邓煜获奖后首次长视频专访：数学、随机性与 AI

Hello, everyone, and welcome back to SAIR. Today we are joined by Yu Deng – thank you so much for taking this interview. Thanks. First of all, congratulations on being awarded the Fields Medal at this week's International Congress of Mathematicians. How has this week been for you? And could you briefly explain your research for us?

大家好，欢迎回到SAIR。今天我们邀请到了邓煜——非常感谢您能接受这次采访。谢谢。首先，祝贺您在本周的国际数学家大会上荣获菲尔兹奖。这一周对您来说感觉如何？能否请您简单介绍一下您的研究工作？

Well, it's been extremely busy. So I remember, on the day of the opening ceremony, I had the whole morning completely occupied. When I finally got back to the hotel, I realized I had nearly 100 unread emails. So, it felt really good, but I was also extremely exhausted at the time. But yeah, and especially I feel that now I'm really getting into it — you still don't know how to talk about this, right?

这周真是忙得不可开交。开幕式那天，我整个上午都排得满满当当。当我终于回到酒店时，我发现有将近100封未读邮件。所以，感觉真的很好，但当时也确实非常疲惫。而且，尤其是现在我真切地投入到这件事中，你仍然不知道该怎么谈论它。

Of course, because obviously the Fields Medal is a huge award. Could you have predicted that you would win it? Was it something you ever thought about while studying math?

因为菲尔兹奖是一个巨大的奖项。你能预料到自己会获奖吗？你在学习数学的过程中有没有想过这件事？

Well, yeah, maybe had some vague thoughts about it, you know, years ago. One time I was really seriously thinking about it. Like in 2024, we finished that work. I was thinking, you know, maybe that's something. But really, I learned this only in January. I really haven't been like thinking too much… these things you cannot control. Although, when the time came close, I was getting a little bit stressed — not too much, but…

嗯，也许几年前有过一些模糊的想法。有一次我真的认真思考过，比如2024年我们完成那项工作的时候。我在想，也许这有点意义。但说实话，我直到今年一月才真正知道。我真的没有想太多。虽然当时间临近时，我有点紧张——但不太多。有时候我不会去想这些事情，因为它们超出了我的控制。

I think that's a great answer. So part of what your work is recognized for is your breakthrough work related to Hilbert's sixth problem — a challenge that has persisted for over a century. So, in terms that a general audience can understand, what did you and your collaborators prove?

您获奖的工作与希尔伯特第六问题有关——一个持续了超过一个世纪的挑战。那么，用普通观众能理解的方式，您和您的合作者究竟证明了什么？

Yes, essentially it concerns this picture of the fluid or the gas kind of that is made up by these tiny particles, tiny molecules. The behavior of these particles is relatively simple — you have Newton's law, you have interactions, kind of binding interactions. From this huge system, you try to find this effective equation that governs the overall behavior of these particles, which will be the Boltzmann equation. And with the second limit, you get to this familiar fluid equation. So this is kind of the main thing that we did, in terms of this bridge between the different scales.

本质上，它涉及这样一幅图景——流体或气体是由这些微小粒子、微小分子组成的。这些粒子的行为相对简单——你有牛顿定律，有相互作用。从这样一个庞大的系统中，你试图找到控制这些粒子整体行为的有效方程，这就是玻尔兹曼方程。然后，用第二个极限，你就能得到这个熟悉的流体方程。这就是我们做的主要工作，建立了不同尺度之间的桥梁。

So how did you go from school to here? Was this something you imagined yourself doing as a child? Or what was your path to this point?

那么你是怎么从学校走到这一步的？你小时候想象过自己会做这些吗？

Yes, so when I was in high school, I was in this IMO International Mathematical Olympiad and I got a gold medal that time. I was kind of thinking that maybe it's the math is the correct thing for me to do. After that MO, I decided to learn some analysis, you know, kind of college math. I went to college, and I keep enjoying doing math… gradually I realized that this is the thing I want to do.

是的，我在高中的时候参加了IMO国际数学奥林匹克竞赛，获得了金牌。那个时候我开始觉得数学可能是正确的道路。之后我决定学习一些分析——就是大学数学。然后我上了大学，一直很享受数学，逐渐意识到这就是我想做的事。

At the heart of your work is the challenge of explaining how order emerges from randomness. You've said that there's an enormous amount of randomness in the world and that mathematics needs a way to capture it. What does it actually mean to track randomness across billions of colliding particles and interacting waves, and then derive predictable laws from it?

