Amid Life’s Chaos, a Meticulous Mathematician Finds Stability
Yu Deng faced waves of uncertainty and doubt before attempting the work that would earn him a Fields Medal.
在尝试为他赢得菲尔兹奖的工作之前，邓宇面临着一波又一波的不确定性和怀疑。
F reedom and restraint seem like opposing forces, but to Yu Deng , they go hand in hand.
自由和克制看似是对立的力量，但对邓宇来说，它们是齐头并进的。
Take the poem he’s just pulled up on his computer screen: a dense forest of Chinese characters corralled into eight lines of equal length — a perfect, symmetric block of text.
以他刚刚在电脑屏幕上调出的一首诗为例：茂密的汉字森林被分成八行等长的文字——一个完美、对称的文本块。
Written in the ninth century by the Tang Dynasty poet Li Shangyin, as China teetered on the brink of civil war, “The Brocade Zither” is famous for its ambiguity.
《锦瑟》由唐代诗人李商隐写于九世纪，当时中国正处于内战的边缘，《锦瑟》以其模糊性而闻名。
“Many people have tried to explain it,” Deng said.
“很多人都试图解释这一点，”邓说。
“But I think this is something that’s not explainable.” A single word might allude to an ancient battle or a classic fairy tale; a simple phrase might have scores of diverging interpretations.
“但我认为这是无法解释的事情。”一个词可能会暗示一场古老的战争或一个经典的童话故事；一个简单的短语可能有多种不同的解释。
“My personal favorite of all time,” Deng said — an example of a highly structured form of Chinese poetry that literally translates to “regulated verse.” It must satisfy all sorts of rules: parallel constructions and contrasting images, lilting tonal patterns and strict rhyme schemes.
邓说：“我个人一直以来最喜欢的。”这是高度结构化的中国诗歌形式的一个例子，字面意思是“规范诗”。它必须满足各种规则：平行结构和对比图像、轻快的音调模式和严格的韵律方案。
“These patterns are part of the beauty,” Deng said.
“这些图案是美的一部分，”邓说。
“But they’re also something that restricts your expression.
“但它们也是限制你表达的东西。
The fact that you can still say so many things under this restriction is another level of beauty.”
在这种限制下还能说这么多话，真是另一种境界的美。”
Deng, a mathematician at the University of Chicago, explores the many things you can say under a mathematical form of regulated verse.
芝加哥大学的数学家邓探索了在规范诗的数学形式下可以表达的许多内容。
He studies equations that describe how complicated systems of waves and particles interact.
他研究描述复杂的波和粒子系统如何相互作用的方程。
Like the Tang poet packing layers of meaning into a handful of couplets, “You are going to summarize, to compress, this whole big system in terms of this single equation,” he said.
就像唐朝诗人将层层含义包装在几句对联中一样，“你要用这个单一的方程来总结、压缩整个大系统，”他说。
The solutions to these equations have a “very rich, natural structure,” full of hidden symmetries and “so many amazing coincidences, so many interpretations.”
这些方程的解具有“非常丰富、自然的结构”，充满隐藏的对称性和“如此多令人惊奇的巧合，如此多的解释”。
Deng, who turned 37 in June, has now been awarded the Fields Medal, math’s highest honor, for pushing the study of these equations — and the physical systems they model — past what anyone thought possible.
今年 6 月就年满 37 岁的邓现在被授予菲尔兹奖，这是数学的最高荣誉，因为他推动了对这些方程以及它们所建模的物理系统的研究，超出了任何人的想象。
He and his colleagues have proved major theorems, some in papers more than 100 pages long, on how randomness moves through systems, and on how larger-scale behaviors emerge from those interactions over time.
他和他的同事已经证明了一些主要定理，其中一些定理长达 100 多页，涉及随机性如何在系统中移动，以及随着时间的推移，这些相互作用如何产生更大规模的行为。
“People thought this was, I don’t know, a decade in the future,” said Benoît Pausader , a mathematician at Brown University and one of Deng’s collaborators.
“人们认为，我不知道，这是未来十年，”布朗大学数学家、邓小平的合作者之一伯努瓦·保萨德 (Benoît Pausader) 说。
At its core, the work is about bringing the flexible nature of probability into the rigid, structured world of wave equations.
这项工作的核心是将概率的灵活本质带入刚性的、结构化的波动方程世界。
Freedom and restraint.
自由与克制。
Hand in hand.
携手。
In a parallel universe, Deng became not a mathematician, but a professional Go player.
在平行宇宙中，邓不是成为一名数学家，而是一名职业围棋棋手。
It was the nearest of misses that set him on his current path.
这是最近的一次失误，让他走上了目前的道路。
Shortly after he was born in 1989, his family moved to Shenzhen, China, at a time when the city — a cluster of fishing villages just 10 years earlier — was growing rapidly.
1989 年他出生后不久，他的家人搬到了中国深圳，当时这座城市——10 年前还是一片渔村——正在迅速发展。
(Today it’s the country’s third-largest metropolis, home to more than 18 million people and one of the world’s biggest tech hubs.) When Deng was young, Shenzhen’s Shekou Industrial Zone, where he grew up, was “a really underdeveloped region” where “you’d see all this waste on the street.
（如今，它是中国第三大城市，拥有超过 1800 万人口，也是世界上最大的科技中心之一。）邓小平年轻时，他长大的深圳蛇口工业区是“一个非常不发达的地区”，“你会在街上看到所有这些废物。
It had a terrible smell.” But within a few years, he said, the district had clean streets, subways, parks, and luxury apartments.
它有一股难闻的气味。”但他说，几年之内，该地区就有了干净的街道、地铁、公园和豪华公寓。
Billboards and posters loudly declared the city’s slogan: TIME IS MONEY!
广告牌和海报大声宣告着这座城市的口号：时间就是金钱！
EFFICIENCY IS LIFE!
