royal Society’s members and often wielded the esteem of his position like an ax. In 1672, he set his sights on Newton’s theory of colors, sending the Royal Society a condescending letter claiming to have performed all the experiments himself, prior to Newton. Personal attacks, the plague, and the fire of London’s near-extinction of the publishing industry are all painted in vivid detail by Bardi, and combined, concludes Bardi, made Newton lose his taste for publishing for decades. He didn’t fully regain his publishing stride until the turn of the century.

Dreamer, Idealist

Leibniz, on the other hand, appears to have no reservations about publishing either the ridiculous or the enlightened. He was a prolific publisher, and Bardi recounts a tale of Leibniz piecing together an absurd manuscript of fancy words and meaningless phrases to impress and gain entrance to an alchemical society, which he soon spurned. To visionary, idealist, and philosopher Leibniz, calculus was but a part of his studies. Bardi writes of Leibniz, “He saw all human ideas, concepts, reasoning, and discoveries to be a combination of a small number of simple, basic fundamental elements – like numbers letters, sounds, colors, and so on. Leibniz hit upon the idea of creating a universal system that would provide a way of representing ideas and the relationships among them – an alphabet of human though with which ideas, no matter how complicated, could be represented and analyzed by breaking them down into their component pieces.”

Leibniz rejected a university position in favor of studying law, worked for legal reform, and then went into service where he thought he could do the most good – counseling and advising the court of Hanover, Germany. Bardi describes Leibniz’s motivation as a desire to “enlighten the princes, dukes, and other rulers of his day so that they could make the right choices.” As an advisor, Leibniz proposed sweeping reforms in everything from agriculture and economics, to politics. He came up with money-making schemes and centralized the information system of the state, and worked towards a grand reunification of Protestants and Catholics.

The Business of Life

Both Newton and Leibniz were plagued with the need to make a living. Leibniz, as a court advisor and historian, volunteered to write a family history of the House of Brunswick. Although Leibniz still managed to make truly important scientific contributions, such as his famous “Discourse on Metaphysics,” the history, says Bardi, “dogged him for the rest of his life, and there was little in his later years that was not clouded by its incomplete assignment.” Newton ended up spending 30 years as first the warden, then master of the mint in London, combating counterfeiters.

The Calculus Wars erupted, says Bardi, when Newton’s work finally made it into print, contained in colleague John Wallis’ math volume. When Wallis asserted that Newton’s methods were better than Leibniz’s, the dispute began smoldering in the scientific community. Each scientist’s friends got their hackles up and attacks began to fly. Then, when Newton published his Optics, with a chapter on calculus, Leibniz used a favorite tactic of his, publishing an anonymous, insulting critique. Years later, when Newton found out about the review, his supporters began to attack Leibniz, not to claim co-inventorship, but implying that Leibniz actually stole calculus from Newton, who clearly had invented it first. Anonymous attacks, and indirect accusations leveled by Newton and Leibniz’s friends on behalf of the two giants led to Leibniz taking the matter to the assumed-impartial Royal Society of London.

A Crushing Verdict

Unfortunately, with Newton as its president, the Society was far from impartial, and ended up compiling and publishing a report described by Bardi as thrusting Newton “into an elevated limelight, casting him as the one who should be rightly recognized as the best mathematician in the last fifty years. It could not have been more damaging to Leibniz’s reputation, painting him as a compulsive plagiarist.” Leibniz’s notation was superior to Newton’s, but even this was ignored by the society. Leibniz died still fighting the Calculus Wars, never able to fully redeem himself and saddled with his never-completed history. Newton outlived Leibniz by ten years, and died an intellectual hero.

Bardi paints a picture of Leibniz and Newton, their strengths and eccentricities, their lives, the demands on them as real people rather than aloof geniuses, and takes us into the heart of the war they waged with each other in the last part of their lives. When the last page is turned, one feels as if we have walked with them through the 17th and 18th centuries, felt the similarities and marked the differences.

Cogito Interview: Jason Bardi, author of The Calculus Wars

Cogito, 03.02.2007

Mr. Bardi, I know studying Calculus is worthwhile if you are pursuing a higher education in math/engineering. But how has Calculus benefited you in your everyday life decisions? (from a Cogito member turning 15 today)

Happy birthday, and thank you for your interest in my book The Calculus Wars.