您工作的核心挑战是解释秩序如何从随机性中涌现出来。你曾经说过世界上有大量的随机性，数学需要找到一种方式去捕捉它。那么，追踪数十亿碰撞粒子和相互作用的波中的随机性，并从中推导出可预测的规律，到底意味着什么？

Randomness means that there is uncertainty. On the other hand, you don't need to keep track of every single trajectory because you have randomness; you only need to keep track of what happens with high probability, and that actually simplifies things a lot. There are always kind of exceptionally bad cases that happen in these dynamics… If you don't need to care about those, you only care about what happened generically, then usually this will be much better. You need to discover some way to estimate it.

You need to justify that these bad things really happen very rarely. There are some things you expect to be true… but on the other hand, you have to be prepared for the unexpected. There are also kinds of very difficult questions that we still do not have enough understanding in terms of randomness, like the aggregation of randomness for long timescales and things like that. It's still something very difficult.

随机性意味着不确定性。但另一方面，因为存在随机性，你不需要追踪每一条具体的轨迹——你只需要关注高概率下会发生什么，这实际上大大简化了问题。在这些动态过程中总会有一些异常糟糕的情况，但如果你不需要在意那些，只关心普遍发生的情况，那么通常情况会好很多。你需要找到某种方法来估算它，证明这些糟糕的事情确实很少发生。有些事情你期望它是真的，但你也必须为意外做好准备。还有一些非常困难的问题，我们对随机性的理解还不够，比如长期时间尺度上的随机性聚合问题，这仍然是很难的事。

So this is almost like probability, yeah?

所以这几乎就是概率论？

It's essentially probability, yes.

本质上是概率论，是的。

So what do you think the applications of your research will be in the future?

那您觉得您的研究未来会有哪些应用？

Well, if we understand it better, who knows? Maybe there will be some applications. For now, we are kind of more from the point of view of pure math, kind of proofs.

如果我们能更好地理解这些结构，也许会有一些应用。但目前我们还是更多从纯数学的角度出发，专注于证明和理论方面。谁知道呢？

How did you reach that level of pure mathematics? And how do you find the problems you're going to work on next?

您是怎么达到那种纯数学水平的？您是如何寻找下一个要解决的问题的？

Math is a combination of these basic objects. It's also a way of understanding nature. You can use this combination of these really simple things to find a way to understand nature, which I think is really amazing. As for next, there are really like many, many interesting questions to look at. One thing I've been thinking about is this longtime kind of almost sure global problem. Also, there are several critical questions in quantum field theory; I hope some methods developed can apply to these.

数学是这些基本对象的组合。它也是一种理解自然的方式。你可以用这些极其简单事物的组合来找到理解自然的途径，我认为这实在令人惊叹。至于下一步，有很多很多有趣的问题。我一直在思考的一个问题是长期时间尺度下的几乎必然全局行为问题。此外，在量子场论中也存在一些临界性问题，我希望我们发展的方法能适用于这些领域。

I'm looking forward to the next ICM in 2030. What would you hope to see, or what discoveries do you hope to be made by then?

我很期待2030年的下一届国际数学家大会。您希望到那时能看到什么样的发现或进展？

I hope in this direction, quantum field theory, I hope to see maybe there will be some significant progress. There are already starting to be some tools in this subject.

我希望在量子场论这个方向上能看到一些显著进展。这个领域已经开始有一些工具了。

Your work shows how new mathematical techniques can make previously unapproachable problems accessible today. AI and formal proof systems are expanding that tool even further. How do you see these technologies changing the way that mathematics work?

您的工作展示了新的数学技术如何让以前无法接近的问题变得可以处理。AI和形式化证明系统正在进一步扩展这个工具。您如何看待这些技术改变数学研究的方式？

AI now can solve some kind of short theorems, can prove some short theorems. If we're working on some big project and we can have a framework that allows us to divide this project into a few individual steps that are relatively short, then AI can probably help us prove it, which would significantly accelerate the progress. I hope that this would make the progress of the doable things faster and will make even currently doable things more doable.

现在AI可以解决一些短的定理、证明一些短的定理。如果我们做一个大项目，有一个框架能将它分解成几个相对简短的独立步骤，那么AI可以帮助我们证明它，这将显著加速进展。我希望这能让可行之事的推进速度更快，并使当前可行之事变得更加可行。

You mentioned you're still learning how to use AI. So what was your initial reaction? The first time you actually used AI in your research, what happened?

