The World’s Best Mathematician (*) – Numberphile

So what did you eat for breakfast- jam sandwich, yeah You had a sandwich?
A jam sandwich?
Yeah yeah I always loved mathematics as a kid, so one of my earliest memories is when I was like two years old and my grandma was cleaning the windows in our home, and I was insisting that she put numbers on the- she put the detergent on windows in the form of numbers so I always liked numbers and patterns and logic and so forth, things are very black-and-white where there's one right answer and everything else is wrong. I didn't like so subjective shades of grey type questions, I would work on math workbooks for fun you know my parents wanted to shut me up she just give me a workbook and I'll just go do sums and so forth so I always liked mathematics, math competitions- doing that was very different from doing research mathematics, the type of problems that that you've given in a problem book and so forth these are very canned problems, things you can do in five minutes or ten minutes and they don't prepare you completely for a research problem where, you know, you have to spend six months, you have to read the literature, talk to people, try something, doesn't work, modify, try it again, and it's a very different experience doing research but I like it a lot better actually than– than all the puzzles I used to do as a kid I don't do these things very much any more.

My mother was a high school math teacher when she was younger so she did help me a little bit you know you know when I was a kid you know, just talking numbers with me and then I had a lot of very good mentors when I was like 10 or 11 there was a retired maths professor in Adelaide which I'd go visit on the weekends, we'd have tea and cookies and you know he would just discuss some some recreational maths problem, and so forth, which was a lot of fun. He would tell me stories about how he'd use maths during World War Two, and so forth, you know, to do ballistics and so forth.

It was kind of fun to actually see maths actually being used for something "He had a PhD in maths from Princeton at 20 and was appointed professor of mathematics at UCLA at 24" I enjoyed it I mean when I was a kid I mostly enjoyed doing math and geeky things and so forth, and you know, being accelerated and going to going to uni that early, I was around people who are older than me but had similar backgrounds so we were at sort of the same level mathematically so you know these people five years older maybe but we're both stuck on the same homework assignments, we both like to talk about various math concepts and so forth so I felt at home you know so I did miss out on maybe sort of the regular high school experience you know sort of– I didn't go to many, you know, high school social events and so forth I did a lot of that actually once I was in grad school in Princeton.

So then I hung around people my own age and you know, then I partied and so forth, so I sort of had a slightly reorganized childhood but it worked out well for me "And I have to say Terence Tao was competing there and he only got one out of seven. Let's actually cut Terrence Tao some slack, the fields medalist, the guy who's like coming close to solving the twin prime conjecture. It's actually pretty awesome because he was only 13 years old. He holds the record as the youngest person to ever have won a gold medal." – It was this question six.
– Oh yeah yeah yeah it's a famous question yes. What's your recollection of it now and why you didn't get it right?
– No I did not get it right – How do you feel about that?
– I well you know you win some you lose some.

I — oh boy — I have — it was so long ago now I don't remember much about it. I do remember once the Olympiad was over I found out that some Romanian woman had solved the question and I remember searching out for her, because it was really bugging me that that I did not know how to solve the question. There's a special trick to solve it, which at the time was not a standard trick that was taught to you. You have to use a method of descent — you had to… I forget the exact question. You had to show that something could not be a perfect square — was always a perfect square, and show that if it wasn't you could find a smaller counter-example and a smaller one and a smaller one. I think nowadays it's become part of standard training in its — they all know the trick now, but… – Were you competitive? Were you the sort of boy that would get upset about it or was it just a fascination? "I just want to know that, I want to know the answer" or when you like angry you didn't get it? – I think I was more obsessive than competitive, yeah, I mean certainly I was a bit angry at myself for not getting it but I wanted more to know the answer than to win, I think.

– There's this famous image of you with Erdős.
– Yes. – It was a great photograph. People want to know what you're discussing in that picture what's going on there.
– Yes, I think he was giving me a maths problem which I did — I think I even know which — what it is because he did send me a postcard afterwards with what may be the same problem.

