The World's Best Mathematician (*) - Numberphile

So what did you eat for breakfast? Jam sandwich, yes. Did you eat a sandwich? Yes. A jam sandwich? Yeah, yeah, I always loved math 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 house, and I was insisting that she put numbers in... she put The detergent. on Windows in the form of numbers, so I always liked numbers, patterns, logic and so on, things are very black and white where there is a right answer and everything else is wrong. I didn't like such subjective questions in shades of gray, I worked on math books for fun, you know, my parents wanted to shut me up, she just gave me a book and I was just going to do sums and stuff, so I always liked them. mathematics, mathematics contests;

Doing that was very different from researching mathematics, the kind of problems that are given in a problem book, etc., these are very canned problems, things that you can do in five minutes or ten minutes and they don't fully prepare you for a problem of research where, you know, you have to spend six months, you have to read the literature, talk to people, try something, it doesn't work, modify it, try it again, and it's Research is a very different experience, but actually I I like it much better than all the puzzles I used to do when I was a kid.

I don't do these things much anymore. My mom was a high school math teacher when I was younger, so she helped me a little bit, you know, when I was a kid, you know, she would just talk numbers with me and then I had a lot of really good mentors when I was like at 10 or 11 there was a retired maths teacher in Adelaide who I would go visit on the weekends, we would have tea and biscuits and, you know, he would just discuss some recreational maths problem etc, which was a lot. of fun. He would tell me stories about how he used mathematics during World War II and so on, you know, to do ballistics and so on.

It was fun to see how math was used for something. "He had a PhD in mathematics from Princeton at age 20 and was appointed professor of mathematics at UCLA at age 24." I enjoyed it, I mean, when I was a kid, what I enjoyed the most was doing math and geeky stuff and stuff, and you know, being accelerated and going to college so early, I was surrounded by people who are older than me but who had backgrounds similar, so we were on the same level mathematically, so you know this. people five years older, maybe, but we're both stuck on the same tasks, we both like to talk about various mathematical concepts, etc., so I felt at home, you know?

So I missed out on maybe a kind of normal high school experience. You know, sort of... I didn't go to a lot of, you know, high school social events and stuff. In fact, I did a lot of that once I was in graduate school at Princeton. So I hung out with people my age and you know, then I went out partying and stuff, so I had a slightly rearranged childhood, but it worked out fine for me." And I have to say, Terence Tao was competing there and I only got one out of seven. In fact, let's cut some slack to Terrence Tao, the Fields medalist, the guy who's about to solve the twin cousins ​​conjecture.

He's actually pretty amazing because he was only 13 years old. youngest person to ever win a gold medal." - It was this question six. - Oh yes, yes, yes, it's a famous question, yes. What is your memory now and why didn't you do it well? - No, I didn't understand it well - How do you feel about that? - I know well that you know that sometimes you win and sometimes you lose. I... oh wow... I did... it was so long ago that I don't remember much about it now. I remember that once the Olympiad was over I discovered that a Romanian woman had solved the question and I remember looking for her, because she really bothered me that she didn't know how to solve the question.

There is a special trick to solve it, which at the time was not a standard trick they taught you. You have to use a descent method... you had to... I forget the exact question. You had to show that something could not be a perfect square, it was always a perfect square, and show that if it was not, you could find a smaller counterexample and a smaller one and another smaller one. I think it's become part of standard training nowadays. Now everyone knows the trick, but... - Were you competitive? Were you the kind of guy who would get angry about that or was it just a fascination? "I just want to know that, I want to know the answer" or when you like to get angry you didn't understand? -I think he was more obsessive than competitive, yes, I mean, I was certainly a little angry at myself for not understanding it, but I wanted to know the answer more than to win, I think. - There is that famous image of you with Erdős. - Yes. - It was a great photograph.

People want to know what you're talking about in that image and what's going on there. -Yes, I think he was giving me a math problem that I did, I think I even know which one, what it is because later he sent me a postcard with what could be the same problem. I have it somewhere. Yeah, I think it was a maths conference in Adelaide, yeah, I was like 10 or something. I don't know why I was there, maybe some mathematics professor at the University of Adelaide told me to come. I get it, he was always very good at talking to mathematically gifted kids and I don't remember much about our conversation except that I remember I really felt like they treated me like an equal, like I wasn't condescending or anything like you. know.