Want to talk about Math? Do problems for fun? Members can discuss and solve in Cogito's forums.

How did you get interested in this subject? Sometimes writers stumble upon a subject by researching something related. Is that what happened to you?

Find out more about the book: read Cogito's Walking with Giants: Synopsis of The Calculus Wars.

Did you have to have a good understanding of mathematics to be able to write a book like this?

But to finally answer your question, I guess I would say that some people might have been able to write this book with no background in mathematics, but I doubt whether I could have.

How did you do the research for this book? How do you determine what is a valid source?

What were most surprised to learn about Newton and Leibniz?

Why did you decide to become a science writer as opposed to a scientist? How was JHU's science writing major?

You were a biophysics major before, right? That's an interesting combination! I haven't heard much about that major. What's it like?

By the way, when I was an undergraduate, I majored in physics and English. A lot of people I knew thought that was a particularly interesting combination.

Bardi signing off: OK that’s all for now. Thank you for the questions and be sure to visit my own web site The Calculus Wars.

Who is Jason S. Bardi?

Wanted: Biologists who can speak 'math,' engineers fluent in genetics

Jennifer Donovan, Howard Hughes Medical Institute Bulletin
Kathleen Cushman, Howard Hughes Medical Institute Bulletin
November 22, 2006

Biologists, computer scientists and engineers speak different languages: Mention "vector" to a molecular biologist and a plasmid (a circular piece of bacterial DNA used in gene cloning) comes to mind. Say "vector" to an engineer, and she thinks of a mathematical concept. Similarly with "expression": To a biologist, it means protein production from a gene; to an engineer, it's an equation.

This communications divide is becoming more of a problem now that research so often requires collaboration across disciplines. One-third of the engineers at MIT now work on biological problems, according to Graham C. Walker, MIT biology professor. Yet it can be challenging for biology and engineering students to understand each other.

The divide, deeper than mere semantics, can touch on basic cultural differences, he says. "Even among top-level scientists, our fundamental ways of conducting inquiry differ, depending on our interests and training."

Teaching introductory biology to MIT undergraduates, Walker experiences the disciplinary disconnect firsthand. "It's a constant challenge," he says, "to find ways to make biology comprehensible and relevant to students who think like engineers."

As a Howard Hughes Medical Institute (HHMI) professor--one of 20 research scientists nationwide who received $1 million each from HHMI to find innovative ways to stimulate undergraduates' interest in science--Walker is ever on the lookout for solutions to this problem. Last spring he invited Mary E. Lidstrom, a fellow HHMI professor, to MIT to discuss how she grapples with it at the University of Washington.

Lidstrom, who conducts an elective biology class for engineers, has found that biologists are motivated by the "what," while engineers are motivated by the "how." She told a room packed with MIT students and faculty that "engineering students tend to view biology as magic because they don't see us using differential equations. And often they don't even necessarily want to understand the 'what' of biology--they just want to use it.

"So we actually teach biology to engineers using a function-based approach, with the idea of nature as the designer and evolution as the design tool," Lidstrom says. "That's real engineering. And that's the way we feel biology should be taught."

To help her engineering students feel comfortable in this strange new territory, she says, "We talk about the functions of life, about information transfer, about adaptability. Engineers understand systems, and ecology is the perfect example of a system."

But while Lidstrom's approach may be useful for engineering students, says Julia Khodor, a graduate student who helps teach Walker's introductory biology course at MIT, it may be limited to engineering students. "Because our lectures need to reach all students, regardless of background," she says, "they are likely to remain mostly in the language of biology."

Lidstrom suggests another option--in effect, double majors. "The new research workforce will always need people firmly based in the core disciplines of biology and engineering," she says, "but it also needs translators who have the understanding and the tools to communicate about the other field."

Douglas A. Lauffenburger, a biological engineer who helped develop MIT's new major in that field, agrees. "The world of science keeps expanding," he says. "For a synthesis to be effective, we have to educate a third kind of person--a 'bilingual' one."

Reprinted with permission. A longer version of this article first appeared in the Dec. 2005 issue of the Howard Hughes Medical Institute Bulletin.

A version of this article appeared in MIT Tech Talk on November 22, 2006 (download PDF).