效率就是生命！
Deng’s path to math took him through the world of competitive Go.
邓的数学之路带他进入了围棋竞技的世界。
Deng was a quiet, contemplative child.
邓小平是一个安静、善于思考的孩子。
He preferred to spend long hours alone, reading whatever he could get his hands on.
他更喜欢独自度过长时间，阅读任何他能找到的东西。
His mother, a gastroenterologist, said that by the time he was 5 or 6, he’d read some of her medical textbooks, frequently asking questions about anatomy and structure.
他的母亲是一位胃肠病学家，她说，当他五六岁的时候，他就读过她的一些医学教科书，经常问有关解剖学和结构的问题。
“Given how young he was, all of my colleagues thought it was quite amusing,” she said in Mandarin.
“考虑到他还年轻，我所有的同事都认为这很有趣，”她用普通话说道。
Soon enough, he had “basically read all the books in our study,” she recalled.
很快，他“基本上读完了我们书房里的所有书籍”，她回忆道。
To get him outside more, she and her husband, a software engineer, took him on hikes or walks along the beach, with the promise that he could read or visit a bookstore as soon as they finished.
为了让他更多地出去活动，她和她的软件工程师丈夫带他去远足或沿着海滩散步，并承诺一旦他们完成，他就可以读书或去书店。
If he misbehaved, “the most effective punishment was to forbid him from reading,” his mother said.
如果他行为不端，“最有效的惩罚就是禁止他读书，”他的母亲说。
On some of those hikes, his father gave him math problems to solve.
在其中一些徒步旅行中，他的父亲给了他数学题来解决。
He still remembers one: You have a square grid divided into a certain number of cells.
他还记得一个：你有一个方形网格，分为一定数量的单元格。
Prove that if you remove any one cell from the grid — it doesn’t matter which — you can always perfectly divide the rest of the grid into L-shaped pieces made from three cells each.
证明如果你从网格中删除任何一个单元格（无论是哪一个单元格），你总是可以将网格的其余部分完美地分成由三个单元格组成的 L 形块。
The solution, which Deng figured out, required a concept known as induction, usually taught in high school.
邓提出的解决方案需要一个称为归纳法的概念，通常在高中教授。
Deng was 7 years old.
邓小平7岁。
His parents grew concerned that Deng’s constant reading would ruin his eyesight, so they bought him a Go set.
他的父母担心邓不断读书会损害他的视力，所以给他买了一套围棋。
He took to the game immediately.
他立即投入到比赛中。
It wasn’t long before he was spending almost all his free time exploring the universes he could create with the black and white stones — solving puzzles, finding optimal strategies, getting lost in thought for hours on end.
不久之后，他几乎把所有的空闲时间都花在了探索他可以用黑白石头创造的宇宙上——解决谜题，寻找最佳策略，连续几个小时陷入沉思。
“Playing Go was something that allows you to focus on a single thing for some time, usually two or three hours,” he said.
“下围棋可以让你在一段时间内专注于一件事，通常是两三个小时，”他说。
“I enjoyed that.”
“我很享受这样。”
He got good quickly, playing at a local Go club and competing in district and city-wide tournaments.
他很快就取得了进步，在当地的一家围棋俱乐部比赛，并参加地区和全市的锦标赛。
“He had this unwillingness to admit defeat,” his mother said.
“他不愿意承认失败，”他的母亲说。
“When he played Go with his friends, if he lost, he would insist on playing again and again until he won.”
“当他和朋友下围棋时，如果输了，他会坚持下一遍又一遍，直到赢。”
People thought this was, I don’t know, a decade in the future.
我不知道，人们认为这是未来十年。
In 2001, when he was in middle school, he competed in the national pro qualification tournament, a grueling competition to enter the world of professional Go.
2001年，还在上中学的他参加了全国职业围棋资格赛，这是一场进入职业围棋世界的艰苦比赛。
By the final three games of the tournament, he needed only one more win to qualify.
到了锦标赛的最后三场比赛，他只需要再赢一场就能获得参赛资格。
He lost all three.
他三个都输了。
He tried again the following year, but his heart was no longer in it.
第二年他再次尝试，但他的心已经不在了。
Ever since those early family hikes, Deng had also been nurturing a passion for math, for the same reasons he’d been drawn to Go: He could spend long periods alone with a single problem, looking for patterns and trying to find the unique path to the right answer.
自从那些早期的家庭远足以来，邓一直在培养对数学的热情，出于同样的原因他被围棋吸引：他可以花很长时间独自解决一个问题，寻找模式并试图找到找到正确答案的独特路径。
And he was good at it.
他很擅长。
“But it wasn’t serious” then, he said.
“但这并不严重”，他说。
After his loss at that Go tournament, though, his assessment was almost clinical: “There was a choice,” he said.
不过，在那场围棋比赛中失利后，他的评估几乎是冷静的：“有一个选择，”他说。
“Either Go or math.
“要么围棋，要么数学。
Since I lost, I decided to move my focus to math.”
自从我输了之后，我决定把注意力转移到数学上。”
He set a goal: to compete in the International Mathematical Olympiad (IMO), the most prestigious math competition for pre-college students in the world.
他设定了一个目标：参加国际数学奥林匹克（IMO），这是世界上最负盛名的大学前学生数学竞赛。
His math teacher at the time was also one of China’s best Olympiad coaches, and they started to train together on evenings and weekends.
他当时的数学老师也是中国最好的奥数教练之一，他们开始在晚上和周末一起训练。
In his last year of middle school (the equivalent of ninth grade), he’d learned the entire high school math curriculum, but he failed to pass the first selection round to compete in the IMO.
在中学的最后一年（相当于九年级），他学完了整个高中数学课程，但他未能通过参加 IMO 比赛的第一轮选拔。
“Then I felt really bad,” Deng said.
