Langlands program Edward Frankel described Mathematicians “Consolidated theory of mathematics. “Robert Langlands founded in 1967, includes several conjections that the program intended to connect the mathematical kingdoms. Theory of numbers and harmonious examination. In the 1990s, a similar link between geometry and harmonic analysis was noticed and the geometric program Langlands was born. Decades later, in 2024, at the Mathmatic Mathematics Institute of Dennis Gaitikory Max Planck, eight of Germany and his colleagues achieved a breakthrough. At Previous scientific preppingsWith almost 1,000 pages, they proved that a large class of geometric objects were associated with the quantities of calculations. Gaitgory has received the Math Advance Award, which includes a $ 3 Billion Award for a notifiable achievement.
American scientificPublication of the German sister Sister Science spectrum He talked to Gaitgory about the Mathare Career, about the achievement of Langlands and the famous progress.
(Continues edited transcript of the conversation.)
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Geometric Langlands program has been working for 30 years. When was you able to prove that you would realize the moment?
It was a very crucial step always a mystery. This mine (mathematician) Sam Raskkin (mathematician) and his graduate students fixed in winter 2022. They proved that something is not. After that, it was clear that it would be able to work on a test.
How did you really feel like you realized it?
I’ve always seen a long-term type of self-entertainment project. So I felt happy, but it wasn’t very hard emotion or anything. There was no time for Eureka.
The conjecture we have proven is something much higher. He has received great care because of a well-formulated thing. But it’s the only step. I was glad that this step was done, but much more needs to be done.
So there was no champagne popping? Did you sit and continue at work?
There was no champagne, but something similar. When (Raskin) said that the decisive part could be proven, we made a bet: he could actually do, I ordered a bottle of Scotch.
The evidence is horrible, almost 1,000 pages. Did you overcome everything?
I wrote 95% of that. (That’s not) not for a good reason, but because I had a skis injury, and I was lying in bed. So what else did he do to do? I was watching Star Wars Write with my son and thing.
You mean you have both done At the same time?
Initially, some parts of our papers were named Star Wars The passages, but in the end, we were removed (that item), especially copyright concerns. But a paper still has a quote Star Wars: “Fear will keep local systems in line.” It was very well adapted to this article, we had to control the space of local systems modules.
One thing is to understand something, but write another exactly. Has a problem appeared?
Of course. We had a road map, but there were still many gaps to meet, many theories to be developed.
But I don’t think it was a moment when there was a real panic. Sometimes I wouldn’t be sure that one thing would need another three pages, 20 pages or 50 more pages. How much more work was to do was just uncertainty.
Did you do all this from bed?
No, it was really a cooperative process. The evidence has nine authors: I was writing to this guy and that guy every day. Different and somewhat different views. Somehow, as if I were lying in bed, and my colleagues were visiting, so I wasn’t bored. I could really talk to my spirit to talk to me.
There are some people who go to a bar; Instead, we talked about mathematics. They talk about football; We talk about mathematics. It is the same thing; It is human interaction.
Do you talk about human interaction, talking to your friends and family about your work?
No. Are not mathematicians. They cannot understand technically. My wife was next to me and knows the story and the development of the theme. They know how to see these things from the outside, but I can’t describe the content (him).
Many people would say that the Langlands program is one of the most complex topics in the world research. Would you agree?
The question is: What do you mean by complex? Yes, it can’t come from the street and just examine that. But the same thing happens what mathematics, such as Peter Scholze (Arithmetic Geometry in Germany and Math Planck Institute of Math Planck). I don’t have a background to talk and say what he understands is because there are many technical details.
It’s the same here. Some efforts to understand how things work, and then can be understood. But that’s not all we’re saying is more self-complex. I think all border math is equally complex. We are all trying to push the boundaries at different points.
How many people can understand the technical parts of your work?
The community is growing because people are learning our evidence. But until last year, in addition to being eight co-authors, it may have been five or six people who will have the ability to understand technical details.
Do you want to involve more people in this type of research?
Yes, definitely. It has been a very small community so far: the people who encouraged boundaries were basically my former students, as well as the Mathematical Professor of DimAkin (Wisconsin-Madison University), who is my age. It has been a friend and an assistant for many years. Some ideas are recycled. It would be nice to have a flow of people from the outside. Could bring a whole new. I was very happy to see new ideas.
What could be done to get more people interested in geometric Langlands’ problem?
More talks and workshops on this topic, I think. There will be a Master Class of Copenhagen, for example, in August. And there will be a talk in Berkeley, Californe. But now our research has more attention because our evidence is outside. I receive emails regularly, mostly from young people.
(At the moment of my interview), for example, I give you the opportunity to talk to a large classmate of Austrian degrees. I will talk about the basics of algebraic geometry derived. So graduate students want to analyze these foundations, and some of them expect to continue to learn the geometric Langlands programs. But they need an algebraic geometry derivatives to understand that. (Editor’s note: This conference was planned for April 2.Diagnies
So you expect to catch the interest of young students by teaching the algebraic geometry. How did you become the Langlands program first?
He came back in the 1990s (Alexander) Sasha Beilinson (now the University of Chicago) came to Tel Aviv (University), where I was a graduate student. Beilinson gave two lectures; It was at the beginning of his work on the subject. And I was completely fascinated. I learned about the classical Langlands program … but in front of his speech, I had no idea that it could be associated with geometry. It was the first time I heard about it. The objects he spoke seemed so attractive. It was the kind of mathematical object that I wanted to learn. And they all gathered in this miracle. And I was like, “Wow”. I had to work on that.
Does it still drive your research yet?
Things evolved, of course. One thing is when you are 20 years old, when you are 50 years old. I don’t know what makes me now. It’s like a real desire. It’s like an appetite to eat. I want to do math. And if I can’t stop doing mathematics, for example, when I am a family holiday with my kids a week, and I can’t do math, I suffer.
Really? Happens after a week?
A week is still well. But after two weeks, I become a terrible man.
Well,“It is wonderful to find such passion in life.
It’s not really passion.
Is it more dependent on a sort?
Yes, perhaps. It’s more: Humans must eat and man has to do math.
What are you working now? Did you dive into the abyss?
I am trying to generalize our work. I have several projects at different stages. There are many theories to be developed, but we have at least one program. We know what we want.
Do you have a new road map?
Suppose we have a path map of desire, but not the way to the road road (describe) In 2013 (and reveal In 2015). Then I knew what was to prove. Now I know what I want, but I don’t know how to get there.
Maybe you will receive new ideas for new researchers.
It would be very nice. But, in a sense, I think it’s like the Darwinian process: If math is valuable, it will be examined. And more people understand and attract. And if math is boring, it’s bad. Time will be displayed.
This article originally appeared Science spectrum and reproduced with permission.