Terence Tao, 31, is one of the world's top mathematicians. (Monica Almeida/The New York Times)

Journeys to the distant fields of prime

By Kenneth Chang

Published: March 13, 2007

LOS ANGELES: Four hundred people packed into an auditorium at U.C.L.A. in January to listen to a public lecture on prime numbers, one of the rare occasions that the topic has drawn a standing-room-only audience.

Another 35 people watched on a video screen in a classroom next door. Eighty people were turned away.

The speaker, Terence Tao, a professor of mathematics at the university, promised "a whirlwind tour, the equivalent to going through Paris and just seeing the Eiffel Tower and the Arc de Triomphe."

His words were polite, unassuming and tinged with the accent of Australia, his homeland. Even though prime numbers have been studied for 2,000 years, "There's still a lot that needs to be done," Tao said. "And it's still a very exciting field."

After Tao finished his one-hour talk, which was broadcast live on the Internet, several students came down to the front and asked for autographs.

Tao has drawn attention and curiosity throughout his life for his prodigious abilities. By age 2, he had learned to read. At 9, he attended college math classes. At 20, he finished his Ph.D.

Now 31, he has grown from prodigy to one of the world's top mathematicians, tackling an unusually broad range of problems, including ones involving prime numbers and the compression of images. Last summer, he won a Fields Medal, often considered the Nobel Prize of mathematics, and a MacArthur Fellowship, the "genius" award that comes with a half-million dollars and no strings.

"He's wonderful," said Charles Fefferman of Princeton University, himself a former child prodigy and a Fields Medalist. "He's as good as they come. There are a few in a generation, and he's one of the few."

Colleagues have teasingly called Tao a rock star and the Mozart of Math. Two museums in Australia have requested his photograph for their permanent exhibits. And he was a finalist for the 2007 Australian of the Year award.

"You start getting famous for being famous," Tao said. "The Paris Hilton effect."

Not that any of that has noticeably affected him. His campus office is adorned with a poster of "Ranma ½," a Japanese comic book. As he walks the halls of the math building, he might be wearing an Adidas sweatshirt, blue jeans and scruffy sneakers, looking much like one of his graduate students. He said he did not know how he would spend the MacArthur money, though he mentioned the mortgage on the house that he and his wife, Laura, an engineer at the NASA Jet Propulsion Laboratory, bought last year.

After a childhood in Adelaide, Australia, and graduate school at Princeton, Tao has settled into sunny Southern California.

"I love it a lot," he said. But not necessarily for what the area offers.

"It's sort of the absence of things I like," he said. No snow to shovel, for instance.

A deluge of media attention following his Fields Medal last summer has slowed to a trickle, and Tao said he was happy that his fame might be fleeting so that he could again concentrate on math.

One area of his research — compressed sensing — could have real-world use. Digital cameras use millions of sensors to record an image, and then a computer chip in the camera compresses the data.

"Compressed sensing is a different strategy," Tao said. "You also compress the data, but you try to do it in a very dumb way, one that doesn't require much computer power at the sensor end."

With Emmanuel Candès, a professor of applied and computational mathematics at the California Institute of Technology, Tao showed that even if most of the information were immediately discarded, the use of powerful algorithms could still reconstruct the original image.

By useful coincidence, Tao's son, William, and Candès's son attended the same preschool, so dropping off their children turned into useful work time.

"We'd meet each other every morning at preschool," Tao said, "and we'd catch up on what we had done."

The military is interested in using the work for reconnaissance: blanket a battlefield with simple, cheap cameras that might each record a single pixel of data. Each camera would transmit the data to a central computer that, using the mathematical technique developed by Tao and Candès, would construct a comprehensive view. Engineers at Rice University have made a prototype of just such a camera.

Tao's best-known mathematical work involves prime numbers — positive whole numbers that can be divided evenly only by themselves and 1. The first few prime numbers are 2, 3, 5, 7, 11 and 13 (1 is excluded).

As numbers get larger, prime numbers become sparser, but the Greek mathematician Euclid proved sometime around 300 B.C. that there is nonetheless an infinite number of primes.

Many questions about prime numbers continue to elude answers. Euclid also believed that there was an infinite number of "twin primes" — pairs of prime numbers separated by 2, like 3 and 5 or 11 and 13 — but he was unable to prove his conjecture. Nor has anyone else in the succeeding 2,300 years.

A larger unknown question is whether hidden patterns exist in the sequence of prime numbers or whether they appear randomly.