“然后我感觉非常糟糕，”邓说。
So he worked harder.
于是他更加努力了。
The next year, he made it through three selection rounds and earned a spot on the national training team.
第二年，他通过了三轮选拔，并获得了国家集训队的一席之地。
But he wasn’t one of the six students ultimately selected to represent China.
但他并不是最终被选为代表中国的六名学生之一。
His focus and ambition never flagged.
他的专注和野心从未减弱。
“I really wanted to win,” he said.
“我真的很想赢，”他说。
The following year he did more than make the team; he won a gold medal.
第二年，他所做的不仅仅是入选球队；他还做了更多的事情。他赢得了一枚金牌。
Deng in his office at the University of Chicago, where he’s been a professor since 2025.
邓在芝加哥大学的办公室里，自 2025 年以来他一直担任该大学的教授。
His performance earned him automatic admission to Peking University, one of China’s top colleges.
他的表现使他自动进入中国顶尖大学之一的北京大学。
At first, “I didn’t even think about choosing math as a career,” he said.
起初，“我什至没有考虑过选择数学作为职业，”他说。
“I kind of gradually realized that this is a good thing for me.”
“我渐渐意识到这对我来说是一件好事。”
The Olympiad experience “made me think, maybe this is something I really have talent for.” So, he figured, “why don’t I just do it?”
奥运会的经历“让我思考，也许这才是我真正有天赋的事情。”所以，他想，“我为什么不这么做呢？”
After two years as a math major at Peking University, he transferred to the Massachusetts Institute of Technology.
在北京大学数学专业学习两年后，他转学到麻省理工学院。
It was common for Chinese math students to attend graduate school abroad, but rare for undergraduates.
中国数学学生到国外读研究生的情况很常见，但对于本科生来说却很少见。
“You are going to go somewhere outside of China eventually,” Deng said — so why wait?
邓说：“你最终会去中国以外的地方。”那么为什么要等呢？
That first year he attended a talk that would set the course of his career.
第一年，他参加了一场决定他职业生涯的演讲。
Andrea Nahmod — a mathematician visiting from the University of Massachusetts, Amherst, who would later become one of Deng’s closest collaborators — presented research on the so-called random data problem.
安德里亚·纳莫德（Andrea Nahmod）是一位来自马萨诸塞大学阿默斯特分校的数学家，后来成为邓小平最亲密的合作者之一，他介绍了所谓随机数据问题的研究。
Deng was hooked.
邓被迷住了。
The random data problem asks about the solutions to certain kinds of partial differential equations, or PDEs, that describe how complex wave patterns — from water waves in the ocean to light waves in fiber-optic cables — change over time.
随机数据问题询问某些类型的偏微分方程（PDE）的解，这些偏微分方程描述了复杂的波浪模式（从海洋中的水波到光纤电缆中的光波）如何随时间变化。
In particular, Nahmod wanted to show that, no matter how your waves look at the beginning, the PDE can accurately model what will happen to them at any point in the future.
特别是，纳莫德想要表明，无论波浪一开始看起来如何，偏微分方程都可以准确地模拟它们在未来任何时刻会发生什么。
There might be cases where the equation breaks down and fails to describe what a wave will do next, but such cases should be extremely rare.
在某些情况下，方程可能会崩溃并且无法描述波浪接下来会做什么，但这种情况应该极其罕见。
If you start with some random wave pattern, the PDE should have a solution that describes its behavior at later times.
如果您从某种随机波形开始，偏微分方程应该有一个解决方案来描述其稍后的行为。
He was brilliant from the beginning …
他从一开始就很出色……
very good at getting at the essence of the thing.
非常善于抓住事物的本质。
The problem is that for most PDEs, the existence of this solution is exceedingly hard to prove.
问题是对于大多数偏微分方程来说，这个解的存在性极难证明。
“I found this idea very interesting,” Deng said.
“我觉得这个想法很有趣，”邓说。
Because the question involves a random starting point, it brings probability theory — used to define that starting point and to prove statements about what happens next — into the deterministic world of PDEs.
因为这个问题涉及一个随机起点，所以它将概率论（用于定义该起点并证明有关接下来发生的情况的陈述）带入了偏微分方程的确定性世界。
“You are seeing your equation from a completely different point of view.”
“你正在从完全不同的角度看待你的方程。”
After the talk, Deng asked Gigliola Staffilani , a mathematician at MIT who was acting as his informal adviser, to give him a related problem to work on over the summer.
谈话结束后，邓小平请他的非正式顾问、麻省理工学院数学家吉廖拉·斯塔菲拉尼(Gigliola Staffilani)给他一个相关的问题，让他在夏天解决这个问题。
She found him one, on the behavior of solutions to a two-dimensional version of the nonlinear Schrödinger equation, a famous PDE that describes how waves move and interact.
她为他找到了一个关于非线性薛定谔方程的二维版本解的行为的问题，这是一个著名的偏微分方程，描述了波如何移动和相互作用。
When Deng returned in the fall, he had a completed paper in hand.
当邓小平秋天回来时，他手里拿着一份已完成的文件。
Staffilani was taken aback by how sophisticated it was.
斯塔菲拉尼对它的复杂程度感到惊讶。
“I was like, ‘Oh my God, this kid, he’s really spectacular,’” she said.
“我当时想，‘天哪，这孩子，他真的太棒了，’”她说。
“I had not seen any work like that, not even from my graduate students, at least not polished in that way.
“我从未见过这样的作品，即使是我的研究生也没有见过，至少没有经过这样的打磨。
It was done.”
已经完成了。”
The paper was published in the journal Analysis & Partial Differential Equations .
该论文发表在《分析与偏微分方程》杂志上。
“I was excited about that,” Deng said.
“我对此感到很兴奋，”邓说。
“And I was thinking, one day I could publish in an even better, top journal.”
