Sometimes Being First Means Seeing the End Before Anyone Else
From an early age, Jacob Tsimerman showed great mathematical promise.
雅各布·齐默曼（Jacob Tsimerman）从小就表现出了巨大的数学潜力。
“My mom told me, ‘You don’t have to be a mathematician just because you’re good at it,’” he said.
“我妈妈告诉我，‘你不必因为擅长数学而成为一名数学家，’”他说。
“I didn’t really take her seriously.”
“我并没有真正认真地对待她。”
J acob Tsimerman knows that many people see mathematics as a search for beauty.
雅各布·齐默曼 (Jacob Tsimerman) 知道许多人将数学视为对美的追求。
Over the course of his career, he has encountered beauty many times.
在他的职业生涯中，他多次遇到过美丽。
But beauty is not what drives him.
但美貌并不是驱动他的动力。
What he loves most is to solve hard problems, and to be the first to do so.
他最喜欢的是解决难题，并且是第一个这样做的人。
“In my experience, and I think many people’s experience, math is very much a goal-oriented endeavor,” he said.
“根据我的经验，我想很多人也有这样的经验，数学在很大程度上是一项以目标为导向的努力，”他说。
“The truth-and-beauty stuff is there when you’re zoomed out, but when you’re zoomed in, you just want to win.”
“当你缩小时，真实与美丽的东西就在那里，但当你放大时，你只想赢。”
Tsimerman , who has slightly wild brown hair and a beard, talks fast and with visible enthusiasm.
齐默尔曼有一头略显狂野的棕色头发和胡须，语速很快，充满热情。
He is open and funny, but also precise — quick to pause and clarify a thought.
他很开放、有趣，但也很精确——很快就会停下来澄清一个想法。
His fierce drive to win has fueled Tsimerman’s efforts at every stage of his career, resulting in teenage glory at the International Mathematical Olympiad, early matriculation at Princeton University, and proofs that reshaped areas of number theory and algebraic geometry, earning him a collection of top prizes.
齐默尔曼对胜利的强烈渴望推动了他职业生涯每个阶段的努力，为他带来了青少年时期在国际数学奥林匹克竞赛中的荣耀，早期被普林斯顿大学录取，以及重塑数论和代数几何领域的证明，为他赢得了一系列最高奖项。
Jacob Tsimerman won a Fields Medal this year in part for his proof of the André-Oort conjecture.
雅各布·齐默尔曼今年获得了菲尔兹奖，部分原因是他证明了安德烈-奥尔特猜想。
The honor reflects a career borrowing techniques from one area of math and leveraging them in another to reveal surprising underlying structure.
这一荣誉反映了一种职业从一个数学领域借用技术并将其运用到另一个领域以揭示令人惊讶的基础结构。
It’s an impulse that Tsimerman has also been wary about.
齐默尔曼也一直对这种冲动保持警惕。
He knew it could consume him, especially when it came to the most entrancing honor of all — the Fields Medal, awarded every four years to the most accomplished mathematicians in the world under the age of 40 in the year the prize is given.
他知道这会让他筋疲力尽，尤其是当涉及到最令人着迷的荣誉时——菲尔兹奖，该奖每四年颁发一次，颁发给颁奖当年世界上 40 岁以下最有成就的数学家。
“I made it an explicit goal for myself not to focus on winning it,” he said.
“我为自己设定了一个明确的目标，不要专注于赢得胜利，”他说。
“I was like, ‘Jacob, you can’t become obsessed with it.’”
“我当时想，‘雅各布，你不能沉迷于此。’”
The mind trick worked, for a time.
头脑戏法一度奏效了。
For nearly two decades, Tsimerman, 38, kept his drive focused on the math itself and became known as one of the most talented problem solvers of his generation.
近二十年来，38 岁的齐默曼一直专注于数学本身，并被誉为他这一代最有才华的问题解决者之一。
But as he approached the last cycle in which he’d be eligible for the Fields Medal, he realized that his successes put him in contention.
但当他接近获得菲尔兹奖资格的最后一个周期时，他意识到他的成功让他成为了竞争者。
He relented and dedicated himself to the pursuit.
他心软了，全身心投入到追求中。
“Two years ago, I knew I was close, and I did let myself become obsessed,” he said.
