/u/NovaNation21 9/9/14

Interview with Terence Tao Professor of Mathematics at University of California ­ Los Angeles

Q: There are a lot of topics in olympiad math that a regular school curriculum will never

cover. A high school curriculum is lucky to end with calculus. As a professor do you think

topics like number theory, combinatorics, and complex algebra/geometry should be taught

in higher­level high school classes or just reserved for the students independently for these

olympiads?

A: The standard high school curriculum traditionally has been focused towards physics and

engineering. So calculus, differential equations, and linear algebra have always been the most

emphasized, and for good reason—these are very important. There is now a trend over the rise

of computer science and also the life sciences, which have become more mathematical. Schools

are slowly transitioning to emphasize things like combinatorics, probability, and statistics. These

things are getting more important. Number theory less so because it has fewer

applications—cryptography is the only really big practical application of the subject. So

mainstream math education is focused on different topics. Most students who take math classes

aren’t going to be mathematicians. They’re going to be engineers, statisticians—in many ways

that’s the more important mission of math education. So it’s sort of complementary.

Math education has changed over the years. In the 19th century, they taught spherical

trigonometry because one of the biggest applications of mathematics was navigating the ocean.

This is no longer so relevant. So it’s great that we have these competitions and that we keep

some classical mathematics like Euclidean geometry and elementary number theory. In

mathematical research, it’s good to know these things. You will learn them eventually, especially

if you take a class in it. At some point you can just learn it yourself and go and read a book on it.

It’s really easy now with the Internet, it’s amazing now. People ask me how I know so many

areas of mathematics. Mostly you just know a little bit and then you look up the rest. It’s so

much easier now than when I was a graduate student. I mean you need a certain amount of base

mathematics so that you can learn everything else quickly. But once you have the foundation,

it’s fairly quick. So it’s great that we have the competitions. I think they complement the main

math education, but I don’t think they need to replace it.

Q: What would you tell an aspiring or current mathematics major about the kind of jobs

available besides teaching?

A: Most people with Bachelor’s of Math don’t go into academics so much. You would get a PhD

for that. There’s many things you can do with a math degree: actuarial science, finance,

computer graphics, and anything quantitative really. You can work in a lab. If you aren’t afraid

of equations or abstract thinking, it’s easy to pick up any kind of STEM subject. You can go into

electrical engineering and there’s always scary equations like Maxwell’s Equations and so forth.

But if you have the basic math training you can pick this up. You can move very easily into

Stats—I mean Stats is huge. You can do insurance and all kinds of medical things. It’s a very

flexible major. Most people with math degrees are not called mathematicians, they might be

analysts. If you want to be an engineer, obviously your best route is to take engineering classes.

If you don’t have mathematical background, the classes you take will help you train to analyze

existing systems and build things that haven’t been built before. If you want to design something

really new, at some point you’ll have to model what you’re doing, which might be different from

previous models, and you have to do some mathematics somewhere.

Q: Do you find more fulfillment in teaching or researching?

A: They are complementary. You only really learn something when you can teach it to someone

else. If you have to do something like abstract geometry, you have to be able to do basic

geometry really quickly. After you’ve taught calculus a few times you can look at differential

equations and say “oh, it should look like this”, which would have taken like half an hour as a

student but now takes a few minutes. You need that speed in order to tackle the harder problems,

so it is complementary.

Q: What do you enjoy most about math?

A: When you finally figure out something that’s bothered you. If there’s something that looks

like you should be able to answer and you can’t answer it immediately, it just sort of bugs you.

And when you figure out the trick that resolves it, then a light bulb goes off. So that’s very

satisfying, when you finally get what’s going on, as well as when you explain it to someone else

and you see the light bulb go on in their head.

Q: What is your favorite number or mathematical constant?

A: The funny thing about mathematics is that you don’t work with regular numbers so much. I

never see a 37, I see ‘n’ –a lot of what I do involves a big number n that goes to infinity. Never

any specific number.

Q: Do you have any other interests besides math?

A: I used to have more. When you work and you have family, it’s tough. When I was younger I

used to watch a lot anime and play computer games and so forth, but I have no time for these

things anymore.

Q: Are you interested in seeing your two kids become mathematicians? Are you going to

push them at all in that direction?

A: Whatever they have the passion for. My son is 11 and he has some talent for math, but he

doesn’t really have a passion for it. What he really likes is acting, theatre and video editing

actually—which is maybe what he’ll become, I don’t know.

Q: What about prime numbers interests you so much?

A: Number theory is one of the things you can actually appreciate. I saw these through high

school and competitions, so I knew about all these numbers and conjecture theories very early, at

8 or 9. I’ve always wanted to make progress in this area. It’s not the most applied area of

mathematics. If you prove the twin­prime conjecture today, it won’t have that much effect in

technology tomorrow. Well, it depends what is proved and how. In many ways, it’s the methods

of proof that are more important. [Conjectures] are just sort of benchmarks to keep score.