I have it somewhere. Yeah I think it was it was some maths conference in Adelaide, yeah I was like 10 or something I don't know why I was there maybe some math professor at the University of Adelaide told me to come. I understand, he was always very good at at speaking to mathematically gifted kids and I don't remember much about our conversation except that I remember I really felt like I was being treated like an equal like it wasn't condescending or anything like you know. It was a very pleasant conversation you know I mean now Erdős has passed you know I mean it does have some sentimental value for me I mean it's… Yeah, I mean I certainly wish I'd paid more attention to him, actually you know I mean like I'd heard of him you know as a child for but you know Erdős is someone…

…to me he was just someone who would like talking math to me, and that was great. He did write me a letter of recommendation for Princeton later on so he did help my career directly.
– Did you ever have people you looked up to and thought they were… they were the top guns?
Not really I mean, you know, I would learn about Euler and Gauss, and Newton, but these are largely just names. I think I didn't really have a sense of… you know so I'd learn all these theorems and tricks and so forth but I didn't really ever have a good sense of what was the most important, or what was the… yeah I didn't learn the why of mathematics until a lot later. I remember when I was learning calculus, I thought that the most important mathematician in the world must be be Taylor, because Taylor's name appears everywhere in undergraduate calculus. Taylor expansion, Taylor's rule and so forth and, you know he was a good mathematician, but you know, there were many other people who did good stuff which is not taught as much at an undergraduate level.

When you're doing mathematics do you use any kind of visualization in your head? What does it look like in your head when you're doing math? It's a bit hard to explain. It's always always a combination of thinking inside your head and speaking out loud and working on the board. You do try to isolate sort of the simplest metaphor or something for for your problem…

How can I explain it… So you know, for instance, I do a lot of estimates I always want X less than Y and sometimes it helps to think of a sort of an economics problem, like "you have a budget of Y, and can you afford X?" and that way you start thinking economically like so the way you work with inequalities like X less than Y is that normally you maybe try to first bound x by z and z by w and then w by y and so forth and this is like you know trading in you know one item for another item and you get a sense of sort of what inequalities are sort of good deals for you that you're getting you're getting you bang for your buck and which ones are really wasting your money.

Sometimes utilizing sort of your financial intuition can be helpful. Algebra and topology… Those have always been my my weakest areas. I've only been able to get a handle on these areas generally by translating them into other types of mathematics, geometry or analysis, I have better handle on… I certainly don't claim any mastery of all of mathematics. I think nobody can do that not since Hilbert. (David Hilbert) The work I'm proudest of is almost all joint work and I think nowadays most of my work is joint. It is really fun to talk over a math problem at a really high level with a co-author who who is really on your wavelength, understands what you're thinking.

It's actually… saying things out loud, it almost forces you to think, like, at a more organized level than in your head where it can be a bit jumbled and vague, and it's just more fun, you know you can go back and forth and if you're stuck maybe your co-author has a suggestion if he or she is stuck you can make suggestions. You at least guarantee one other person is interested in what you're doing. You know when you write something, when you write a paper by yourself you know there's always somehow the nagging fear at the back of your mind that maybe you know no one will care about this… but you know you at least have one person to talk about it with.
What do you think or feel, what's your impression of those mathematicians that go the opposite way, Andrew Wiles is an obvious example, the mathematician who works in solitude.

What do you… how does that impress you?
What he did was very impressive… you need both. You need people who focus very very hard on one very narrow problem working for years, become a very deep expert. But then you need the people who can connect things in fragmented fields. I make my living you know by understanding one field X and taking some ideas from that and applying it to field Y, but I couldn't do that if there weren't people who are very deeply working in in field X, and so forth, so I think it's great that there's a huge diversity in mathematics you know if we all thought the same way we all had similar philosophy it would be a much poorer environment.

You know you can't really call your shots in mathematics. Some problems, the tools are not there. It doesn't matter how smart or quick you are. The analogy I have is like climbing, if you want to climb a cliff that's 10 meters high you can what we do it with the right tools and equipment, but you know if it's a sheer cliff face, you know, a mile high and there's no handholds whatsoever, you know just forget it. It doesn't matter how strong you are or whatever, you have to wait until there's some sort of breakthrough, like some opening occurs like halfway through, halfway up the cliff and now you have some easier sub-goal. You know there's some speculation, there's some possible ways to attack the conjecture but nothing is really promising currently.
You're not climbing that cliff, but if few foot holes appear you might run and try and climb it as well? Yeah, yeah yeah! You know it would…

this is the way it works, whenever there's an exciting breakthrough like everyone just sort of nearby in the area just sort of takes a look at their favorite list of open problems, "okay maybe this new trick can give you some advance". It's very hard to rule out that there's some major breakthrough in something which seemed impossible suddenly becomes very very feasible. This has happened many times…that is not the same thing as the full Kakeya problem because maybe as the direction varies smoothly, maybe the pole would have to jump around..

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