🎙️ 邓煜菲尔兹奖获奖专访 · 中英对照

Thank you.

谢谢。

It's a really busy week. So I remember on the day of opening ceremony, I had the whole morning completely occupied. And when I finally got back to my hotel, I realized I have almost 100 unread emails. So it was really nice feeling, but really exhausting at that time. And especially I feel that now I'm getting really famous on Chinese internet, which is — I don't know how to say about it, but it's yeah.

这周非常忙。我记得开幕式那天，整个上午都排满了。当我终于回到酒店时，发现几乎有100封未读邮件。那种感觉很好，但当时也很累。尤其是我现在在中国互联网上变得非常出名——我不知道该怎么说，但确实如此。

Well, yeah, I maybe had some vague thoughts about it, you know, years ago. I was really seriously thinking about it. In 2014 we finished our work, so thinking maybe that's something. But really, I learned this only in the generalities here, so I really haven't been thinking too much. When the time came close, I was getting a little bit stressed, but not too much. I don't think about these things because these things you cannot control.

嗯，是的，多年前我可能有过一些模糊的想法。我确实认真考虑过。2014年我们完成了工作，所以觉得也许有可能。但实际上，我只是在大概层面了解，并没有想太多。临近时我有点紧张，但不算太多。我不去想这些事情，因为这些都是无法控制的。

Yes, essentially, it concerns this picture of the fluid or the gas, kind of that is made up by these tiny particles, tiny molecules. So what you want to do here is that you want to connect this kind of behavior of these particles — which is very relatively simple, like you have Newton's law, you have interactions, kind of binding interactions — and from this huge system, try to find the effective equation that governs the statistical behavior of these particles, which will be the Boltzmann equation, and then with the second limit you get to this familiar fluid equation. So this is the main thing that we did, building this bridge between the different scales.

是的，这主要涉及由微小粒子、微小分子组成的流体或气体的图像。这里要做的是将这些粒子的行为——相对简单，比如牛顿定律和相互作用——与整个巨大系统联系起来，试图找到支配这些粒子统计行为的有效方程，即Boltzmann equation，然后通过第二个极限得到熟悉的流体方程。这就是我们主要的工作，在不同尺度之间建立桥梁。

I was in this IMO and I got a gold medal that time. I was thinking that maybe math is the correct thing for me to do. I don't think there is a particular moment where I decided on that, but after IMO I decided to learn some analysis — college math. And then I went to college and I kept enjoying doing math. So gradually I realized that this is the thing I want to do.

我参加了IMO并获得了金牌。那时我觉得数学可能是我该做的事。我不认为有某个特定的时刻我做了决定，但IMO之后我决定学一些分析——大学数学。然后我上了大学，一直享受做数学的过程。渐渐地我意识到这就是我想做的事。

Yeah, so there is a lot of randomness. On the one hand, 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 exceptionally bad cases in this dynamics, but if you don't need to care about those, you only care about what happens generically, then usually this will be much better. Of course you need to discover ways to estimate it — you need to justify that these bad things really happen very rarely. But on the other hand, there are also very difficult questions. I still do not have enough understanding in terms of propagation of randomness for long-time scales. That's still something very difficult.

是的，随机性很多。一方面，随机性意味着不确定性。另一方面，你不需要追踪每一条轨迹——因为有随机性，你只需要关注高概率发生的事件，这实际上简化了很多。动力学中总是有一些特别糟糕的情况，但如果你不需要关心那些，只关心一般情况，那么通常情况会好很多。当然，你需要找到估计的方法——你需要证明这些坏事确实很少发生。但另一方面，也有一些非常困难的问题。对于长时间尺度上随机性的传播，我还不够理解。这仍然是非常困难的。

Yeah, it's essentially probability, yes.

是的，本质上就是概率论，对。

Applications. Well, I mean, yeah, if there is a better understanding of these structures that we are seeing in these derivations, maybe there could be some applications. For now, we are more from a technical point of view of pure math proofs. But who knows, maybe there will be some applications if you understand it better.

应用。嗯，我的意思是，是的，如果对这些推导中看到的结构有更好的理解，也许会有一些应用。目前，我们更侧重于纯数学证明的技术层面。但谁知道呢，如果你理解得更好，也许会有应用。

Math is a combination of these basic objects. It's also a way of understanding nature. This is what's amazing — you can use this combination of really simple things to find a way to understand nature. For the next step, well, there are many questions. One question I've been thinking about is this propagation of randomness for long times. And there are also a number of so-called critical problems in PDEs, SPDEs, quantum field theory, that hopefully the methods developed will be able to be applicable. Many, many interesting questions to look at.

数学是这些基本对象的组合。它也是一种理解自然的方式。令人惊奇的是，你可以用这些非常简单的东西的组合来理解自然。至于下一步，嗯，有很多问题。我一直在思考的一个问题是随机性在长时间上的传播。还有PDEs, SPDEs, 量子场论中的许多所谓临界问题，希望已经开发的方法能够适用。有很多很多有趣的问题值得研究。

I hope in this direction of quantum field theory, I hope to see maybe significant progress, because I think they are already starting to have some tools in this subject. So I would hope to see significant progress in this direction.

