The Quietest Mathematician Has Always Been Worth Listening To
John Pardon is known to his colleagues and students as quiet but insightful.
约翰·帕登（John Pardon）被他的同事和学生认为是一个安静但富有洞察力的人。
As one former student put it, “You really have to go fishing for his knowledge.”
正如一位以前的学生所说，“你真的必须去挖掘他的知识。”
S everal eminent mathematicians tell different versions of the same story about John Pardon.
几位杰出的数学家讲述了关于约翰·帕登的同一个故事的不同版本。
Fifteen years ago, János Kollár was seated next to Pardon at a Phi Beta Kappa dinner.
十五年前，在 Phi Beta Kappa 晚宴上，János Kollár 坐在 Pardon 旁边。
Kollár held an endowed chair as a full professor at Princeton University, where he’d been teaching for over a decade after stints at the University of Utah and at Harvard University as a member of its Society of Fellows, arguably the country’s most prestigious postdoctoral fellowship.
科拉尔在普林斯顿大学担任正教授，担任教授，在犹他大学和哈佛大学担任研究员协会成员（可以说是美国最负盛名的博士后奖学金）成员之后，他已在该大学任教十多年。
Which is to say, Kollár is not bad at math.
也就是说，Kollár 的数学并不差。
Pardon was an undergraduate, a few months into his senior year.
帕登当时是一名本科生，刚进入大四几个月。
Kollár had gotten stuck on a question in the field of topology.
科拉尔被拓扑学领域的一个问题困住了。
At the dinner, he mentioned it to Pardon in passing.
晚宴上，他顺便向帕登提到了这件事。
Two weeks later, he got an email from Pardon — with the solution.
两周后，他收到了 Pardon 发来的一封电子邮件，其中包含解决方案。
“And it’s not just that he solved the question I needed, but he did a much more general case in a very nice way,” Kollár recalled.
“他不仅解决了我需要的问题，而且以非常好的方式解决了一个更普遍的案例，”科拉尔回忆道。
He found working with the 21-year-old a pleasure.
他发现和这位 21 岁的年轻人一起工作很愉快。
It felt, he said, “like working with a postdoc who had very good ideas.”
他说，感觉“就像和一个有很好想法的博士后一起工作。”
When Pardon graduated the following May, it was at the top of the Princeton class.
第二年五月，帕登毕业时，他在普林斯顿大学的班级中名列前茅。
In his valedictory address, he asked his fellow graduates, rhetorically, “When is the last time you had a truly original idea?” He proceeded to answer his own question: “The single most basic form of expression that humans draw upon is imitation of others, and so I think having an original idea may qualify you as being partially insane.”
在告别演说中，他反问道，“你们最后一次有真正原创的想法是什么时候？”他接着回答了自己的问题：“人类最基本的表达形式就是模仿他人，所以我认为拥有一个原创的想法可能会让你有部分疯狂的资格。”
By young Pardon’s definition, present-day Pardon, who turned 37 in June, is more than a little crazy, in that he has had not one but many original ideas.
按照年轻的帕登的定义，今年 6 月就年满 37 岁的如今的帕登有点疯狂，因为他的原创想法不止一个，而是很多。
For these, the International Mathematical Union has now awarded him the Fields Medal, the most prestigious prize in math.
为此，国际数学联盟现已授予他菲尔兹奖，这是数学界最负盛名的奖项。
“He started off solving well-known spectacular problems,” said Tobias Ekholm of Uppsala University.
乌普萨拉大学的托比亚斯·埃克霍尔姆 (Tobias Ekholm) 说：“他一开始就解决了众所周知的重大问题。”
“At a slightly older age, he’s doing this same spectacular work, but he has this ability: He solves a famous problem but does it in a way that creates a framework, a whole new package, that will be useful to other people.”
“在年龄稍大的时候，他正在做同样令人惊叹的工作，但他有这种能力：他解决了一个著名的问题，但以创建一个框架、一个全新的包的方式完成，这对其他人有用。”
He is silent, but he says very useful things.
他沉默寡言，但说了一些非常有用的话。
Pardon, now at Stony Brook University in New York, is not particularly prolific.
帕登现在就职于纽约石溪大学，他的作品并不多产。
“His publication list is not very long,” said Kai Cieliebak of the University of Augsburg.
“他的出版物清单不是很长，”奥格斯堡大学的 Kai Cieliebak 说。
“But every paper he wrote is some kind of breakthrough paper.
“但他写的每一篇论文都是某种突破性的论文。
All of them appeared in very top high-level journals.” Furthermore, Cieliebak said, Pardon “has done work in really a whole number of, to my understanding, completely separate areas of math.
它们都出现在非常顶级的高水平期刊上。”此外，Cieliebak 说，Pardon“据我了解，在许多完全独立的数学领域进行了工作。
Probably somewhere in his brain these things might be connected.” As his former student Mohan Swaminathan , now at the Tata Institute of Fundamental Research, put it, “If John has a problem, he goes really deep and learns whatever is necessary for it.”
