The Langlands program is described by mathematician Edward Frankel as “the grand unified theory of mathematics.” The program, conceived by Robert Langlands in 1967, contains numerous speculations aimed at connecting the theory of different mathematical domains: numbers with harmony analysis. The 1990s saw similar connections between geometry and harmonic analysis, resulting in the geometric Langland program. Decades later, in 2024, Dennis Gatesgory and eight of his colleagues at the Max Planck Institute for Mathematics in Bonn, Germany achieved a breakthrough. in 5 Scientific Preprint Paperswhich consists of about 1,000 pages, proved that a large class of geometric objects is related to the amount from the calculation. Gaitsgory is currently receiving a groundbreaking mathematics award, including a $3 million award for this outstanding achievement.
Scientific AmericanGerman Sister Publications Spektrum der wissenschaft We spoke to Gaitsgory about his mathematics career, his accomplishments in Langlands and his prestigious Breakthrough Award.
[An edited transcript of the interview follows.]
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I have been working on the geometric Langlands program for 30 years. When was the moment when you realized you could prove it?
There was a very important step that was always a mystery. This was solved by my former graduate student, [mathematician] Sam Ruskin and his graduate student in the winter of 2022. They proved that something was not zero. It was clear that the evidence could be resolved after this.
How did you feel when you realised that you could really do that?
I have always seen it as some kind of long-term project for self-entering. So I was clearly happy, but it wasn’t like a very strong emotion or anything. It wasn’t an Eureka moment.
The speculation we have proved is one of the specific cases where something is much bigger. It attracts a lot of attention because it is one well-formed thing. But that’s just a step. I was delighted that this step was taken, but there’s more to do.
So there was no champagne pop? Have you just sat down and continued working?
There was no champagne, but there was something similar. when [Raskin] He said he could prove this important part, we made a bet: if he could really do it, I promised him a bottle of scotch.
The proof is huge, about 1,000 pages. Did you oversee everything in it?
I wrote 95% of it. [That was] Not for a good reason, but because I was injured from skiing and was lying in the bed. So, what else were you there to do? I was watching Star Wars I’m writing this with my son.
Do you mean you did both? at the same time?
Originally named after several sections of our paper Star Wars This is an episode, but I deleted it at the end [that element]mainly due to copyright concerns. But one paper still has citations Star Wars: “Fear will line up local systems.” In this paper, it was a really good fit, as there was a need to control the modular space of the local system.
Understanding something is about writing everything down in detail. Has there been a problem?
of course. There was a roadmap, but there were still many gaps and many theories being developed.
But I don’t think there was a real moment of panic. Sometimes I wasn’t sure if I needed three more pages, 20 more pages, or 50 more pages. There was uncertainty that I had to do more work.
Have you done all this from your bed?
No, it was actually a collaborative process. The proof has nine co-authors. Every day I would write to this man and that man. They all have different perspectives and slightly different kinds of expertise. In a way, it was as if I was lying in bed and my colleagues were visiting me, so I wasn’t bored. Being able to speak to them via email really embraced my spirit.
Some people go to the bar and drink. Instead, I’ll talk about mathematics. They talk about football. I’ll talk about mathematics. That’s the same thing. It is human interaction.
Speaking of human interaction, do you talk about your work with your friends and family?
no. They are not mathematicians. They don’t understand technically. My wife is close to my side and knows the story and development of the topic. She knows how these things look from the outside, but I can’t explain the content [to her].
Many people will say that the Langlands program is one of the most complex research topics in the world. do you agree?
The question is, what does complex mean? Yes, come from the street and can’t study this. But the same applies to other mathematicians such as Peter Scholze. [who studies arithmetic geometry at the University of Bonn in Germany and the Max Planck Institute for Mathematics]I’m doing it. I just come to the story he’s saying and understand what he’s saying and don’t understand as there are a lot of technical details.
The same goes for here too. You need to invest some effort into understanding how things work, and then you should be able to understand. But it doesn’t say that anything we do is inherently more complicated. I think all frontier mathematics is just as complicated. We are all trying to push boundaries at various points.
How many people can understand the technical aspects of your job?
Now, our community is growing as people study our evidence. However, until last year, other than that [my] The eight co-authors may have five or six with the ability to understand technical details.
Would you like more people to be involved in this type of research?
Yes, definitely. So far, it was a very small community: the people who pushed the boundaries were basically my former student and Dima Alinkin. [a math professor at the University of Wisconsin–Madison]it’s my age. He has been a close friend and collaborator for many years. So some ideas are recycled. It would be nice to see an influx of people from outside. They can bring something entirely new. I’m very excited to see new ideas.
What can you do to win more people on the issue of geometric Langlands?
I guess more lectures and workshops on that topic. For example, Copenhagen will have a masterclass in August. And then, a conference will be held in Berkeley, California, and now our evidence is being published, so our research is attracting more attention. I receive regular emails, mainly from young people.
[At the time of my interview]for example, [I am set to give] Talk to a large audience of graduate students from Glaz, Austria. We will talk about the fundamentals of derived algebraic geometry. Therefore, graduate students want to study these foundations and hopefully some of them will proceed to study the geometric Langlands program. But they need derived algebraic geometry to understand this. [Editor’s Note: This talk was scheduled for April 2.]
Therefore, you want to capture the interest of young students by teaching them derived algebraic geometry. How did you become interested in the Langlands program in the first place?
I’ve returned to the 1990s [Alexander] Sasha Bailinson [a mathematician now at the University of Chicago] I’ve come to Tel Aviv [University]I was a graduate student. Bailinson gave two lectures. He was at the beginning of his own work on the subject. And I was totally fascinated. I learned about the classic Langlands program… but before he told me I never thought it was related to geometry. It was the first time I’d heard about it. The objects he spoke to seemed very appealing to me. It was exactly the type of mathematical object I wanted to study. And they all came together miraculously. And I was like, “Amazing.” I had to work on that.
Is the same appeal still driving your research?
Of course, things have evolved. That was when you were 20 years old. It’s another one when I was 50 years old. I don’t know what’s driving me right now. It’s like a real desire. It’s like an appetite. I want to do math. And if I can’t, if it’s preventing me from doing math, such as when I’m on a family vacation with my kids for a week, and when I can’t do math, I’ll suffer.
Really? Will that happen in a week?
It might still be okay for a week. But two weeks later, I’m going to be a terrible person.
Well, that‘It’s great to find such a passion in life.
It’s not really passionate.
Maybe it’s kind of addiction?
Yes, maybe. It’s like this: People need to eat, humans need to do mathematics.
What are you working on now? Have you entered the deep by?
I’m trying to generalize our work. There are several projects at different stages. There are many theories to develop, but at least there are programs. We know what we want.
Do you have a new roadmap?
Let’s say we have a roadmap of desire, but we don’t have a roadmap of the way [described] 2013 [and published in 2015]. At the time I knew exactly what I needed to prove. Now I know what I want, but I don’t know how to get there.
Maybe you’ll get new ideas from new researchers.
That would be very good. But in a way, I think it’s like a Darwinian process. If mathematics is worth it, it will be studied. And more people will understand and attract. And if math is boring, it’s a shame. The time will be displayed.
This article was originally published Spektrum der wissenschaft Reproduction with permission.