“两年前，我知道我已经很接近了，而且我确实让自己着迷了，”他说。
The final push has paid off, as Tsimerman has been named one of four winners of the Fields Medal.
最后的努力得到了回报，齐默尔曼被任命为菲尔兹奖的四位获得者之一。
The honor reflects a career built around borrowing techniques from one area of math and leveraging them in another to reveal surprising underlying structure.
这一荣誉反映了他的职业生涯是围绕从一个数学领域借用技术并将其运用到另一个数学领域来揭示令人惊讶的基础结构的。
“In my experience, and I think many people’s experience, math is very much a goal-oriented endeavor,” Tsimerman said.
“根据我的经验，我想很多人也有这样的经验，数学在很大程度上是一项以目标为导向的努力，”齐默尔曼说。
“He’s a brilliant problem solver.
“他是一位出色的问题解决者。
He can learn anything,” said Peter Sarnak , a professor at Princeton and the Institute for Advanced Study and Tsimerman’s graduate school adviser.
他可以学到任何东西。”普林斯顿大学和高等研究院教授、齐默尔曼研究生院顾问彼得·萨纳克 (Peter Sarnak) 说。
Following each milestone in his career, Tsimerman has taken stock, asking himself whether math is really how he wants to spend his life, and whether he’s in fact good enough to succeed in it.
在他职业生涯的每一个里程碑之后，齐默曼都会进行盘点，问自己数学是否真的是他想要度过一生的方式，以及他是否真的足够优秀以取得成功。
And at every previous stage the answer has come back yes.
在之前的每个阶段，答案都是肯定的。
But this time may be different.
但这一次可能会有所不同。
As Tsimerman contemplates what’s next with a Fields Medal around his neck, he does so as a new form of problem solver rises — one that no human may be able to beat, and which has him questioning his future in math.
当齐默曼脖子上挂着菲尔兹奖思考下一步要做什么时，他这样做是因为一种新形式的问题解决者的崛起——一种可能没有人能够击败的人，这让他对自己在数学领域的未来产生了质疑。
Tsimerman was born in 1988 in Kazan, in southwest Russia.
Tsimerman 1988 年出生于俄罗斯西南部的喀山。
When he was 3, his family moved to Israel, where his mother took a job as a high school math teacher.
当他三岁时，他的家人搬到了以色列，他的母亲在那里担任高中数学老师。
Around that time, his grandfather, a physicist, began giving him little puzzles.
大约在那个时候，他的物理学家祖父开始给他出一些小难题。
There was one about how long it takes to fill a bathtub with two taps running, and another about two speeding trains on a collision course with a junction between them.
其中一个是关于在两个水龙头同时运行的情况下需要多长时间才能注满浴缸，另一个是关于两列超速行驶的火车在相撞路线上，它们之间有一个交汇处。
His mother also began handing him textbooks.
他的母亲也开始给他递课本。
Even as she nurtured his mathematical ability, she was careful to let Tsimerman know he didn’t need to go down this path if he didn’t want to.
即使在培养他的数学能力的同时，她也小心翼翼地让齐默曼知道，如果他不愿意，他就不需要走这条路。
“My mom told me, ‘You don’t have to be a mathematician just because you’re good at it,’” he said.
“我妈妈告诉我，‘你不必因为擅长数学而成为一名数学家，’”他说。
“I didn’t really take her seriously.”
“我并没有真正认真地对待她。”
The truth-and-beauty stuff is there when you’re zoomed out, but when you’re zoomed in, you just want to win.
当你缩小时，真实与美丽的东西就在那里，但当你放大时，你只想赢。
When Tsimerman was 9, his family moved to Toronto, and he began entering math competitions.
齐默曼 9 岁时，全家搬到了多伦多，他开始参加数学比赛。
His father, a computer scientist, took him to the contests and brought him roast beef sandwiches midway through the long exams.
他的父亲是一名计算机科学家，带他去参加比赛，并在漫长的考试中途给他带来了烤牛肉三明治。
In seventh grade, he participated in the Canadian Mathematical Olympiad for the first time.
七年级时，他第一次参加加拿大数学奥林匹克竞赛。
There were five problems, and Tsimerman got most of them wrong.
有五个问题，齐默曼大部分都错了。
Afterward, he went over to the University of Toronto, near his school, and found a blackboard.