也许在他大脑的某个地方，这些东西可能是有联系的。”正如他以前的学生、现供职于塔塔基础研究所的莫汉·斯瓦米纳坦 (Mohan Swaminathan) 所说，“如果约翰遇到问题，他会深入研究并学习解决该问题所需的一切。”
Pardon is tall and quiet.
帕登身材高大，安静。
He has a gentle, unassuming affect and a reputation for kindness.
他性格温和、谦逊，以善良而闻名。
He’s likely to be the best mathematician in any room he walks into, but he wears this lightly.
他很可能是他走进的任何房间里最好的数学家，但他对此的态度很轻松。
He’s helpful to colleagues, sporadically answering graduate students’ questions on the online bulletin board MathOverflow.
他对同事很有帮助，偶尔在在线公告牌 MathOverflow 上回答研究生的问题。
He sometimes weighs in on other subjects as well, with an invariable polite decisiveness.
有时他也会以一贯的礼貌果断的态度参与其他话题的讨论。
When one user, exasperated by their inability to make restaurant-quality pancakes, asked for help, Pardon offered advice: “The difference between ‘fluffy and fall-apart crumbly’ and ‘thinner, chewy, and sort of dense,’” he wrote, “is precisely governed by baking powder/soda.
当一位用户因无法制作餐厅品质的煎饼而恼火并寻求帮助时，帕登提出了建议：“‘蓬松易碎’和‘稀薄、有嚼劲、有点稠密’之间的区别，”他写道，“正是由泡打粉/苏打水决定的。
…
……
If you want thin and chewy, omit the baking powder.” Pardon is as likely to ask for help as he is to offer it — airline ticketing, Schengen visas, and Wi-Fi passwords are as frustrating and confusing to him as they are to anybody.
如果你想要又薄又耐嚼，就不要加泡打粉。”帕登寻求帮助的可能性与他提供帮助的可能性一样大——机票、申根签证和 Wi-Fi 密码对他来说和对任何人一样令人沮丧和困惑。
He is a father of two sons.
他是两个儿子的父亲。
He plays the cello well and learned to speak Chinese fluently in college, eventually winning a Chinese-language debate tournament held in Singapore.
他拉得很好，在大学里学会了说一口流利的中文，并最终赢得了在新加坡举行的华语辩论赛的冠军。
Pardon declined to speak on the record for this profile.
帕登拒绝就这篇个人资料公开发表言论。
Many people who know him remark on his reluctance to say just about anything, least of all about himself.
许多认识他的人都表示他不愿透露任何事情，尤其是关于他自己的事情。
“He will only say something when he feels it is definitely correct,” said Shaoyun Bai , a mathematician at the Massachusetts Institute of Technology who got his doctorate under Pardon.
“只有当他认为某件事绝对正确时，他才会说一些话，”在帕登的指导下获得博士学位的麻省理工学院数学家白少云说。
Pardon, though not particularly prolific, has proved major theorems in many different areas of math.
帕登虽然不是特别多产，但已经证明了许多不同数学领域的主要定理。
“He refrains from saying too much,” said Thomas Massoni of Stanford University, another former student.
“他没有说太多，”另一位前学生、斯坦福大学的托马斯·马索尼 (Thomas Massoni) 说。
“He knows so much that you really have to go fishing for his knowledge.” Multiple students recounted meetings with him where they had to do almost all the talking — but he was always available, glad to meet with them, and a patient listener.
“他知道得太多了，你真的必须去探寻他的知识。”多名学生回忆起与他的会面，几乎所有的谈话都是由他们进行的——但他总是有空，很高兴与他们见面，并且是一位耐心的倾听者。
As Barış Kartal , who worked with Pardon while doing a postdoctoral fellowship, said, “He is silent, but he says very useful things.”
正如在做博士后研究期间与帕登一起工作的巴里什·卡尔塔尔（Barış Kartal）所说，“他很沉默，但他说的都是非常有用的事情。”
“I think John knows who he is and probably doesn’t want to come across as a cartoon of ‘Here’s a boy genius who’s done such-and-such a thing,’” said David Gabai , a former chairman of the Princeton math department.
普林斯顿大学数学系前系主任戴维·加拜 (David Gabai) 表示：“我认为约翰知道自己是谁，他可能不想给人留下‘这是一个做了某事的天才男孩’的漫画印象。”
This is a real, reasonable concern.
这是一个真实、合理的担忧。
Nonetheless, when mathematicians speak about him, it’s hard to avoid the conclusion that he is a once-in-a-generation mathematical talent.
尽管如此，当数学家谈论起他时，难免得出这样的结论：他是千载难逢的数学天才。
Like a generational talent in sports — Ohtani, Jordan, Messi — Pardon appears to be playing a different game than his colleagues, all of whom are themselves exceptionally skilled.