“我在想，有一天我可以在一本更好的顶级期刊上发表文章。”
He’d heard about a student his age named John Pardon — another Fields Medal recipient this year — who published a paper in the Annals of Mathematics , the field’s top journal, as an undergraduate.
他听说一位名叫约翰·帕登（John Pardon）的同龄学生——今年的另一位菲尔兹奖获得者——本科时在该领域顶级期刊《数学年鉴》上发表了一篇论文。
Was he envious?
他嫉妒吗？
“A little bit,” Deng laughed.
“有一点。”邓笑道。
Deng often gets his best ideas when he’s alone in open spaces — on hikes, near the ocean.
邓独自一人在空旷的地方——徒步旅行、靠近大海时，常常会得到最好的想法。
He finds a similar feeling in the Tang poetry he’s drawn to.
他在自己所喜爱的唐诗中也找到了类似的感觉。
“It’s a lonely kind of atmosphere,” he said of one line.
“这是一种孤独的氛围，”他谈到其中的一句话时说道。
Of another, “I like the view.
另一位则说：“我喜欢这里的景色。
By yourself, hearing these sounds, looking at the sky, seeing the sky reflected in the river …
独自一人，听到这些声音，看着天空，看到天空倒映在河里……
Everything is by yourself.
一切都是靠你自己。
You stand there and think.” The stars are distant, he said, even your thoughts are distant, as you contemplate historical events from long ago; you are completely, utterly alone, the world quiet around you.
你站在那里思考。”他说，星星很遥远，甚至你的思想也很遥远，当你沉思很久以前的历史事件时；你完全、完全孤独，你周围的世界很安静。
“It’s really a nice feeling.”
“这真的是一种很好的感觉。”
Deng has turned to nature during times of stress and uncertainty.
在面临压力和不确定性的时期，邓小平转向了大自然。
He lingers over such forlorn, isolated scenes when he writes his own poetry, too.
当他写自己的诗时，他也对这些荒凉、孤立的场景流连忘返。
His living room is pretty much empty.
他的客厅几乎是空的。
There’s a desk, a whiteboard, a small bookshelf, and an armchair.
有一张桌子、一块白板、一个小书架和一张扶手椅。
No couch, no TV, no art on the walls, no rug on the floor.
没有沙发，没有电视，墙上没有艺术品，地板上没有地毯。
“It’s good for me,” he said.
“这对我有好处，”他说。
“It’s a feeling of freedom.
“这是一种自由的感觉。
I can just walk around and think.” Even though he’s lived there for a year and a half, it looks like he just moved in.
我可以一边走一边思考。”虽然他已经在那里住了一年半了，但看上去就像是刚搬进来一样。
His biggest adjustment in his undergraduate years, he said, was living with roommates.
他说，他本科期间最大的调整是和室友住在一起。
Even when he’s around the people he’s closest to, Deng is humble and reserved.
即使在他最亲近的人身边，邓也是谦虚、内敛的。
He’s eager to discuss math — when he does, he talks a bit faster, a bit louder — but according to his colleagues, when the conversation moves to other topics, his responses are short and pointed.
他渴望讨论数学——当他讨论时，他会说得更快、更大声——但据他的同事说，当谈话转向其他话题时，他的回答简短而尖锐。
He doesn’t see the need to elaborate or steer the discussion in new directions.
他认为没有必要详细阐述或将讨论引向新的方向。
(When asked if he has any favorite poets: “Yes.”)
（当被问到他是否有最喜欢的诗人时：“是的。”）
Deng doesn’t drive — never even got a license — because he fears that his mind will drift to math while he’s on the road.
邓不开车——甚至从来没有拿到驾照——因为他担心在路上他的注意力会转移到数学上。
As an undergraduate, he once went line-by-line through a highly technical PDE paper , all 75 pages.
作为一名本科生，他曾经逐行阅读过一篇技术含量很高的 PDE 论文，共 75 页。
It took him a week, eight hours per day.
他花了一周的时间，每天八个小时。
“I just checked every single detail,” he said.
“我只是检查了每一个细节，”他说。
While he no longer has quite that level of stamina, he’s still able to devote long hours to reading a paper thoroughly, or to working on a problem.
虽然他不再有那样的耐力，但他仍然能够花很长时间来彻底阅读论文或解决问题。
“If I don’t make good progress, I will be really disappointed, and I will keep thinking about it,” he said.
“如果我没有取得好的进展，我会非常失望，我会继续思考，”他说。
Why should the sad zither have fifty strings?
哀琴为何要有五十弦？
Each string, each strain evokes but vanished springs:
每根弦、每一种张力都会唤起却又消失的春天：
Dim morning dream to be a butterfly;
晨昏梦化蝶；
Amorous heart poured out in cuckoo’s cry.
多情的心在杜鹃的叫声中倾泻而出。
In moonlit pearls see tears in mermaid’s eyes;
月光下的珍珠看到人鱼眼中的泪水；
With sunburnt mirth let blue jade vapor rise.
带着晒伤的欢笑，让蓝色的玉气升起。
Such feeling cannot be recalled again;
这种感觉再也无法回忆；
It seemed long lost e’en when it was felt then.
当它被感觉到时，它似乎已经消失了很久。
He finished his degree at MIT in 2011, then went to graduate school at Princeton University, followed by a postdoc at the Courant Institute at New York University.
2011年，他在麻省理工学院获得学位，随后前往普林斯顿大学攻读研究生，随后在纽约大学库朗研究所攻读博士后。
“He was brilliant from the beginning,” said Alexandru Ionescu , Deng’s doctoral adviser.
“他从一开始就非常出色，”邓小平的博士生导师亚历山德鲁·约内斯库 (Alexandru Ionescu) 说。
“Even as a graduate student, without experience …
“即使作为一名研究生，没有经验......
he was very good at getting at the essence of the thing, picking up what matters in a problem.”