随后，他去了学校附近的多伦多大学，发现了一块黑板。
For hours, the 12-year-old toiled over one of the geometry questions he had failed to solve during the competition.
这位 12 岁的孩子花了几个小时苦苦思考他在比赛中未能解决的一道几何题。
Eventually, he got it.
最终，他得到了。
“I was super happy I’d solved an Olympiad geometry problem,” he said.
“我非常高兴我解决了奥林匹克几何问题，”他说。
Mathematicians debate the utility of competition math as preparation for a research career.
数学家们争论竞赛数学作为研究生涯准备的效用。
Many argue that it encourages the wrong mindset, because competition problems are guaranteed to have answers, while research requires coming to terms with the fact that many problems may not be solvable at all.
许多人认为，这会鼓励错误的思维方式，因为竞争问题肯定会有答案，而研究则需要接受这样一个事实：许多问题可能根本无法解决。
Tsimerman doesn’t see it that way.
齐默曼不这么认为。
To him, competition math teaches the skill that has mattered most to him: how to stick with a problem.
对他来说，数学竞赛教会了他最重要的技能：如何坚持解决问题。
“Competition math teaches you to sit with one problem for many hours, even though you’re almost certainly stuck,” he said.
他说：“数学竞赛教会你花几个小时解决一个问题，即使你几乎肯定会陷入困境。”
“If that’s not research math, I don’t know what is.”
“如果这不是研究数学，我不知道什么才是。”
Tsimerman at the Institute for Advanced Study in Princeton, New Jersey, where he spent the last academic year.
齐默曼在新泽西州普林斯顿高等研究院度过了最后一个学年。
After his failed effort at the Olympiad, Tsimerman devoted himself to improving.
在奥林匹克运动会失败后，齐默曼致力于提高自己。
He spent five hours every day practicing Olympiad problems after school rather than doing his regular homework.
他每天放学后花五个小时练习奥数题，而不是做常规作业。
It paid off.
它得到了回报。
In 2003, when he was 15, Tsimerman won a gold medal at the International Mathematical Olympiad in Tokyo, correctly answering four of six problems.
2003年，15岁的齐默曼在东京举行的国际数学奥林匹克竞赛中正确回答了六道题中的四道，赢得了金牌。
The following year, in Athens, he notched a perfect score.
次年，他在雅典获得了满分。
After that, Tsimerman conducted what would be the first of several reassessments at critical points in his math career.
此后，齐默曼在他数学生涯的关键时刻进行了几次重新评估中的第一次。
He decided to retire from competition math, reasoning that after receiving a perfect score, he could only blemish his record with further participation.
他决定退出数学竞赛，理由是在获得满分后，他只能通过进一步参加来玷污自己的记录。
At that level at least, he had nothing left to win.
至少在那个层面上，他已经没有什么可赢的了。
Tsimerman quit high school at 16 to enroll at the University of Toronto.
齐默曼 16 岁时从高中退学，进入多伦多大学就读。
Over the next two years, he devoured every available math course and earned an undergraduate degree at 18.
在接下来的两年里，他学完了所有可用的数学课程，并在 18 岁时获得了本科学位。
In 2006, he began graduate school at Princeton like other teen phenoms before him, including Terry Tao (Fields medalist in 2006) and Akshay Venkatesh (Fields medalist in 2018).
2006 年，他开始在普林斯顿大学攻读研究生，就像他之前的其他青少年天才一样，包括 Terry陶（2006 年菲尔兹奖得主）和 Akshay Venkatesh（2018 年菲尔兹奖得主）。
Soon after arriving, he sought out Sarnak, a renowned mathematician with a reputation as a skilled mentor.
抵达后不久，他找到了著名数学家萨纳克（Sarnak），他以技艺高超的导师而闻名。
“I was told Peter and I would mesh well and that he’s at the center of the universe and would guide me the right way,” Tsimerman said.
“我被告知彼得和我会很好地配合，他位于宇宙的中心，会引导我走上正确的道路，”齐默曼说。
“It was good advice on both counts.”
“从这两方面来说，这都是很好的建议。”
I think AI will be better than mathematicians at doing math within two years.
我认为两年内人工智能在数学方面将比数学家做得更好。
Sarnak gave Tsimerman a long list of books and papers to read.