就像大谷、乔丹、梅西等一代代体育天才一样，帕登似乎与他的同事们玩着不同的游戏，而他的同事们本身都拥有非凡的技术。
But his achievements aren’t as easy to appreciate as a star athlete’s.
但他的成就并不像明星运动员那样容易被欣赏。
Without years of study, it’s tough to understand just how surprising Pardon’s proof of a conjecture about six-dimensional manifolds really is.
如果没有多年的研究，很难理解帕登对六维流形猜想的证明到底有多么令人惊讶。
An earlier Fields Medal had been awarded in part just for making the conjecture.
早期菲尔兹奖的颁发部分就是因为提出了这一猜想。
So, with every intention of avoiding caricature, here’s a glimpse of Pardon’s work in knot theory, topology, and symplectic geometry.
因此，出于避免讽刺的目的，这里是帕登在纽结理论、拓扑和辛几何方面的工作的一瞥。
Pardon was already the star of Princeton’s math department when he sat next to Kollár at that dinner during his senior year.
当帕登在大四那次晚宴上坐在科拉尔旁边时，他已经是普林斯顿大学数学系的明星了。
He’d grown up in North Carolina, and by the time he was in high school, he was taking math courses at Duke University, where his father was a math professor.
他在北卡罗来纳州长大，上高中时，他在杜克大学学习数学课程，他的父亲是该大学的数学教授。
Once in college, he set out to solve an open problem in knot theory that he’d first encountered in high school.
进入大学后，他开始解决他在高中时第一次遇到的纽结理论中的一个开放问题。
As he later told his Stony Brook colleague Simon Donaldson , now at Imperial College London, he had more or less given up on it by the end of his junior year.
正如他后来告诉他在石溪分校的同事西蒙·唐纳森（现就职于伦敦帝国理工学院）的那样，到大三结束时，他或多或少已经放弃了这个想法。
But a line of attack occurred to him while he was walking in an English park that summer.
但那年夏天，当他在英国的一个公园散步时，他突然遭遇了一系列袭击。
A few months later, he had his proof.
几个月后，他得到了证据。
It’s a paper you can sit down and read.
这是一篇你可以坐下来阅读的论文。
It has this genius touch to it.
它有这种天才的感觉。
Knot theory is one of those areas of math that are just what they sound like, but also somehow far deeper and more complicated than they seem.
纽结理论是数学领域之一，正如它们听起来的那样，但在某种程度上比它们看起来更深入、更复杂。
It is the study of knots — literally, given a length of rope, how can you tie it?
这是对结的研究——从字面上看，给定一段绳子，你如何打结它？
(Mathematicians often splice the ends of the rope together.) But this question generalizes to higher dimensions and differently structured spaces, and in the end, the structure of knots is a powerful tool for understanding the geometry and topology of the spaces in which they can be embedded.
（数学家经常将绳子的末端拼接在一起。）但是这个问题可以推广到更高的维度和不同结构的空间，最终，结的结构是理解它们可以嵌入的空间的几何和拓扑的强大工具。
In 1983, the prominent mathematician Mikhael Gromov made a conjecture about a property of knots called the distortion.
1983 年，著名数学家米哈埃尔·格罗莫夫 (Mikhael Gromov) 对纽结的一种称为扭曲的性质做出了猜想。
Consider two points on a knot.
考虑结上的两个点。
You can measure the distance between them in two different ways.
您可以通过两种不同的方式测量它们之间的距离。
Either you can travel along the rope itself, as though you were a tiny ant, or you can take a straight-line shortcut across space, as a (small) crow flies.
你要么可以沿着绳子本身旅行，就像你是一只小蚂蚁，要么你可以像一只（小）乌鸦一样在空间中采取直线捷径。
It’s easy to intuit that the length you travel along the rope will always be at least as long as the straight-line distance.
很容易凭直觉看出，您沿着绳子行进的长度将始终至少与直线距离一样长。
For any given knot, there will be a pair of points for which the ratio between the distance along the rope and the straight-line distance is the biggest.
对于任何给定的结，都会有一对点，它们沿绳子的距离与直线距离之间的比率最大。
That ratio is the knot’s distortion.
该比率就是结的变形。
Knots have the advantage of being flexible: So long as you don’t tear the rope, you can move it around to minimize the distortion.
结的优点是灵活：只要不撕裂绳子，就可以移动它以最大程度地减少扭曲。
Gromov asked a question about “torus knots,” which can be drawn on the surface of a doughnut without crossing themselves.
格罗莫夫提出了一个关于“环面结”的问题，“环面结”可以画在甜甜圈的表面上，而不会交叉。
You can think of the doughnut as a sort of mold that helps make the structure of the knots clear, as seen below:
您可以将甜甜圈视为一种模具，有助于使结的结构变得清晰，如下所示：
Gromov wanted to know if, as you wrap the rope more and more times around the doughnut to make your knot, there might be an upper bound to the distortion.