我希望在量子场论这个方向能看到重大进展，因为我认为这个领域已经开始有一些工具了。所以我希望在这个方向看到显著进展。

AI has been more and more helpful in math research. For the future, I'm still in the process of learning how to use AI effectively. One of the things is I now can prove some short theorems with AI. If we're working on some big project, we can have a framework that allows us to divide this project into individual sub-lemmas that are relatively short, then AI can probably help us prove them, which would significantly accelerate progress. I hope this would make kind of doable things faster and make currently undoable things doable.

AI在数学研究中越来越有帮助。对于未来，我仍在学习如何有效使用AI。其中一件事是，我现在可以用AI证明一些简短的定理。如果我们正在做一个大项目，可以有一个框架，将项目分解成相对较短的子引理，然后AI可能帮助我们证明它们，这将大大加速进展。我希望这能让可行的事情更快，让目前不可行的事情变得可行。

The first time I seriously used AI in my research was — there was a lemma in my research project recently, and I spent like five days finding the proof, a four-page proof. But I asked GPT, and it gave me a one-page proof within one hour. So I was really, really impressed. In the end, it turned out that that proof cannot be generalized because of the way the question was asked — it was a special case. So we didn't actually include it in the paper. But at least it provided some ideas that were very useful.

我第一次在研究中认真使用AI是——最近我的研究项目中有一个引理，我花了大约五天找到证明，有四页长。但我问了GPT，它在一小时内给出了一个一页的证明。所以我非常、非常印象深刻。最后发现那个证明不能被推广，因为问题的提法是一个特例。所以我们没有把它写进论文。但至少它提供了一些非常有用的想法。

I'm sure AI will get stronger — which means it will be more helpful. There are also some concerns: if AI can produce even longer, more complicated proofs, then what do mathematicians do? My point of view is: right now we prove theorems, we rely on the theorems that we know. If AI can prove more theorems, then we will rely on those theorems as well, and we'll build our arguments and thoughts on those theorems and investigate further things. So in this aspect, I think AI will always be something that is helpful. I wouldn't say it's a helper, assistant, or replacement. I don't want to feel that way. It's not about competing against AI. 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.

我相信AI会变得更强——这意味着它会更有帮助。也有一些担忧：如果AI能产生更长、更复杂的证明，那么数学家做什么？我的观点是：现在我们证明定理，依赖已知的定理。如果AI能证明更多定理，那么我们也会依赖那些定理，并在这些定理上构建我们的论证和想法，研究进一步的问题。因此在这方面，我认为AI总是有帮助的。我不会说它是一个助手、帮手或替代品。我不想那样认为。这不是与AI竞争。我们与AI合作来实现以前无法实现的目标。只要能做到这一点，就没有什么可担心的。

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

我希望AI的加速将使以前不可能的问题变得可能。这样我们就能真的取得比之前想象的更大的进步。

One question I mentioned earlier is this long-time propagation of randomness — the global hypoellipticity problem, which I think there are some big ideas, but I'm not sure how to make that work. And another thing is this quantum field theory, which I really expect to see at least some progress in the near future.

我之前提到的一个问题是随机性的长时间传播——全局次椭圆性问题，我认为有一些大想法，但不确定如何实现。另一件事是量子场论，我真的很期待在不久的将来至少看到一些进展。

It concerns the construction of certain kind of infinite-dimensional measures. The most famous problem here would be the Yang-Mills construction in an elliptic version — it's a construction of a measure, but it's really complicated. There are different ways to approach it. And the reason it's so difficult is because it's a so-called critical problem. I hope that the methods we have been working on — with turbulence in some sense and also critical problems — might be helpful. Of course it's still pretty far away, but I think at least we have a roadmap and we're starting to have some of the first basic tools for that. I wouldn't say we are close to it, but with AI, maybe there is nothing that is impossible.

它涉及某种无限维测度的构造。这里最著名的问题可能是椭圆版本的Yang-Mills构造——这是一个测度的构造，但非常复杂。有不同的方法可以接近它。它之所以如此困难，是因为它是一个所谓的临界问题。我希望我们一直在研究的方法——在某种意义上涉及湍流和临界问题——可能会有所帮助。当然它还很远，但我认为至少我们有了路线图，并且开始有一些最初的基本工具。我不会说我们已经接近了，但有了AI，也许没有什么是不可能的。

In finite dimensions, your point is determined by finitely many coordinates. Now in infinite dimensions, your point is determined by infinitely many coordinates, which means you think of your point as a function. This is the basic point of view, going back to the beginning of functional analysis — you view a function as a point, and the set of functions as a space. It is indeed complicated, especially in a nonlinear setting. But this is something that is really important in physics, so you need to understand it.

在有限维中，一个点由有限个坐标确定。而在无限维中，一个点由无穷多个坐标确定，这意味着你把点看作一个函数。这是基本的观点，可以追溯到泛函分析的早期——你将函数视为一个点，函数集视为一个空间。这确实很复杂，尤其是在非线性背景下。但这是物理学中非常重要的东西，所以你需要理解它。