Q: Is there a result that you consider the most beautiful?

A: I don’t think much in those terms. The funny thing is anything you prove yourself, you think

“oh, that was a lot easier than I thought.” You know, once you actually see it, you realize it

wasn’t that hard after all.

Q: How did you learn to teach and what is your teaching philosophy?

A: Sink or swim basically. When I was a graduate student, I didn’t teach, I was a grader. I

assisted one or two classes. When I first came to UCLA that was my first class. I had a mentor

teaching another session for the class—they do that for the first class. He showed me what kinds

of homework, midterms, and syllabi he used. When you start out, you’re given a syllabus and a

textbook and basically told to use the textbook—it’s not actually that difficult.

There were a couple things I had to learn. I remember the first exam I gave out I put a lot of

effort into designing very cute questions where the answer to the first question would be useful

in the second question, and so forth. I didn’t realize this would be a nightmare to grade. When

each question has a single answer, you’re either right or wrong. You don’t want questions where

someone makes a mistake in the first step but everything else is correct, so you have to check. So

it took me a while to figure out what are good questions and that sort of thing. You should make

clear at the beginning what your policies are for homework. If you’re vague you always get kids

that say, “I didn’t realize this midterm was worth this much. Can you reconsider it?”

Q: What can teachers do to make kids like math—not just computational math but the

ideas of math like problem solving and exploration?

A: It depends on the teacher’s style. Some teachers are really entertaining. You know, they tell

jokes. Not everyone can do that. Some can make really elaborate presentations and experiments.

Some are good at finding really relevant videos and things on the Internet. Sometimes making a

class more enjoyable or entertaining is not the same as making it more educational. I remember

once when I taught calculus, one of the sections was quadric surfaces, like ellipsoids, paraboloids

and so forth. So there’s this thing called a hyperbolic paraboloid, and I wanted to demonstrate

this. It turns out that a Pringle has the shape of a hyperbolic paraboloid. So I brought in a pack of

Pringles to class and said, “this is a hyperbolic paraboloid, this is what it looks like. I ate the

Pringle as I was writing equations on the board. Well many years later, I ran into someone on the

street who said, “Oh, I know you. I took one of your classes. I forget what it was, but there was a

Pringle.” So I thought, “you didn’t remember any of the math, but you remember the Pringle.”

So it didn’t really work.

Q: What do you hope to accomplish through your research? Do you have any specific

goals?

A: No really short­term objectives. There's always questions which I’d love to solve (Riemann

Hypothesis, Navier Stokes, whatever). But usually these things are so out of reach. We

understand the tools that we have pretty well. We know that without an extra or original idea you

can’t really answer them. We don’t focus directly on these long­shot goals. A lot of the focus is

incremental. A lot of mathematicians liken mathematical research to climbing a cliff. You’re at a

certain point on the cliff, and your first goal is to get one foot higher, and you just keep doing

that. Every time you solve a problem, naturally other problems appear. What was extremely

difficult now just looks moderately difficult. You use all of your notes and get a sense of what is

a promising direction—basically anytime there’s a phenomenon which looks interesting and you

can’t explain it, but looks like you should be able to analyze it. That’s what you want to study.

Q: What do you feel are the 2 or 3 most important open problems are in mathematics?

A: Depends whether you weight them by how easy they are to solve. I mean if you solved P=NP,

it could have huge applications, but it’s so difficult to narrow down. It’s hard to measure the

importance.

Q: Which of the millennium problems do you think we are furthest from solving? In 2007

you mentioned in a talk at UCLA that you expect a solution to P vs. NP to come last, with

the Riemann hypothesis just before it. Do you hold that same view today?

A: I would say so. These I think are decades before we can solve them for sure.

Q: What about Navier­Stokes?

A: So there I think we have a chance. That one I think we are closest to proving or disproving.

One direction [within the problem] might be solvable. A lot of conjectures are like this. Take the

Goldbach conjecture: every even number is the sum of two primes. If it’s false, it could be

disproven because if it turns out that some humongous number is not the sum of two primes after

you check all the possibilities, then you’ve disproven the Goldbach conjecture. But no one

believes that. In principle it could be easy to disprove. But probably not. So maybe there’s

chance with Navier­Stokes, but it’s a longshot.

Q: You have three IMO appearances, and you won one of each medal, so you have

Olympiad experience. Do you notice anything different between how math contests are

now compared to how they were in the late 80s when you participated in them?