格罗莫夫想知道，当你将绳子越来越多地缠绕在甜甜圈上以打结时，扭曲是否可能存在上限。
He asked if any torus knot might be rearranged so as to make the ratio smaller than a fixed upper bound: 100.
他询问是否可以重新排列任何环面结，以使比率小于固定上限：100。
One day in 2010, Gabai remembers Pardon — still an undergraduate — walking into his office and saying, “Well, I’ve proven such-and-such a theorem.” He’d shown that the answer to Gromov’s question was no: There are knots that have an arbitrarily large distortion, no matter how cleverly you try to rearrange them.
2010 年的一天，加拜记得帕登——还是一名本科生——走进他的办公室说：“好吧，我已经证明了这样那样的定理。”他证明了格罗莫夫问题的答案是否定的：有些结会产生任意大的扭曲，无论你如何巧妙地重新排列它们。
Since the distortion is one measure of how tangled a knot is, he essentially proved that it’s possible to create knots of this particular type, once thought to be relatively tame, that are arbitrarily tangled.
由于扭曲度是衡量结缠结程度的一种指标，因此他基本上证明了有可能创建这种特定类型的结，曾经被认为相对温和的结，实际上是任意缠结的。
Pardon’s proof was later published in the Annals of Mathematics , the field’s top journal — a rare achievement for an undergraduate.
帕登的证明后来发表在该领域顶级期刊《数学年鉴》上——这对于本科生来说是一项罕见的成就。
Cieliebak speaks of the paper in a wistful tone.
西利巴克以一种渴望的语气谈到这篇论文。
“It’s a paper you can sit down and read,” he said.
“这是一篇你可以坐下来阅读的论文，”他说。
“It has this genius touch to it.”
“它有这种天才的感觉。”
If the knot distortion paper was the valedictory to Pardon’s collegiate career, he was just getting started.
如果说结扭曲论文是帕登大学生涯的告别词，那么他才刚刚开始。
He arrived at Stanford as one of several standouts in a strong class of graduate students, according to Yakov Eliashberg , one of the most prominent geometers of his generation.
据当代最杰出的几何学家之一雅科夫·埃利亚什伯格 (Yakov Eliashberg) 称，他来到斯坦福大学时，是研究生班中的几位佼佼者之一。
There, Pardon could often be seen on the steps in front of the math department building, lost in thought.
在那里，经常可以看到帕登站在数学系大楼前的台阶上，陷入沉思。
“He’d sit there for the longest time,” said the Stanford mathematician Ralph Cohen .
“他在那里坐的时间最长，”斯坦福大学数学家拉尔夫·科恩说。
“You’d go by and say hello.
“你会过去打招呼。
Sometimes he’d recognize you and nod hello.
有时他会认出你并点头打招呼。
Sometimes he’d be deep in thought and wouldn’t know you’re there.”
有时他会陷入沉思，不知道你在那里。”
At first, Pardon worked on some problems in low-dimensional topology.
起初，Pardon 研究了低维拓扑中的一些问题。
Then, after asking Eliashberg to be his adviser, he began to work in symplectic geometry.
然后，在请埃利什伯格担任他的顾问后，他开始研究辛几何。
He solves a famous problem but does it in a way that creates a framework, a whole new package, that will be useful to other people.
他解决了一个著名的问题，但以创建一个框架、一个全新的包的方式完成，这对其他人有用。
In the web of mathematical disciplines, symplectic geometry sits somewhere between topology and geometry.
在数学学科网络中，辛几何位于拓扑和几何之间。
Historically, it originated in the study of “phase space” — an abstract, high-dimensional space in which each possible state of a physical system (for instance, the positions and momenta of atoms bouncing around in a gas) gets represented as a single point.
从历史上看，它起源于对“相空间”的研究——一个抽象的高维空间，其中物理系统的每个可能状态（例如，原子在气体中弹跳的位置和动量）都被表示为一个点。
Position and momentum are related in a particular mathematical way: If you change one, you change the other.
位置和动量以一种特定的数学方式相关：如果你改变一个，你就会改变另一个。
Symplectic geometry begins with this relationship, which has geometric consequences, and abstracts it to other, mathematically similar objects.
辛几何始于这种具有几何后果的关系，并将其抽象为其他数学上相似的对象。
These objects have more flexibility than traditional geometric objects, such as a rectangle or a sphere, but have more rigidity than topological objects, which can be stretched and compressed at will.
这些对象比传统的几何对象（例如矩形或球体）具有更大的灵活性，但比拓扑对象具有更大的刚性，可以随意拉伸和压缩。
Pardon’s doctoral dissertation on the “virtual cycle problem” made him instantly famous among symplectic geometers, said Leonid Polterovich of Tel Aviv University, who spoke of Pardon with amazement.
特拉维夫大学的列昂尼德·波尔特罗维奇 (Leonid Polterovich) 惊讶地谈到帕登，他说，帕登关于“虚拟循环问题”的博士论文使他立即在辛几何学家中出名。
Many of the top people in the field, he remembers, had been trying to solve the problem for years.