It was a very nice conversation, you know, I mean, now that Erdős has passed away, you know, I mean, it has some sentimental value for me, I mean, it's... Yeah, I mean, I certainly wish I'd paid more attention to it, actually you . I know, I mean, like I heard about him, you know, when I was a kid, but you know, Erdős is someone... ...to me, he was just someone who would like to talk to me about mathematics, and that it was great. He later wrote me a letter of recommendation to Princeton, so he directly helped my career. - Did you ever have people you admired and thought were... were the

best

?

Not really, I mean, you know, I would learn about Euler, Gauss and Newton, but these are largely just names. I think I didn't really have a sense of... you know, so I was learning all these theorems and tricks and stuff, but I never really had a good idea of ​​what was the most important thing, or what was the... yeah, I didn't learn. the reason for mathematics until much later. I remember when I was learning calculus, I thought the greatest

mathematician

in the

world

must be Taylor, because Taylor's name appears everywhere in college calculus. Taylor expansion, Taylor rule, etc., you know, he was a good

mathematician

, but you know, there were a lot of other people doing good things that aren't taught as much at the university level.

When you do math, do you use some type of visualization in your head? What does it look like in your head when you are doing math? it's a little hard to explain. It's always a combination of thinking inside your head, speaking out loud, and working on the whiteboard. You try to isolate the simplest metaphor or something for your problem... How can I explain it? So you know, for example, I do a lot of estimating. I always want and that way you start to think economically, so the way you work with inequalities like By exchanging, you know one item for another and you have an idea of ​​which inequalities are good deals for you, which are the

best

deals for you, and which are really wasting your money.

Sometimes it can be helpful to use a kind of financial intuition. Algebra and topology... Those have always been my weakest areas. In general, I have only been able to master these areas by translating them to other types of mathematics, geometry or analysis, I handle them better... I certainly do not claim any mastery of all mathematics. I don't think anyone can do that since Hilbert. (David Hilbert) The work I'm most proud of is almost all collaborative work, and I think most of my work today is collaborative. It's a lot of fun to talk about a math problem at a really high level with a co-author who is really on your wavelength and understands what you're thinking.

It's actually...saying things out loud, it almost forces you to think, on a more organized level than in your head, where it can be a little confusing and vague, and it's more fun, you know you can go back. and go ahead and if you're stuck maybe your co-author has a suggestion. If you are stuck, you can make suggestions. At least you guarantee that someone else is interested in what you're doing. You know when you write something, when you write an article by yourself, you know there's always somehow that lingering fear in the back of your mind that maybe you know no one will care about this... but you know that at least you have a person. to talk about it.

What do you think or feel, what is your impression of those mathematicians who go the opposite way? Andrew Wiles is a clear example, the mathematician who works alone. What... how does that impress you? What he did was very impressive...you need both. You need people who focus very, very hard on a very specific problem and work for years, so that they become very deep experts. But then you need people who can connect things in fragmented fields. I make a living by understanding a field So I think it's great that there is enormous diversity in mathematics. If we all thought the same way, we all had a similar philosophy, the environment would be much poorer.

You know you can't really make decisions in math. Some problems, the tools are not there. It doesn't matter how smart or fast you are. The analogy I have is like climbing, if you want to climb a 10 meter high cliff, you can do it with the right tools and equipment, but you know, if it's a steep cliff, you know, a mile high and there's no there are no handholds whatsoever, you know, just forget it. No matter how strong you are or whatever, you have to wait until there's some kind of breakthrough, like some opening occurs halfway up, halfway up the cliff and now you have an easier secondary objective.

You know there is some speculation, there are some possible ways to attack the conjecture, but nothing is really promising currently. You're not going to climb that cliff, but if some foot holes appear, could you run and try to climb it too? Yes Yes Yes! You know what... that's how it works, whenever there's an exciting breakthrough like everyone else in the area is around, they just take a look at their favorite list of open issues, "okay, maybe this new trick I can give you some advance." It is very difficult to rule out that there is some important advance in something that seemed impossible suddenly becomes very feasible.

This has happened many times. ...that's not the same as Kakeya's whole problem because maybe as the direction varies smoothly, maybe the pole would have to jump.