A: I’m not involved in them anymore, but they look much more professional. It was really an

amateur thing. I mean people would train for maybe 2 weeks. Now there are some countries who

have been training for years and are very systematic where you study 50 years’ worth of

problems. It’s become much more like a professional sport. It’s like youth baseball; it starts off

amateur and eventually some of the professionals take over, which makes it different. In some

ways it’s less fun, in some ways it’s better. One nice thing though is that many other

competitions have sprung up. When I was a kid it was basically just the Olympiads and things

that fit into the Olympiads. Now there’s these other activities too if you don’t want to be so

competitive. There’s other things like Math Circles and other online websites where you can

discuss problems in friendly ways. So it’s more diverse.

Q: At IMO, why do you think that China, Russia, Japan, and the Koreas always seem to

win the unofficial team competition and win the most medals as a team, often with perfect

scores? Do you think this says anything about English­speaking cultures like Australia, the

US, and the UK and their emphasis on math education?

A: I think it's a cultural emphasis. Asian countries often place a larger premium on international

recognition on any sphere. In America, this is less important as America is a global leader in so

many areas, it doesn’t matter as much to them. In many of these countries, they’re actually

trying to make them less competitive. There are many students who have over­trained. They go

and get three gold medals, but they sacrifice their regular education, spending years doing

nothing but working on these problems. They finish the Olympiad and they’re burnt out.

They’re not properly trained, even to do a math degree or they’ll be sick of maths by then if

that’s what they’ve been doing. There’s a balance. You can get a little obsessed with the medal

tallies. After a while it’s sort of pointless to optimize. It’s sort of like the SATs. Getting a 600 or

something is pretty good. When you obsess about getting that perfect score, you start skipping on

your social activities and do nothing but squeeze those last few points, when it doesn’t really

matter in the long run. There’s a great observation called Goodhart’s Law that basically says any

metric becomes useless once you start using it for control purposes. So the SAT, for example, is

a good general test of academic aptitude. But since it’s used so much for admission to college,

kids are trained and coached. They spend lots of time and effort, specifically to improve their

SAT score at the expense of a well­rounded education, to the point where [the SAT] may not be

such a good guide to general academic excellence, even though it used to be before students

started optimizing.

Q: Do you feel like just anyone with enough hard work and dedication can reach the level

of mathematics that you have, or do you think it requires a certain mental disposition that

you're born with in order to do the type of maths done at the IMO or in graduate studies?

A: It depends on what your goal is. If your goal is to be good, as opposed to obsessed with

getting a perfect score or being at the absolute top, or just doing decently. If you have the

enthusiasm, you also have to spend a lot of time. I think one of the things you have to do is you

have to play with the subject. When I was a kid, I just loved doing math on my own, I would try

to solve equations—it was a hobby. I would try to find the sum of the first n­cubes by myself.

It’s not a very serious level of mathematics, but it’s only by tinkering with a subject that you

really get good at it. Same as anything, like building things. You need a certain baseline

intelligence and be able to think and write, but I think it is mostly actually your enthusiasm and

your time put in given the opportunity.

Q: What has been your opinion on the Polymath projects that you've been involved in, and

do you think similar kinds of semi­massively collaborative mathematics has a future?

A: I think we’ll see more of them. For a long time they’re going to be a niche. There are certain

types of problems that they are very well suited for: problems which are very modular, that can

be split up into different pieces that a set of people can work on. You need a good leader who

can organize everything. There needs to be some clear measure of progress by members of your

group. It has to be accessible enough and interesting enough to get people involved. There’s a

critical mass, though. There are some projects where only about 3 people get involved and it gets

converted into normal, traditional collaboration. There are projects where you expect the answer

to come from a lot of little ideas rather than a few really big ideas. But I don’t think we’ll ever

solve the Riemann hypothesis by this sort of crowdsourcing thing, like “let’s get 1000 people

together, throw ideas around and see what sticks.” That’s not how these problems are going to be

solved.

Q: What is your opinion on strong AI?

A: The funny thing about AI is that it’s a moving target. In the seventies, someone might ask

“what are the goals of AI?” And you might say, “Oh, we want a computer who can beat a chess

master, or who can understand actual language speech, or who can search a whole database very

quickly.” We do all that now, like face recognition. All these things that we thought were AI, we

can do them. But once you do them, you don’t think of them as AI. It has this connotation of

some mysterious magical component to it, but when you actually solve one of these problems,

you don’t solve it using magic, you solve it using clever mathematics. It’s no longer magical. It

becomes science, and then you don’t think of it as AI anymore. It’s amazing how you can speak

into your phone and ask for the nearest Thai restaurant, and it will find it. This would have been

called AI, but we don’t think about it like that anymore. So I think, almost by definition, we will

never have AI because we’ll never achieve the goals of AI or cease to be caught up with it.

Q: Would you consider doing an AMA on reddit?

A: (He’s familiar with reddit and will look into it. Fingers crossed!)