他记得，该领域的许多顶尖人士多年来一直在努力解决这个问题。
“Some young guy comes and just does this,” Polterovich said.
“一些年轻人来了，就这么做了，”波尔特罗维奇说。
“He’s very precise, obviously has a fantastic skill of developing language.
“他非常精确，显然拥有出色的语言发展能力。
It’s really special.
这真的很特别。
It’s not just tricks.”
这不仅仅是技巧。”
To understand the virtual cycle problem, it helps to consider a concept called transversality.
要理解虚拟循环问题，需要考虑一个称为横向性的概念。
Two curves intersect in a transverse way if they cross like so:
如果两条曲线像这样交叉，则它们以横向方式相交：
If, on the other hand, they just barely “kiss” each other — intersecting at a tangent — then their intersection is not transverse.
另一方面，如果它们只是勉强“亲吻”彼此——以切线相交——那么它们的相交就不是横向的。
Transverse and non-transverse intersections also arise with surfaces, which are, in the end, higher-dimensional analogues of curves.
横向和非横向相交也出现在曲面上，它们最终是曲线的高维类似物。
In symplectic geometry, the basic object of study is called a symplectic manifold.
在辛几何中，研究的基本对象称为辛流形。
One central way in which mathematicians in the field try to understand a symplectic manifold is by studying mappings that send points on other, well-studied surfaces to points on that manifold.
该领域的数学家试图理解辛流形的一种主要方法是研究将其他经过充分研究的曲面上的点发送到该流形上的点的映射。
The mappings that symplectic geometers are interested in have to satisfy certain mathematical conditions that make it possible to count them.
辛几何学家感兴趣的映射必须满足某些数学条件，从而可以对它们进行计数。
“Rather rapidly it was understood that this counting doesn’t work very well,” Polterovich said, because the presence of non-transverse intersections in the relevant calculations can lead to incorrect counts.
波尔特罗维奇说：“很快人们就意识到这种计数效果不太好，因为相关计算中非横向交叉点的存在可能会导致错误的计数。”
In his dissertation, Pardon “succeeded in developing a novel language …
在他的论文中，帕登“成功地开发了一种新颖的语言......
which led to the resolution of this problem,” Polterovich said.
这导致了这个问题的解决，”波尔特罗维奇说。
By coming up with a new way to think about so-called virtual cycles — a technique for circumventing the issues caused by non-transverse intersections — he was able to count rigorously.
通过提出一种新的方式来思考所谓的虚拟循环（一种规避非横向交叉点引起的问题的技术），他能够进行严格的计数。
As Polterovich put it, “It was clean.
正如波尔特罗维奇所说，“它很干净。
It was powerful.”
它很强大。”
The result came at a time when symplectic geometry was in something of a slow-burn crisis , rife with disputes over whether key results had been established rigorously.
这一结果出现之际，辛几何正陷入一场缓慢燃烧的危机，关于关键结果是否严格建立的争论充斥着。
“We really didn’t properly prove everything,” Eliashberg said.
“我们确实没有正确证明一切，”埃利什伯格说。
Pardon’s approach helped get around some of these foundational obstacles.
帕登的方法有助于克服其中一些基本障碍。
“With some magic things he invented,” Polterovich said, “Pardon succeeded in gluing all this information together.” His wasn’t the first, or only, technique for dealing with transversality, but it was more widely applicable than any of the previous approaches.
“帕登发明了一些神奇的东西，”波尔特罗维奇说，“帕登成功地将所有这些信息粘合在一起。”他的方法不是第一个，也不是唯一的处理横向性的技术，但它比以前的任何方法都更广泛适用。
Now, Eliashberg said, “a kind of foundation for certain parts of symplectic field theory are built on his machinery.”
现在，埃利什伯格说，“辛场论某些部分的基础是建立在他的机器上的。”
As Eliashberg’s Stanford colleague Mohammed Abouzaid said, Pardon’s thesis “made it possible for many people to stop worrying about things.
正如埃利什伯格在斯坦福大学的同事穆罕默德·阿布扎伊德所说，帕登的论文“让许多人不再担心事情”。
…
……
It was immediately impactful.”
它立即产生了影响。”
He’s very precise, obviously has a fantastic skill of developing language.
他非常精确，显然具有出色的语言发展能力。
It’s really special.
这真的很特别。
It’s not just tricks.
这不仅仅是技巧。
As a mark of just how impactful it was, even before Pardon officially received his doctorate, Princeton began recruiting him to return as a full professor — unheard of for someone fresh out of graduate school.
作为其影响力的一个标志，甚至在帕登正式获得博士学位之前，普林斯顿大学就开始招募他作为正教授回来——这对于刚从研究生院毕业的人来说是闻所未闻的。
It required some explaining to the university’s dean, Kollár remembered.