萨纳克给了齐默曼一长串需要阅读的书籍和论文。
Tsimerman spent the first eight months of graduate school working through it.
齐默曼在研究生院的前八个月都在研究这个问题。
The more he read, the more overwhelmed he became.
他读得越多，就越感到不知所措。
As he read page after page of theory, the field seemed to grow vaster, and he felt more anxious about his own place within it.
当他一页一页地阅读理论时，这个领域似乎变得越来越广阔，他对自己在其中的地位感到更加焦虑。
Finally, he went back to Sarnak and pleaded for a concrete task, anything that would allow him to gain a foothold in the field and take stock of his chances of succeeding within it.
最后，他回到沙纳克，恳求一项具体的任务，任何能让他在该领域站稳脚跟并评估自己在该领域取得成功的机会的事情。
“I said, ‘I need a problem.
“我说，‘我需要一个问题。
I need a problem to work on because I can’t keep reading theory,’” he said.
我需要解决一个问题，因为我无法继续阅读理论，”他说。
Sarnak gave him a problem in an area of math called analytic number theory, which had been Tsimerman’s least favorite topic as an undergraduate.
萨纳克给了他一个数学领域的问题，称为解析数论，这是齐默曼本科时最不喜欢的话题。
He had come to Princeton hoping to avoid the area completely.
他来到普林斯顿希望完全避开这个地区。
But here was Sarnak, offering him the problem he had asked for, and he knew there was no way he could say no.
但萨纳克在这里，向他提出了他所要求的问题，他知道他无法拒绝。
“It took me three years to call Peter by his first name,” Tsimerman said.
“我花了三年时间才直呼彼得的名字，”齐默曼说。
“I wasn’t about to say, ‘Give me a different problem.’”
“我不想说，‘给我一个不同的问题。’”
He solved the problem, and he and Sarnak wrote up the proof together as Tsimerman’s first result.
他解决了这个问题，并且他和 Sarnak 一起写出了证明，作为 Tsimerman 的第一个结果。
It changed his relationship with research.
这改变了他与研究的关系。
Tsimerman had arrived at Princeton knowing how to solve contest problems.
齐默曼来到普林斯顿大学时知道如何解决竞赛问题。
Now, with Sarnak, he had gained confidence that he could succeed as a researcher too.
现在，在 Sarnak 的帮助下，他有了信心，相信自己也能成为一名成功的研究员。
Tsimerman on the Princeton University campus.
普林斯顿大学校园内的齐默曼。
“For the first two years I was extremely intimidated and kind of worried,” Tsimerman said.
“头两年我非常害怕，而且有点担心，”齐默曼说。
“Your first paper seems impossible to write.
“你的第一篇论文似乎不可能写出来。
It seemed impossible to me until I’d done it, then I was like, ‘OK, I can do this.’”
对我来说这似乎是不可能的，直到我做到了，然后我就想，‘好吧，我能做到。’”
After that, Sarnak gave Tsimerman a steady diet of hard problems.
此后，萨纳克给齐默曼带来了一系列难题。
Tsimerman would go off, think about the problems for a few months, then come back and report that he’d gotten nowhere.
齐默曼会离开，思考几个月的问题，然后回来报告说他一无所获。
Sarnak would say, “Great,” and then hand him another impossible task.
萨纳克会说：“太好了”，然后交给他另一个不可能完成的任务。
In Sarnak’s mind, the goal was not for Tsimerman to solve the problems.
在萨纳克看来，齐默尔曼的目标不是解决问题。
It was for him to understand what exactly makes hard problems hard, the way he might appreciate the specific technical difficulties and treacherous places that make it hard to scale a mountain — a kind of prerequisite knowledge for eventually making it to the top.
他要了解到底是什么让困难的问题变得困难，就像他理解导致难以攀登一座山的具体技术困难和危险的地方一样——这是最终登上顶峰的先决知识。
“The way you solve a problem is, you have to understand where the difficulties are, and you have to fail and develop an intuition of where the answers are,” Sarnak said.
萨纳克说：“解决问题的方法是，你必须了解困难在哪里，你必须在失败后培养出对答案在哪里的直觉。”
“It can be learned, but it’s much better learned when you try something, fail, form an intuition, and try again.”
