Cogito Interview: Terence Tao, Mathematician and Fields Medalist

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.

Further Reading

Read more from Terence Tao in his blog, http://terrytao.wordpress.com, which includes answers to more questions about being a mathematician.

And watch a video about him produced by UCLA.

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