In 2004, Tao, along with Ben Green, a mathematician now at the University of Cambridge in England, solved a problem related to the Twin Prime Conjecture by looking at prime number progressions — series of numbers equally spaced. (For example, 3, 7 and 11 constitute a progression of prime numbers with a spacing of 4; the next number in the sequence, 15, is not prime.) Tao and Green proved that it is always possible to find, somewhere in the infinity of integers, a progression of prime numbers of any spacing and any length.

"Terry has a style that very few have," Fefferman said. "When he solves the problem, you think to yourself, 'This is so obvious and why didn't I see it? Why didn't the 100 distinguished people who thought about this before not think of it?' "

Tao's proficiency with numbers appeared at a very young age. "I always liked numbers," he said.

A 2-year-old Terry Tao used toy blocks to show older children how to count. He was quick with language and used the blocks to spell words like "dog" and "cat."

"He probably was quietly learning these things from watching 'Sesame Street,' " said his father, Dr. Billy Tao, a pediatrician who immigrated to Australia from Hong Kong in 1972. "We basically used 'Sesame Street' as a babysitter."

The blocks had been bought as toys, not learning tools. "You expect them to throw them around," said the elder Tao, whose accent swings between Australian and Chinese.

Terry's parents placed him in a private school when he was 3 ½. They pulled him out six weeks later because he was not ready to spend that much time in a classroom, and the teacher was not ready to teach someone like him.

At age 5, he was enrolled in a public school, and his parents, administrators and teachers set up an individualized program for him. He proceeded through each subject at his own pace, quickly accelerating through several grades in math and science while remaining closer to his age group in other subjects. In English classes, for instance, he became flustered when he had to write essays.

"I never really got the hang of that," he said. "These very vague, undefined questions. I always liked situations where there were very clear rules of what to do."

Assigned to write a story about what was going on at home, Terry went from room to room and made detailed lists of the contents.

When he was 7 ½, he began attending math classes at the local high school.

Billy Tao knew the trajectories of child prodigies like Jay Luo, who graduated with a mathematics degree from Boise State University in 1982 at the age of 12, but who has since vanished from the world of mathematics.

"I initially thought Terry would be just like one of them, to graduate as early as possible," he said. But after talking to experts on education for gifted children, he changed his mind.

"To get a degree at a young age, to be a record-breaker, means nothing," he said. "I had a pyramid model of knowledge, that is, a very broad base and then the pyramid can go higher. If you just very quickly move up like a column, then you're more likely to wobble at the top and then collapse."

Billy Tao also arranged for math professors to mentor Terry.

A couple of years later, Terry was taking university-level math and physics classes. He excelled in international math competitions. His parents decided not to push him into college full time, so he split his time between high school and Flinders University, the local university in Adelaide. He finally enrolled as a full-time college student at Flinders when he was 14, two years after he would have graduated had his parents pushed him only according to his academic abilities.

The Taos had different challenges in raising their other two sons, although all three excelled in math. Trevor, two years younger than Terry, is autistic with top-level chess skills and the musical savant gift to play back on the piano a musical piece — even one played by an entire orchestra — after hearing it just once. He completed a Ph.D. in mathematics and now works for the Defense Science and Technology Organization in Australia.

The youngest, Nigel, told his father that he was "not another Terry," and his parents let him learn at a less accelerated pace. Nigel, with degrees in economics, math and computer science, now works as a computer engineer for Google Australia.

"All along, we tend to emphasize the joy of learning," Billy Tao said. "The fun is doing something, not winning something."

Terry completed his undergraduate degree in two years, earned a master's degree a year after that, then moved to Princeton for his doctoral studies. While he said he never felt out of place in a class of much older students, Princeton was where he finally felt he fit among a group of peers. He was still younger, but was not necessarily the brightest student all the time.

His attitude toward math also matured. Until then, math had been competitions, problem sets, exams. "That's more like a sprint," he said.

Tao recalled that as a child, "I remember having this vague idea that what mathematicians did was that, some authority, someone gave them problems to solve and they just sort of solved them."

In the real academic world, "Math research is more like a marathon," he said.

As a parent and a professor, Tao now has to think about how to teach math in addition to learning it.

An evening snack provided him an opportunity to question his son, who is 4. If there are 10 cookies, how many does each of the five people in the living room get?

William asked his father to tell him. "I don't know how many," Tao replied. "You tell me."