“它是可以学习的，但当你尝试某件事，失败，形成直觉，然后再试一次时，学习效果会更好。”
Fail enough, learn where trouble lies, and maybe eventually you’ll succeed.
失败得够多了，了解问题所在，也许最终你会成功。
That was Sarnak’s idea, and it was what eventually happened when Sarnak told Tsimerman to look at another difficult problem called the André-Oort conjecture .
这是萨纳克的想法，当萨纳克告诉齐默尔曼研究另一个名为安德烈-奥尔特猜想的难题时，这最终发生了。
The André-Oort conjecture, posed in limited form by Yves André in 1989 and in more general terms by Frans Oort in 1995, grows from a central idea in modern mathematics — that arithmetic and geometry are deeply connected.
安德烈-奥尔特猜想由伊夫·安德烈 (Yves André) 于 1989 年以有限形式提出，并由弗朗斯·奥尔特 (Frans Oort) 于 1995 年以更笼统的形式提出，它源于现代数学的一个中心思想——算术和几何紧密相连。
To take a simple example, picture the graph of the solutions to an equation.
举一个简单的例子，画出方程解的图形。
The graph might pass through a few points whose coordinates are whole numbers.
该图可能会经过几个坐标为整数的点。
That could happen by accident.
这可能是偶然发生的。
But if it passes through many points with whole-number coordinates, mathematicians take this as a sign of hidden structure in the equation that explains why it repeatedly intersects with these special points.
但如果它经过许多具有整数坐标的点，数学家就会将此视为方程中隐藏结构的标志，这解释了为什么它会反复与这些特殊点相交。
André-Oort is a much more sophisticated version of that idea.
安德烈-奥尔特是这个想法的一个更为复杂的版本。
Instead of the two-dimensional graph of an equation, it concerns higher-dimensional geometric spaces whose points can represent entire mathematical objects.
它不是方程的二维图，而是涉及更高维的几何空间，其点可以代表整个数学对象。
Some of those points correspond to arithmetic objects with unusually rich numerical properties.
其中一些点对应于具有异常丰富的数值属性的算术对象。
André-Oort makes an assertion about cases when many of these special arithmetic points appear together inside the same geometric region.
安德烈-奥尔特对许多特殊算术点同时出现在同一几何区域内的情况做出了断言。
“If you have an arithmetic phenomenon that you don’t expect to happen, then there should be a geometric reason for it,” Tsimerman said.
“如果你有一个你没有预料到会发生的算术现象，那么它应该有一个几何原因，”齐默曼说。
When Sarnak first introduced Tsimerman to André-Oort, the conjecture was far out of reach.
当萨纳克第一次将齐默曼介绍给安德烈-奥尔特时，这个猜想还遥不可及。
But eventually Tsimerman found one hard piece of the problem he could attack.
但最终齐默曼发现了他可以解决的一个难题。
To prove André-Oort, mathematicians needed a way to show that special points did not appear in isolation.
为了证明安德烈-奥尔特，数学家需要一种方法来证明特殊点并不是孤立出现的。
As with ants, where there’s one, there are often many.
就像蚂蚁一样，有一只的地方往往有很多只。
Tsimerman likes to move between various areas of math.
齐默尔曼喜欢在数学的各个领域之间移动。
“You want to learn expertise in one area and bring it to another area where it’s useful but not yet used,” he said.
“你想学习一个领域的专业知识，并将其带到另一个有用但尚未使用的领域，”他说。
“That’s the golden goose.”
“这就是金鹅。”
One way to get from one special point to others is via something called a Galois orbit.
从一个特殊点到达其他特殊点的一种方法是通过伽罗瓦轨道。
That is the complete collection of points produced when you start at one point and apply the symmetries of the number system in which the point is defined.
这是当您从一点开始并应用定义该点的数字系统的对称性时产生的点的完整集合。
Find a special point and apply the symmetries, and they’ll take you to another special point.
找到一个特殊点并应用对称性，它们会带你到另一个特殊点。
Apply the symmetries again, get a third special point, and so on, until you’ve collected every point that can be reached in this way.
再次应用对称性，获得第三个特殊点，依此类推，直到收集到可以通过这种方式到达的所有点。
For his dissertation, Tsimerman set out to prove that in the setting of André-Oort, those orbits had to contain at least a certain number of points.
