Grigori Perelman and the Poincare Conjecture | Jordan Ellenberg and Lex Fridman

we talked about the Poncare

conjecture

, there is a guy, he is the Russian Grigori Perlman, he proved the pancreas

conjecture

, if you can comment on the proof itself, if that highlights something interesting to you or human history, that is, he rejected the Fields medal for the test. Is there something that you find inspiring or enlightening about the test itself or about the man? Yes, I mean, one thing I really like about the test and part of it is because it's something that happens over and over again in this book, I mean, what I'm writing about. geometry and the way it appears in all these kinds of real world problems, but it happens so often that the geometry that you think you're studying is somehow not enough, you have to go one level higher in abstraction and study a level The highest point of Geometry and the Way It Develops is that, you know, Poincaré asks a question about a certain kind of three-dimensional object: is it the usual three-dimensional space that we know or is it some kind of exotic thing?

So, of course, this sounds like it's a question about the geometry of three-dimensional space but no parallel understands it and certainly in a tradition that involves richard hamilton and many other people as really important mathematical advances this doesn't just happen, it doesn't happen in a void happens as the culmination of a program that involves many people with wiles by the way, I mean we talked about wiles and I want to emphasize that starting with kumar who I mentioned in the 19th century but um gerhard frye and mazer and ken ribbit and like Many other people are involved in building the other pieces of the arch before the cornerstone is laid, will you stand on the shoulders of giants?

Yeah, so what is this idea? The idea is that well, of course, the geometry of the three-dimensional object itself is relevant, but the real geometry you have to understand is the geometry of the space of all three-dimensional geometries. Wow, you're moving up to a higher level because when you do that you can say now, let's trace a path in that space yeah, um, there's a mechanism called Richie flow and again we're outside my area of ​​research so for all the geometric analysts and differential geometers listening to this, please, I'm doing my best and I say approximately So Ritchie flow allows you to say: Okay, let's start from some mysterious three-dimensional space that Poincaré would conjecture is essentially the same as our three-dimensional space familiar, but we don't know it and now you let it flow. you like to let it move in its natural path according to some almost physical process and you ask where it ends and what you find is that it always ends, you have continually deformed it, that word deformation appears again and what you can prove is that the process does not stops until it reaches the usual three-dimensional space and since one can go from the mysterious thing to the standard space through this process of continuous change and without having abrupt transitions, then the original shape must have been the same as the standard shape, that is The nature of the test now, of course, is incredibly technical.

I think, as I understand it, I think the hard part is proving that, AI people's favorite word, you don't get any uniqueness along the way. um, but of course, in this context, singularity just means acquiring a sharp curvature, it just means becoming non-smooth at some point, so I'm just saying something interesting about uh formal about the smooth trajectory through this strange space, yeah , but yes, then, what I like. is that it's just one of many examples of it not being the geometry you think it is, but the geometry of all geometries, so to speak, and it's just kind of like being ripped off flat ground, the same idea is Only by seeing everything globally at once is that you can really make progress in understanding that the only thing you thought you were seeing is a romantic question, but what do you think of him rejecting metal fields? is that just our Nobel prizes and sentiment medals are just the icing on the cake and really the mathematics itself, the process of curiosity of pulling the thread of the mystery before us, that is the cake and then the prizes are just the cherry and, clearly, I've been fasting and I'm hungry, but do you think it's tragic or just a little curiosity that I lit the medal?

Well, it's interesting because, on the one hand, I think it's absolutely true, in some ways. from a vast spiritual sense, prizes are not important, they are not important, the way you understand the universe is important, um, on the other hand, most people who are offered that prize accept it, you know, It is, so there's something unusual about your uh. his choice there um, I wouldn't say I see it as tragic, I mean, maybe if I don't really feel like I have a clear idea of ​​why he decided not to take it, I mean, it's not like I'm not alone in doing things like These People sometimes rejects awards for ideological reasons.

Probably most often in math. I mean, I think I'm right in saying that Peter Schulze turned down some kind of big monetary award because, you know, I think at some point. you have a lot of money and maybe you think it sends the wrong message about what the point of doing math is. I find that most people accept, you know, most people are given an award, most people accept it, I mean, people like to be appreciated, but like I said, we are people, yes, not that different from most people, but the important reminder that rejecting the prize serves me well is not that there is something wrong with the prize and that there is something wonderful about the prize, I think not, the Nobel.

The award is more complicated because many Nobel Prizes are awarded. First of all, the Nobel Prize winner often forgets many of the important people throughout history. Secondly, there are strange rules that it is only three people and some projects have a large number of people. it's like that, um, I don't know, it doesn't highlight the way that science is done in some of these projects in the best way possible, but overall the awards are great, but what this teaches me and reminds me. Sometimes in your life there will be times when what you would really like to do, that society would really like you to do, is something that goes against something that you believe in, whatever it is, that is some kind of principle. and stand firm.

In light of that, it's something that I think most people will have a few moments like that in their life, maybe a moment like that and you have to do it, that's what integrity is, so it doesn't have to make sense. for the rest of life. world, but staying in that like saying no, it's interesting because I think you know he turned down the award in service of some principle because I don't really know, yeah, that seems to be the clue, but he's never done super. sure, but the indication is that he had some problems with the whole mathematics process, which includes awards like this, hierarchies and reputations and all that kind of stuff and the individualism that is fundamental to American culture, probably because he visited the United States quite a bit.

Joined. You probably know that it's all about experiences and I may have let you know that some parts of academia, some sectors of academia can be less inspiring, maybe sometimes because of the individual egos involved, not the academic people in Generally, smart people with egos and if you interact with certain types of people, you can become cynical too easily. I'm one of those people where I've been very lucky to interact with amazing people at MIT and in academia in general, but I've met a few and I tend to. It's like when I meet difficult people, I just smile and send them all my love and just go with it, but for others those experiences can be sticky, like they can become cynical about the world when people like that. exists, so he is the one who may have become a little cynical about the process of science that you