他非常善于抓住事物的本质，找出问题中的关键所在。”
Deng wanted to make progress on the random data problem, specifically when PDEs start with “rough” initial waves — random waves that oscillate wildly or have discontinuities.
Deng 希望在随机数据问题上取得进展，特别是当偏微分方程以“粗糙”初始波（剧烈振荡或不连续的随机波）开始时。
If he could show that even under these conditions, a PDE must have a solution, he’d have a deeper understanding of the equation and of a particular property of physical systems called the Gibbs measure.
如果他能够证明即使在这些条件下，偏微分方程也必须有一个解，那么他就会对该方程以及物理系统的一个称为吉布斯测度的特定属性有更深入的理解。
But he soon realized that the tools available at the time weren’t up to the task.
但他很快意识到当时可用的工具无法胜任这项任务。
“We could get some results, but they were not satisfactory,” he said.
“我们可以获得一些结果，但并不令人满意，”他说。
He moved on to other questions, but they didn’t grip him in the same way.
他继续讨论其他问题，但这些问题并没有以同样的方式吸引他。
He began to feel aimless and, at times, “a bit depressed.”
他开始感到漫无目的，有时“有点沮丧”。
“The work I did at that time wasn’t the best,” he said.
“我当时所做的工作并不是最好的，”他说。
“I think it hurt him,” said Nahmod, who would later become a close confidante and friend.
“我认为这伤害了他，”纳莫德说，他后来成为了亲密的知己和朋友。
“He’s very sensitive.”
“他非常敏感。”
For comfort, Deng turned to manga.
为了寻求安慰，邓转向漫画。
“I remember him at Courant.
“我记得他在库朗。
He was always holding a manga,” said Jalal Shatah , a professor there who formerly chaired the math department.
他总是拿着一本漫画，”曾担任数学系主任的该校教授贾拉勒·沙塔 (Jalal Shatah) 说道。
Deng particularly liked stories about deep friendship and romance and found himself drawn to a genre known as yuri , which focuses on those kinds of relationships between women — stories that “feel nice and warm and beautiful,” as he put it.
邓特别喜欢有关深厚友谊和浪漫的故事，并发现自己被一种名为《尤里》的类型所吸引，该类型专注于女性之间的关系——用他的话说，这些故事“感觉良好、温暖和美丽”。
They helped him be kinder to himself, “more comfortable with life.”
他们帮助他对自己更加友善，“对生活更加舒适”。
In 2017, as the end of his NYU postdoc drew near, Deng hadn’t gotten tenure-track offers from his preferred institutions.
2017年，随着纽约大学博士后生涯的临近，邓还没有从他喜欢的机构获得终身教授职位。
For several months he considered going back to China or taking a job in finance with one of the companies that had been recruiting him.
几个月来，他一直在考虑是否要回到中国，或者是在一家一直在招聘他的公司中找到一份财务工作。
“I told them I still want to do math, but if I realize I cannot reach what I’m aiming for, then maybe I could consider this,” he said.
“我告诉他们我仍然想做数学，但如果我意识到我无法达到我的目标，那么也许我可以考虑这个，”他说。
Deng still liked being alone — preferred it, in fact — but for the first time, he felt lonely.
邓仍然喜欢独处——事实上更喜欢独处——但他第一次感到孤独。
He envisioned the days and years ahead.
他展望了未来的日子和岁月。
“In the end, as a mathematician, I’m just going to do math every day,” he said.
“最后，作为一名数学家，我每天都会做数学，”他说。
“Go to school, go home, go to school, go home.” But he felt overwhelming uncertainty about that future.
“上学，回家，上学，回家。”但他对未来感到巨大的不确定性。
Would he be able to produce great results?
他能打出出色的成绩吗？
Would he get recognized for them?
他会因为他们而得到认可吗？
He recalled reading a manga, “one of these idealized stories where people get together, and they work, and they have these complicated lives.
他回忆起读过一本漫画，“这是一个理想化的故事，人们聚在一起，工作，过着复杂的生活。
Eventually, they lead a good life.
最终，他们过上了美好的生活。
So I was reading this, and suddenly I got pretty emotional.
所以我正在读这篇文章，突然我很激动。
I was thinking, ‘OK, then what about myself?’”
我在想，‘好吧，那我自己呢？’”
That winter, feeling “very stressed and anxious,” he took a train to nearby Long Beach, on the south shore of Long Island, where he booked a hotel room.
那年冬天，他感到“非常紧张和焦虑”，于是乘火车前往长岛南岸附近的长滩，在那里订了一间酒店房间。
There, he roamed the coastline and wrote poetry, pouring himself onto the page in eight lines of restricted verse, the same style that Li Shangyin had used.
在那里，他漫步海岸线写诗，以八行小诗倾注于书页，与李商隐所用的风格相同。
He wrote about how he wanted to escape.
他写下了他如何想逃跑。
How doing so would be a betrayal of his dream to do math.
这样做会背叛他做数学的梦想。
How the tides seemed anxious, unsettled.
潮汐显得多么焦虑、不安。
Deng’s focus on math is so absolute that he doesn’t drive, as he worries his mind will drift to math when he’s on the road.
邓对数学的专注程度如此之高，以至于他不开车，因为他担心在路上时自己的注意力会转移到数学上。
Once he’d gotten it all down, “then I could say, ‘OK, this is done.’ I could go back to my life,” he said.
一旦他把一切都记下来，“然后我就可以说，‘好吧，完成了。’我就可以回到我的生活了，”他说。
He returned to Manhattan.
他回到了曼哈顿。
He remained uneasy, but after a week without work, he “automatically felt a need to work, to do math.”
他仍然感到不安，但一周没有工作后，他“自动感到需要工作，需要做数学。”
He kept grinding, and tried to push the doubts out of his mind.
他不断地思考，试图将心中的疑虑赶出去。
Then, in 2018, he got a tenure-track offer from the University of Southern California.