科拉尔记得，这需要向大学院长做出一些解释。
Shortly after finishing his doctorate, Pardon teamed up with Sheel Ganatra and Vivek Shende to prove a series of other landmark results in symplectic geometry.
完成博士学位后不久，Pardon 与 Sheel Ganatra 和 Vivek Shende 合作，证明了辛几何领域的一系列其他里程碑式的结果。
“Working with John has been one of the great professional highlights of my career,” Ganatra said.
“与约翰一起工作是我职业生涯中最伟大的职业亮点之一，”加纳特拉说。
“It has been wonderful for me.” He remembers visiting Pardon at Princeton and spending an afternoon eating much of a bag of “what must have been 100 peaches” while working through details of what would become their second paper together.
“这对我来说太棒了。”他记得在普林斯顿拜访 Pardon 时，花了一个下午吃了一袋“大概有 100 个桃子”，同时研究他们将要写的第二篇论文的细节。
On another occasion, they spent a week in Beijing, eating durian and doing math.
还有一次，他们在北京呆了一周，一边吃榴莲，一边做数学。
“He loves fresh fruit,” Ganatra said.
“他喜欢新鲜水果，”加纳特拉说。
“He really loves fresh fruit.”
“他真的很喜欢新鲜水果。”
“He can be quiet at first, but his mind is always thinking,” Ganatra added.
“他一开始可能很安静，但他的头脑总是在思考，”加纳特拉补充道。
“We’ll have moments when we are talking, and all of the sudden we’re leaping forward.”
“我们有时会在谈话时突然向前迈进。”
In 2016, Pardon left Stanford for Princeton, where he spent six years as a professor before moving to the Simons Center for Geometry and Physics at Stony Brook University.
2016 年，帕登离开斯坦福大学前往普林斯顿大学，在那里担任了六年教授，然后转到石溪大学西蒙斯几何和物理中心。
There, he solved his biggest problem yet, proving a 20-year-old conjecture about counting curves.
在那里，他解决了迄今为止最大的问题，证明了 20 年前关于计算曲线的猜想。
The unifying thread to Pardon’s work is not a particular mathematical area or theme; he simply solves problems that are, to him, interesting.
帕登作品的统一主线并不是某个特定的数学领域或主题；而是一个数学领域。他只是解决对他来说有趣的问题。
As in symplectic geometry, curve counting gives mathematicians deep insight into the spaces where those curves live.
与辛几何一样，曲线计数使数学家能够深入了解这些曲线所在的空间。
On a flat piece of paper, if you choose two points, you can only draw one line (a kind of curve) between them.
在一张平坦的纸上，如果选择两个点，则只能在它们之间绘制一条线（一种曲线）。
This seems obvious enough, but it says something fundamental about the structure of 2D space.
这似乎很明显，但它说明了二维空间结构的一些基本内容。
Though you can draw infinitely many lines on a 2D plane or in 3D space, there’s a classic result in geometry that a shape called a quintic threefold contains exactly 2,875 lines: The geometry of the shape constrains the number of lines that are possible.
尽管您可以在 2D 平面或 3D 空间中绘制无限多条线，但几何学中有一个经典结果，即称为五次三重的形状恰好包含 2,875 条线：形状的几何形状限制了可能的线数。
Pardon is drawn to particular problems, rather than broader fields or unifying themes.
帕多恩被特定的问题所吸引，而不是更广泛的领域或统一的主题。
Quintic threefolds are an example of a type of six-dimensional shape called a Calabi-Yau manifold , which is not only mathematically intriguing but also useful in theoretical physics.
五次三重是一种称为卡拉比-丘流形的六维形状的一个例子，它不仅在数学上很有趣，而且在理论物理学中也很有用。
Geometers are particularly interested in counting curves on these shapes as a way to understand them.
几何学家对计算这些形状上的曲线特别感兴趣，以此作为理解它们的一种方式。
Many versions of string theory predict that six extra dimensions are “curled up” into microscopic sizes, in the shape of Calabi-Yau manifolds.
许多版本的弦理论都预测六个额外维度会“卷曲”成微观尺寸，呈卡拉比-丘流形的形状。
If this is true, the shapes form an inextricable, if invisible, part of our world.
如果这是真的，那么这些形状就构成了我们世界不可分割的、看不见的一部分。
(In this formulation, gravity-like forces acting in the hidden dimensions are what we see as electricity, magnetism, and the strong and weak nuclear forces.)
（在这个公式中，作用在隐藏维度中的类似重力的力就是我们所看到的电、磁以及强核力和弱核力。）
Calabi-Yau manifolds are still not fully understood.
Calabi-Yau 流形仍未被完全理解。
In 2003, three mathematicians and a physicist formulated the MNOP conjecture, which states that two different ways of counting curves on Calabi-Yau manifolds are equivalent in a deep and unexpected way.
2003 年，三位数学家和一位物理学家提出了 MNOP 猜想，该猜想指出，在 Calabi-Yau 流形上计算曲线的两种不同方式在深刻且意想不到的方面是等效的。
For a given Calabi-Yau manifold, mathematicians are interested in counting not just one type of curve (such as lines) but many of them, all at once.