With a little more prodding, William divided the cookies into five stacks of two each.

Tao said a future project would be to try to teach more non-mathematicians how to think mathematically — a skill that would be useful in everyday tasks like comparing mortgages.

"I believe you can teach this to almost anybody," he said.

But for now, his research is where his focus is.

"In many ways, my work is my hobby," he said. "I always wanted to learn another language, but that's not going to happen for a while. Those things can wait."

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Thứ Sáu, 18 tháng 7, 2008

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COGITO.ORG

About Cogito

Melanie Wood: The Making of a Mathematician

Interviews

Interview with Kiran Kedlaya, Mathematician and Puzzler

Questions and Answers

What are your primary fields of study?

Can you remember a particular event or experience that made you want to become a mathematician?

How has your experience with puzzles helped you in your mathematical career?

Can you say anything about the kinds of math questions you’re working on now?

What do you think the compelling questions in mathematics will be in five years?

What kind of puzzle got you into solving?

What is your favorite puzzle type, and do you do Sudoku puzzles?

How long does it take you to do the New York Times crossword?

How long did it take you to create your New York Times-published puzzle?

How do you manage to balance work vs. play in your life?

For more information

News & Views

Math Whiz, 17, Hits the Big Time with Research

Math's Architect of Beauty

Interviews

Cogito Interview: Terence Tao, Mathematician and Fields Medalist

What kind of math were you doing when you were 8?

What type of math questions intrigued you most when you were in high school and why?

Were you good at other fields than math, such as writing, reading, history, etc?

When you were in junior high or high school, what kind of summer programs did you participate in?

Do you feel that your achievements at the International Math Olympiad were largely based on extensive and arduous study or mainly because of your superior intellectual capacity?

You were obviously way ahead of everyone in your age group. How were you educated?

What advice would you give someone who isn't challenged in math class but has no real opportunity to move ahead because her school cannot accommodate her? What activities would recommend?

What mathematical magazines or journals are good for someone in high school?

Do you ever get to see applications of your discoveries in other fields of study? If so, is there any specific application which is your favorite?

What has motivated you the most? Who has supported you the most other than yourself?

What does your current research deal with?

Do you plan to extend your work on prime numbers in the future to possibly find a way to find them?

In your spare time, what kind of hobbies do you have?

If you weren't a mathematician, what else would you want to be?

Further Reading

News & Views

Terence Tao, "Mozart of Math," Is Awarded the Fields Medal

Further Reading

News & Views

Terence Tao Named a MacArthur "Genius" Fellow

About the MacArthur Fellows Program:

Interviews

Cogito Interview: Keith Devlin, The Math Guy

When did you begin to like math? Did you like math in high school?

What do you mean when you say that humans are born with "spatial reasoning abilities"? Wouldn't that be contradicted by the fact that some people (like me) are incapable of determining what 3D shape a given 2D cutout would fold into?

I know Pi is a Greek letter, but do you know how and why was it selected for the mathematical formula Pi?

You’ve said that animals have some mathematical ability. What symbols do chimps have for numbers? Can my cat do math? What kind and how?

As a teacher of high school mathematics, how can I connect with my students to make them interested enough to stay with the algebra or geometry long enough to realize the benefits of their perseverance? How do you reach those students who struggle?

I am very curious as to what the life of a mathematician is like - do you try to create new theorems or teach math?

Being more of a language person, I'm not really the most interested in math. I like the concept behind it, but doing it doesn't really engage me. Are there any ways to change this? Do you know of any books that could help me become more at ease with mathematical operations?

37 Under 36: America's Young Innovators

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Cogito Interview: Harold Reiter, Problem Spinner

Q & A with Dr. Reiter

More about Harold Reiter…

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Cogito Interview: Adam Hesterberg, 2007 ILO Winner

I enjoyed the problems at namclo.linguistlist.org. Is this what linguists do for a living?

I really like logic. Is linguistics something I would enjoy?

I don't understand the term “computational linguistics." Where does the computing come in?

Which was more satisfying: your success at the USAMO or the ILO?

Do you think there's a connection between the skills used for linguistics and those used for math? They seem like such different fields, using different parts of the brain.

How did you prepare for the linguistics competitions other than doing practice problems? It said you hadn't ever studied linguistics, except just a little bit in a Latin class.

What did you enjoy most about the Olympiad?

What advice would you give someone interested in competing?