然后，2018 年，他获得了南加州大学的终身教职录取通知书。
He accepted.
他接受了。
Another mathematician who was feeling lost drew Deng out of this period of anxiety — and back to the random data problem.
另一位感到失落的数学家将邓从这段焦虑中拉了出来，回到了随机数据问题上。
Haitian Yue had just started his postdoctoral research at USC and was having a hard time.
岳海天刚刚开始在南加州大学的博士后研究，日子过得很艰难。
“I didn’t find a really good project.
“我没有找到一个真正好的项目。
I didn’t have a goal,” Yue said.
我没有目标，”岳说。
While driving Deng back to his apartment after a group dinner — everyone knew Deng didn’t drive, even in Los Angeles — Yue confided his worries and mentioned that he’d done his doctoral work on the random data problem.
集体晚餐后，岳开车送邓回公寓——每个人都知道邓不开车，即使是在洛杉矶——岳吐露了他的担忧，并提到他已经完成了随机数据问题的博士论文。
Deng thought it was a good time to return to it and suggested that they collaborate with Nahmod, whose talk had first drawn Deng to the subject.
邓认为现在是回到这个话题的好时机，并建议他们与纳赫莫德合作，纳赫莫德的演讲首先吸引了邓的关注。
Deng, Nahmod, and Yue turned to the two-dimensional Schrödinger equation, and set out to show that even when you start with a random initial wave that’s particularly rough, the equation still has a well-defined solution.
邓、纳莫德和岳转向二维薛定谔方程，并着手证明，即使从特别粗糙的随机初始波开始，方程仍然有一个明确定义的解。
If they could prove this, they’d then be able to show that the equation has a hidden statistical structure.
如果他们能够证明这一点，那么他们就能够证明该方程具有隐藏的统计结构。
Say you randomly choose an initial wave from infinitely many possibilities, according to a well-known rule called the Gibbs measure.
假设您根据称为吉布斯测度的众所周知的规则，从无限多种可能性中随机选择一个初始波。
That rule, which is supposed to model a system in a state of equilibrium, says that some fraction of possible initial waves has one property, and some fraction has another, and so on.
该规则应该对处于平衡状态的系统进行建模，它规定可能的初始波的某些部分具有一种属性，而某些部分具有另一种属性，依此类推。
Any particular wave might change over time and end up exhibiting very different properties.
任何特定的波都可能随着时间的推移而发生变化，并最终表现出截然不同的特性。
But Deng, Nahmod, and Yue wanted to show that if you look at all the waves you end up with, they are still distributed according to the same rule that you started with.
但邓、纳莫德和岳想要表明，如果你看看最终得到的所有波浪，它们仍然按照你开始时的相同规则分布。
That, as mathematicians put it, the Gibbs measure is “invariant.”
正如数学家所说，吉布斯测度是“不变的”。
Traditionally, mathematicians would approach the problem by splitting the solution to the PDE into two pieces and analyzing them separately.
传统上，数学家会通过将偏微分方程的解分成两部分并分别分析来解决这个问题。
But that didn’t work in this case: One of the pieces was still too complicated.
但这在这种情况下不起作用：其中一件作品仍然太复杂。
So Deng, Nahmod, and Yue split up that troublesome piece even further.
因此，邓、纳莫德和岳进一步分解了这个麻烦的部分。
They then developed tools called random tensors to analyze the pieces and explore how they influenced one another.
然后，他们开发了一种称为随机张量的工具来分析这些片段并探索它们如何相互影响。
Yue recalled that when they met in person to work on the project, Deng would often go quiet, lost in thought for five, 10, 15 minutes at a time.
岳回忆说，当他们亲自见面做这个项目时，邓经常安静下来，一次陷入沉思五分钟、十分钟、十五分钟。
“Andrea and I would keep discussing, while Yu Deng kept silent and tried to think by himself,” Yue said.
“我和安德里亚会一直讨论，而邓宇则保持沉默，试图自己思考，”岳说。
“In his mind he does a lot of computations.
“在他的脑海里，他做了很多计算。
Then suddenly he’d erase what we wrote down and say, ‘We should do it like this.’”
然后他突然会擦掉我们写下的内容并说，‘我们应该这样做。’”
“It was just magical,” Nahmod said.
“这真是太神奇了，”纳莫德说。
“Collaborating with him is exhilarating.
“与他合作令人兴奋。
When he is interested in something, he focuses like a laser beam.”
当他对某件事感兴趣时，他会像激光束一样集中注意力。”
As Pierre Germain , his postdoctoral adviser, put it, “When you work with him, it’s like you are running after a train.”
正如他的博士后导师皮埃尔·热尔曼所说，“当你和他一起工作时，就像你在追火车一样。”
This skill ultimately allowed the trio to understand the solution to the two-dimensional Schrödinger equation, and to prove that the Gibbs measure is invariant.
这项技能最终让三人组理解了二维薛定谔方程的解，并证明了吉布斯测度是不变的。
The proof was later published in the Annals , fulfilling the goal Deng had set for himself years before.
该证明后来发表在年鉴上，实现了邓小平多年前为自己设定的目标。
Around the same time that Deng started to work with Nahmod and Yue, he met Zaher Hani — an encounter he credits with also helping to dispel his period of doubt and depression.
大约在邓开始与纳莫德和岳合作的同时，他遇到了扎赫·哈尼——他认为这次相遇也帮助他消除了一段时期的怀疑和抑郁。
Hani, a mathematician at the University of Michigan, was also studying solutions to PDEs with random initial waves, but with a different goal in mind.
密歇根大学的数学家哈尼也在研究具有随机初始波的偏微分方程的解，但有着不同的目标。
Take the Schrödinger equation, which reveals how individual waves will interact and influence each other over time.