对于给定的卡拉比-丘流形，数学家不仅感兴趣的是计算一种类型的曲线（例如直线），而且感兴趣的是同时计算许多类型的曲线。
For example, a given curve has a property called a genus, a number that provides information about its structure.
例如，一条给定的曲线有一个称为“属”的属性，这是一个提供有关其结构信息的数字。
You might want to know how many curves you can draw on your Calabi-Yau manifold that are of a particular class and have a genus of zero — and how many curves from that class have genus 1, genus 2, genus 3, and so on.
您可能想知道可以在 Calabi-Yau 流形上绘制多少条特定类别且亏格为零的曲线，以及该类别中有多少条曲线具有亏格 1、亏格 2、亏格 3 等。
You usually can’t count the number of curves with each genus directly, for a number of technical reasons.
由于多种技术原因，您通常无法直接计算每个属的曲线数量。
In order to get meaningful counts, mathematicians instead have to deploy an arsenal of tricks — resulting in an infinite sequence of numbers, which mathematicians call the “virtual” number of curves.
为了获得有意义的计数，数学家必须部署一系列技巧，从而产生无限的数字序列，数学家将其称为“虚拟”曲线数。
Working with John has been one of the great professional highlights of my career.
与约翰一起工作是我职业生涯中最伟大的职业亮点之一。
It has been wonderful for me.
这对我来说太棒了。
“The word ‘virtual’ is doing a lot of work,” notes Jim Bryan , a mathematician at the University of British Columbia who has been studying the MNOP conjecture for the past several decades.
“‘虚拟’这个词做了很多工作，”过去几十年来一直在研究 MNOP 猜想的不列颠哥伦比亚大学数学家吉姆·布莱恩 (Jim Bryan) 指出。
“These numbers are far from the literal count.” The numbers, which help count curves of ever-increasing genus, are called Gromov-Witten invariants.
“这些数字与字面数字相去甚远。”这些数字有助于计算不断增加的属的曲线，被称为 Gromov-Witten 不变量。
(They are named after the same Gromov whose conjecture about knots Pardon disproved, and the physicist Ed Witten.)
（它们是以格罗莫夫和物理学家埃德·维滕的名字命名的，格罗莫夫关于结的猜想被帕登反驳了。）
These invariants emerged as an object of study in the late 1980s and early 1990s.
这些不变量在 20 世纪 80 年代末和 90 年代初成为研究对象。
While working on his dissertation with Simon Donaldson in the late 1990s, Richard Thomas , now of Imperial College London, described another infinite sequence of numbers associated with Calabi-Yau manifolds.
20 世纪 90 年代末，在与西蒙·唐纳森 (Simon Donaldson) 合作撰写论文时，现任伦敦帝国理工学院的理查德·托马斯 (Richard Thomas) 描述了另一个与 Calabi-Yau 流形相关的无限数序列。
These numbers, called Donaldson-Thomas invariants, help count curves organized not by genus but by a different property.
这些数字称为唐纳森-托马斯不变量，有助于计算不是按属而是按不同属性组织的曲线。
These two ways of counting curves are mathematically very different.
这两种计算曲线的方法在数学上是非常不同的。
“These numbers look like they have nothing to do with each other,” Bryan said.
“这些数字看起来彼此无关，”布莱恩说。
If you take a particular Calabi-Yau manifold and a class of curves you’re interested in, the Gromov-Witten and Donaldson-Thomas invariants will give you two completely different sequences of numbers.
如果您采用特定的 Calabi-Yau 流形和一类您感兴趣的曲线，Gromov-Witten 和 Donaldson-Thomas 不变量将为您提供两个完全不同的数字序列。
But the MNOP conjecture posits that they both correspond to properties of the same underlying function, called a partition function.
但 MNOP 猜想认为它们都对应于同一个底层函数（称为配分函数）的属性。
“MNOP says these two sequences are the coefficients of the same function expanded in two different ways,” Bryan said.
“MNOP 表示，这两个序列是以两种不同方式扩展的同一函数的系数，”布莱恩说。
“The conjecture is pretty strange.”
“这个猜想很奇怪。”
If mathematicians could prove it was true, it would imply that both sets of invariants were indeed counting the same thing, an extremely useful equivalence.
如果数学家能够证明这是真的，那就意味着两组不变量确实在计算同一件事，这是一个非常有用的等价性。
Depending on the context, mathematicians might want to use one set of invariants over the other; the MNOP conjecture, if true, would let them move between the two.
根据具体情况，数学家可能希望使用一组不变量而不是另一组； MNOP 猜想如果成立，将使他们在两者之间移动。
“That is why people care about MNOP — it effectively doubles the number of tools one has to study the partition function,” Bryan said.