您第一次在研究中实际使用AI的经历是怎样的？

The first time I actually used AI in my research. There was a problem in my research project recently, and I spent about five days finding a proof — it was about four pages long. But I asked G, and within about an hour it came up with a one-page proof. So I was really, really impressed. In the end, it turns out that proof cannot be generalized; the question I asked was a special case. So we didn't actually include it in the paper, but at least it provides some ideas that are very useful.

If AI can generate longer proofs — even more complicated than what mathematicians do… we rely on the tools that we know will be relied upon. AI can prove some more, then we will rely on those theorems as well, build our arguments, build our thoughts on those. I wouldn't say it's a helper, assistant, or replacement or anything. I don't want to view it in this way. It's not competing against AI. Now we are working with AI to achieve things that we previously cannot achieve.

As long as we can do this, there is nothing to worry about.

我第一次在研究中实际使用AI，是在最近的一个研究项目中。有一个问题我花了大约五天找到一个证明——大概四页长。然后我问了G（某AI模型），大约一小时内它就给出了一个一页的证明。我真的很受震撼。但最后发现那个证明不能被泛化，我问的是一个特例。所以我们没有把它写进论文里，但它提供了一些非常有用的想法。如果AI能生成更长的证明——甚至比数学家所做的还要复杂——我们会依赖这些定理来构建我们的论证和思想。我不会说它是助手、助理或替代品。这不是在跟AI竞争。我们现在和人工智能一起工作，去实现之前无法实现的事情。只要能做到，就没有什么可担心的。

If AI proves more theorems, where do you think mathematics is headed in the future?

如果AI证明了更多的定理，数学的未来会走向哪里？

I hope the acceleration by AI will make the previously impossible questions possible. So then we'll really be able to advance much further than we had imagined before.

我希望AI的加速能让之前不可能的问题变得可能。那么我们就真的能取得比想象中更远的进展。

After solving a problem linked to a challenge that has persisted for over 125 years, where do you go from here? What question or larger mystery would you most want to tackle next?

在解决了一个延续了125年以上的问题之后，您下一步最想解决什么更大的问题？

This longtime kind of almost sure global problem — there are some ideas, some big ideas, but I'm not sure how to make that work. And another thing was this quantum field theory. The most famous problem here would be the Yang-Mills theory in elliptic version, it's a construction of a measure, but it's really complicated. The reason it's complicated is because it's a so-called critical problem. The problem we had been working on — wave turbulence — in some sense also critical problems.

So I hope the method developed might be kind of helpful. At least we have a kind of a roadmap, and we probably have started to have some first basic tools for that. I wouldn't say like we are close to it. But with AI, maybe nothing is truly impossible.

一个是长时间尺度下的几乎必然全局行为问题——有一些宏大的想法，但我还不确定如何让它们真正发挥作用。另一个是量子场论。最著名的问题是杨-米尔斯理论，在椭圆版本中构造一个测度，但这真的非常复杂。复杂的原因在于它被称为「临界问题」。而我们在波动湍流中一直致力解决的问题在某种意义上也是临界问题。所以我希望发展的方法可能有所帮助。至少我们已经有了一种路线图，并开始拥有一些初步的基础工具。我不会说我们已经接近了。但是有了AI，也许没有什么是不可行的。

This is really interesting and crazy. How do you even conceptualize things like that? How do you think about concepts like this?

这真的太有趣了。你是如何构想出那种东西的？你是如何思考这类概念的？

In finite dimensions, your point is determined by many coordinates. In infinite dimensions, your point is determined by infinitely many coordinates, which means you think of your point as a function. You view a function as a point, and then the set of functions as a space. It is indeed complicated, especially when you have the nonlinear. But this is also something that is really important in physics, so you need to understand it.

在有限维中，你的点由许多坐标确定。在无限维中，你的点由无穷多个坐标确定，这意味着你把一个点看作一个函数。你把一个函数看作一个点，然后把函数的集合看作一个空间。这确实很复杂，尤其是在涉及非线性的时候。但它在物理学中非常重要，所以你需要理解它。

Everything is connected — physics, mathematics, even my basic high school math. That's really inspiring for me to hear. I think it's a great place to end off. Thank you so much for joining me for this interview, and it's really been a pleasure to talk to you. Thank you.

我现在意识到一切都是有联系的——物理、数学，甚至我那些基础的高中数学。这对我是很大的启发。非常感谢您接受采访。

Thank you.

谢谢。