以薛定谔方程为例，它揭示了各个波如何随着时间的推移相互作用和相互影响。
Physicists have long accepted that from this “microscopic” description of a system, it’s possible to derive a higher-level description of the system’s average behavior, which is given by a different equation — the so-called wave kinetic equation.
物理学家长期以来一直认为，从系统的这种“微观”描述中，可以得出系统平均行为的更高层次的描述，该描述由另一个方程（即所谓的波动动力学方程）给出。
Mathematicians like Hani hoped to offer a rigorous proof of that derivation.
像哈尼这样的数学家希望为这一推导提供严格的证明。
Deng’s attention to detail is legendary.
邓小平对细节的关注是有口皆碑的。
“He knew everything, 200 pages of computations, off the top of his head,” said a colleague.
“他什么都知道，200 页的计算全都在他的脑海里，”一位同事说。
In 2019, Hani and some colleagues had just proved that the Schrödinger equation converges to the wave kinetic equation on short time scales.
2019年，Hani和一些同事刚刚证明了薛定谔方程在短时间尺度上收敛于波动动力学方程。
But on longer time scales, waves have the chance to interact in far more complicated ways that seemed impossible to deal with.
但在更长的时间尺度上，波有机会以更加复杂的方式相互作用，而这似乎是无法处理的。
Hani realized that insights from research on the random data problem might help.
哈尼意识到，对随机数据问题的研究得出的见解可能会有所帮助。
“At that time, I had already asked several other experts in the field.
“当时我已经询问过该领域的其他几位专家。
And I didn’t get any satisfactory answer,” he said.
但我没有得到任何满意的答复，”他说。
“Until I asked Yu Deng.
“直到我问于邓。
He had precisely the right idea that I needed.”
他的想法正是我所需要的。”
The two teamed up and extended Hani’s previous result to a time scale just shy of what physicists cared about.
两人联手，将哈尼之前的结果延长到了一个距离物理学家所关心的时间尺度。
Deng was ready to stop there.
邓小平准备就此打住。
“I was worried that it might not be doable, that the tools weren’t enough,” he said.
“我担心这可能不可行，工具还不够，”他说。
But Hani persuaded him to keep going.
但哈尼说服他继续前进。
They ended up abandoning previous approaches and developing their own technique “from scratch,” Germain said.
杰曼说，他们最终放弃了以前的方法并“从头开始”开发自己的技术。
“It’s completely crazy …
“这完全是疯了……
It’s nice, or scary, or however you want to put it, that they ignored everything and just went vroom .”
这很好，或者很可怕，或者不管你怎么说，他们忽略了一切，只是继续前进。”
First they represented the longer-term solution to the Schrödinger equation as a complicated sum of diagrams that resembled trees.
首先，他们将薛定谔方程的长期解表示为类似于树的复杂图表之和。
To prove that this sum converged to the wave kinetic equation, they needed to estimate how much each tree contributed to the sum.
为了证明这个总和收敛于波动动力学方程，他们需要估计每棵树对总和的贡献有多大。
“These estimates were beyond anyone’s ability,” Shatah said.
“这些估计超出了任何人的能力，”沙塔说。
As Ionescu put it, “it requires a very strong mind to do it.”
正如约内斯库所说，“这需要非常坚强的意志才能做到。”
When you work with him, it’s like you are running after a train.
当你和他一起工作时，就像你在追火车一样。
Deng has the right kind of mind.
邓的想法是正确的。
For one, he’s very good at combinatorics, a skill that his colleagues say sets him apart.
首先，他非常擅长组合数学，他的同事说这是一项让他与众不同的技能。
And “he’s fearless,” Hani said.
“他无所畏惧，”哈尼说。
“I mean, sometimes there are computations that might take me half a day to psych myself up to do them, and then take me the other half of the day in order to do them.
“我的意思是，有时有些计算可能需要我花半天的时间来让自己做好准备，然后又需要我花另一半天的时间来完成它们。
He can do them in an hour or two.”
他可以在一两个小时内完成这些任务。”
The duo showed that some of the trees cancel each other out, contributing nothing to the overall sum.
两人的研究表明，有些树相互抵消，对总和没有任何贡献。
They then found a clever way to cut the remaining trees up into smaller pieces that were easier to analyze.
然后，他们找到了一种巧妙的方法，将剩余的树木切成更容易分析的小块。
It required what Hani described as almost surgical precision.
它需要哈尼所说的近乎外科手术的精确度。
“My mind was exploding,” Deng said.
“我的心爆炸了，”邓说。
By working with these smaller pieces, they proved that, indeed, the Schrödinger equation gives rise to the wave kinetic equation over the right time scales — a result that they later extended to even longer (and more realistic) time scales.
通过研究这些较小的部分，他们证明了薛定谔方程确实在正确的时间尺度上产生了波动动力学方程——这一结果后来被扩展到更长（也更现实）的时间尺度。
That last proof, he and Hani realized, might hold the key to another problem — one that was so ambitious, and that so many other researchers were already trying to solve, that they felt they had to work on it in secret.
他和哈尼意识到，最后一个证明可能是解决另一个问题的关键——这个问题是如此雄心勃勃，许多其他研究人员已经在试图解决这个问题，他们觉得必须秘密地解决这个问题。
At the turn of the 20th century, the mathematician David Hilbert asked a question, his so-called “sixth problem,” about bridging microscopic and macroscopic descriptions of gases.
在 20 世纪初，数学家大卫·希尔伯特提出了一个问题，即他所谓的“第六个问题”，即连接气体的微观和宏观描述。
It was essentially the same problem that Deng and Hani had just solved for waves, but for particles in specific settings.
这本质上与邓和哈尼刚刚解决的波问题相同，但适用于特定环境下的粒子。
These problems only seem the same “if you look at things from afar,” said Isabelle Gallagher , a mathematician at Paris Cité University who had spent years working on Hilbert’s sixth problem.