“这就是人们关心 MNOP 的原因——它有效地使研究配分函数的工具数量增加了一倍，”Bryan 说。
But for two decades, the conjecture remained open, despite many efforts to prove it.
但二十年来，尽管人们做出了许多努力来证明这一猜想，但它仍然存在。
Then, in the summer of 2023, Pardon shared a proof online.
然后，2023 年夏天，Pardon 在网上分享了一个证明。
“It came out of nowhere,” said Bryan, who has spent a good chunk of the past few years studying the proof.
“它不知从何而来，”布莱恩说，他在过去几年中花了很大一部分时间研究这个证明。
He would have dismissed it entirely if not for Pardon’s reputation.
如果不是考虑到帕登的名声，他可能会完全驳回它。
To show that the two different ways of counting were equivalent, Pardon used tools from a completely different area of math and created a whole new mathematical structure, now called a Pardon algebra.
为了证明这两种不同的计数方式是等效的，帕登使用了完全不同的数学领域的工具，并创建了一个全新的数学结构，现在称为帕登代数。
“The method of the proof is so different than what we’ve seen before that it really rewires the way I think about the entire subject,” Bryan said.
“证明的方法与我们之前看到的非常不同，它确实改变了我对整个主题的思考方式，”布莱恩说。
“This is the biggest result in enumerative algebraic geometry for the last 20 years.”
“这是过去20年来枚举代数几何领域的最大成果。”
Pardon is now writing a book about the foundations of symplectic geometry.
帕登现在正在写一本关于辛几何基础的书。
He’s made his work in progress available online, since many have asked him to do so.
由于许多人要求他这样做，他已将正在进行的工作发布在网上。
The book’s preface captures the gap between how Pardon discusses his own work and how others think of what he does.
这本书的序言抓住了帕登如何讨论他自己的工作和其他人如何看待他所做的事情之间的差距。
“The interesting material is spread a bit thin,” Pardon writes.
“有趣的材料传播得有点薄弱，”帕登写道。
He hopes “to formulate statements and proofs which are as simple and down-to-earth as possible.” He allows that he may have failed in this aim.
他希望“提出尽可能简单、务实的陈述和证明”。他承认他可能未能实现这一目标。
The book reworks the foundations of symplectic geometry using ideas from a branch of math called higher category theory.
这本书利用数学分支（称为高范畴论）的思想重新构建了辛几何的基础。
“Many mathematicians view it as a little esoteric,” said Hiro Lee Tanaka , who studies higher category theory at Texas State University.
“许多数学家认为它有点深奥，”德克萨斯州立大学研究高范畴论的 Hiro Lee Tanaka 说。
“John’s speaking my language, which is not the language he spoke 10 years ago.”
“约翰说的是我的语言，这不是他十年前说的语言。”
“John is not afraid of big machinery, of abstract nonsense,” Abouzaid said.
“约翰并不害怕大型机械和抽象的废话，”阿布扎伊德说。
“He isn’t afraid of it; he just uses it.” But, Abouzaid added, Pardon’s technical facility is not what’s most impressive.
“他并不害怕它；他只是利用它。”但是，阿布扎伊德补充说，帕登的技术设施并不是最令人印象深刻的。
“That’s not the point — the point is that there is geometric insight.” It’s like one musician praising another not only for their impeccable technique but for the feeling it allows them to bring to a performance.
“这不是重点——重点是有几何洞察力。”这就像一位音乐家赞扬另一位音乐家不仅因为他们无可挑剔的技术，而且因为他们为表演带来的感觉。
“Most of the real work has been in finding the ‘right’ formalism, after which the proofs fall into place with little resistance,” Pardon continues in the preface.
“大部分真正的工作都是寻找‘正确的’形式主义，之后证明就几乎没有阻力，”帕登在序言中继续说道。
“This work is largely hidden from the view of the consumer, and so the main results may appear deceptively trivial.”
“这项工作很大程度上隐藏在消费者的视野之外，因此主要结果可能显得微不足道。”
Editor’s note: The Simons Center for Geometry and Physics at Stony Brook University is funded by the Simons Foundation, which also funds this editorially independent magazine .
编者注：石溪大学西蒙斯几何和物理中心由西蒙斯基金会资助，该基金会也资助这本编辑独立的杂志。
Simons Foundation funding decisions have no influence on our coverage.
西蒙斯基金会的资助决定对我们的报道没有影响。
Get highlights of the most important news delivered to your email inbox
获取发送到您的电子邮件收件箱的最重要新闻的摘要
Quanta Magazine moderates comments to facilitate an informed, substantive, civil conversation.
《Quanta》杂志对评论进行审核，以促进知情、实质性、文明的对话。
Abusive, profane, self-promotional, misleading, incoherent or off-topic comments will be rejected.
辱骂、亵渎、自我推销、误导、语无伦次或偏离主题的评论将被拒绝。
Moderators are staffed during regular business hours (New York time) and can only accept comments written in English.
版主在正常工作时间（纽约时间）有工作人员，并且只能接受英文评论。