巴黎西岱大学数学家伊莎贝尔·加拉格尔 (Isabelle Gallagher) 多年来一直在研究希尔伯特的第六个问题，他说，只有“从远处看”，这些问题才看起来是一样的。
“They’re physically very different.”
“他们在身体上非常不同。”
Deng and Hani set their sights on the problem anyway, which had been resolved on shorter time scales only.
无论如何，邓小平和哈尼都把目光投向了这个问题，但这个问题只是在较短的时间内得到了解决。
Joined by Xiao Ma of the University of Michigan, they worked day and night, analyzing new diagrams and figuring out how to cut them up in even more complicated ways.
在密歇根大学的小马的帮助下，他们夜以继日地工作，分析新图表并找出如何以更复杂的方式将它们分割。
They came close to burnout.
他们已经快要精疲力竭了。
Deng’s office, like his home, is austere.
邓小平的办公室和他的家一样，十分简朴。
Deng recalled chugging energy drinks and barely going outside as he worked out the final details.
邓回忆说，他一边喝着能量饮料，一边敲定最后的细节，几乎没有出门。
He was visiting his parents in China at the time.
当时他正在中国探望父母。
They’d be eating or talking, his mother said, when “suddenly a math problem would flash into his mind.
他母亲说，他们正在吃饭或聊天，“突然，他的脑海里闪现出一道数学题。
We’d know at that moment we should stop talking, as he had already entered his own mathematical world.”
那时我们就知道我们应该停止说话，因为他已经进入了自己的数学世界。”
By August 2024, Deng, Hani, and Ma had finished their 200-page paper solving Hilbert’s 124-year-old problem for long time scales.
到 2024 年 8 月，邓、哈尼和马云完成了 200 页的论文，解决了希尔伯特长达 124 年的长期问题。
It took months for the rest of the mathematical community to even start to digest it.
数学界的其他人花了几个月的时间才开始消化它。
“We were so lost and confused,” Gallagher said.
“我们非常迷失和困惑，”加拉格尔说。
Deng flew out to Paris to walk her and her colleagues through the work.
邓飞往巴黎，带领她和她的同事完成工作。
“He really wanted us to understand,” she said.
“他真的希望我们能够理解，”她说。
“He was incredible.
“他太不可思议了。
During the whole week, I don’t think he took out his notes one single time.
整整一周，我想他一次都没有拿出过笔记。
He knew everything, 200 pages of computations, off the top of his head.”
他什么都知道，200页的计算全都在他的脑子里。”
Deng, Hani, and Ma later rewrote their paper to make it more readable.
邓、哈尼和马云后来重写了他们的论文，使其更具可读性。
This revised version is now set to appear in the Annals , too .
这个修订版现在也将出现在年鉴中。
Deng still holds himself to extremely high standards.
邓小平仍然以极高的标准要求自己。
When he found out he’d won the Fields Medal in January 2026, he felt more relieved than excited.
2026 年 1 月，当他得知自己获得菲尔兹奖时，他感到更多的是如释重负，而不是兴奋。
He hadn’t really cared about winning the award, but as rumors started to spread that he was on the short list, “I was a bit stressed,” he said.
他其实并不关心是否能获奖，但随着有关他入围候选名单的谣言开始传播，“我有点压力，”他说。
Now he no longer worries so much about his future.
现在他不再那么担心自己的未来了。
He’s doing work he cares about; he’s widely recognized for it.
他正在做他关心的工作；他因此而受到广泛认可。
And he sees beauty in the questions he’s exploring.
他在他正在探索的问题中看到了美。
That beauty might not be obvious: There’s an ad hoc, brute-force aspect to it.
这种美丽可能并不明显：它有一个特别的、蛮力的方面。
Each time he analyzes a different set of equations, he has to redo all the details, even if the overall approach stays the same.
每次他分析一组不同的方程时，即使总体方法保持不变，他也必须重做所有细节。
“But each time,” Deng said, “we have something new, which is also interesting.”
“但每次，”邓说，“我们都有新的东西，也很有趣。”
There’s also something satisfying about the way combinatorics ended up playing such a massive role in his work.
组合数学最终在他的工作中发挥如此重要的作用，这也令人感到满意。
It echoes his childhood — spent on Go, one of the world’s most combinatorially rich games, and mathematical Olympiads, which tend to be highly focused on combinatorics.
这与他的童年相呼应——围棋是世界上组合数学最丰富的游戏之一，而数学奥林匹克竞赛往往高度关注组合数学。
It echoes what first drew him to poetry, too: the seemingly endless proliferation of meanings made possible by a simple combination of words.
这也呼应了他最初对诗歌的兴趣：通过简单的单词组合，看似无穷无尽的意义扩散成为可能。
His research has in that way become familiar, a place where he can feel at home.
他的研究就这样变得熟悉起来，让他有宾至如归的感觉。
His life now looks much the same as it has for the past decade.
现在他的生活看起来和过去十年没什么两样。
But he’s comfortable.
但他很舒服。
According to Nahmod, although Deng is still quiet, he’s no longer painfully shy.
纳莫德表示，尽管邓仍然很安静，但他不再害羞了。
He reads and writes poetry.
他读诗、写诗。
He plays Go from time to time.
他时不时下围棋。
He’s thinking of getting a pet snake because “it looks good.
他正在考虑养一条宠物蛇，因为“它看起来不错。
It’s beautiful,” he said.
很漂亮，”他说。
He reads manga, but purely for enjoyment, not because he’s trying to be less hard on himself.
他看漫画，但纯粹是为了享受，并不是因为他想对自己放松一点。
“I have less pressure,” he said.
“我的压力较小，”他说。
Which means that “you can just think about math, as you want.”
这意味着“你可以随心所欲地思考数学。”
“We’ll see how much I can get to.”
“我们会看看我能做到什么程度。”
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