在他的论文中，齐默尔曼着手证明，在安德烈-奥尔特的背景下，这些轨道必须至少包含一定数量的点。
That would allow mathematicians to begin turning isolated arithmetic coincidences into evidence of a larger geometric pattern.
这将使数学家能够开始将孤立的算术巧合转化为更大几何图案的证据。
By 2010, when Tsimerman was approaching the end of his graduate studies, other mathematicians had made partial progress toward solving the conjecture.
到 2010 年，当齐默曼的研究生学业即将结束时，其他数学家在解决这个猜想方面已经取得了部分进展。
A group based in France had proved the conjecture if the generalized Riemann hypothesis is true, and that famously remains one of math’s great unsolved problems.
法国的一个小组已经证明了广义黎曼假设是否成立的猜想，众所周知，这仍然是数学中尚未解决的重大问题之一。
Separately, a mathematician at the University of Oxford named Jonathan Pila had found a different strategy.
另外，牛津大学一位名叫乔纳森·皮拉（Jonathan Pila）的数学家发现了一种不同的策略。
Pila’s approach was powerful enough to prove the simplest case of the conjecture but no more.
皮拉的方法足以证明该猜想的最简单情况，但仅此而已。
“We quickly saw that Pila’s method has a chance to generalize, but there were millions of hurdles along the way,” Sarnak said.
“我们很快发现皮拉的方法有机会推广，但一路上遇到了数百万的障碍，”萨纳克说。
Pila had also been a student of Sarnak’s, and in 2010 Sarnak invited him to Princeton to share his work.
皮拉也是萨纳克的学生，2010 年萨纳克邀请他到普林斯顿分享他的工作。
During that visit, Sarnak made sure to introduce him to Tsimerman.
在那次访问期间，萨纳克一定要把他介绍给齐默尔曼。
In the last few years, Tsimerman committed himself to projects with the kind of depth and impact he thought would matter most to those judging his work.
在过去的几年里，齐默曼致力于那些具有深度和影响力的项目，他认为这些项目对于评判他的作品的人来说最为重要。
“Peter asked me to explain the paper to him and said he has this very good student, would I mind if he sat in?
“Peter让我给他解释一下这篇论文，并说他有一个非常好的学生，我介意他坐在里面吗？
That student was Jacob,” Pila said.
那个学生就是雅各布，”皮拉说。
The timing was lucky.
时机很幸运。
Pila’s method needed a way to produce many special points, and Tsimerman’s work on Galois orbits offered exactly that kind of possibility.
皮拉的方法需要一种产生许多特殊点的方法，而齐默尔曼在伽罗瓦轨道上的工作恰恰提供了这种可能性。
In his dissertation, Tsimerman combined Pila’s methods with his work on Galois orbits to prove an important and difficult special case of the André-Oort conjecture.
在他的论文中，齐默尔曼将皮拉的方法与他在伽罗瓦轨道上的工作结合起来，证明了安德烈-奥尔特猜想的一个重要且困难的特例。
A proof of the full conjecture was still far off — it would take more than a decade, and additional collaborators, to get there — but even the partial result established Tsimerman as a top-flight researcher, not just a competition phenom.
完整猜想的证明还很遥远——需要十多年的时间以及更多的合作者才能实现这一目标——但即使是部分结果也证明了齐默尔曼是一位顶级研究人员，而不仅仅是一个竞争天才。
At that point, Tsimerman was confident he could do research and knew he could get a good job.
那时，齐默曼对自己能够从事研究充满信心，并且知道自己能够找到一份好工作。
And just as he had when he was younger, when he recorded a perfect score at the Math Olympiad, he checked in with himself, this time to assess whether the career opening up before him was one he really wanted.
就像他年轻时一样，当他在数学奥林匹克竞赛中获得满分时，他审视了自己，这一次是为了评估摆在他面前的职业是否是他真正想要的。
“The only doubt I had about becoming a mathematician was at the end of graduate school,” he said.
“我对成为一名数学家唯一的怀疑是在研究生毕业时，”他说。
“I asked myself, ‘Do I really want this?’”
“我问自己，‘我真的想要这个吗？’”
While in graduate school, Tsimerman did a two-month internship at Jane Street, the elite quantitative trading firm, and worked part time for another investment firm.
在读研究生期间，齐默曼在精英量化交易公司 Jane Street 实习了两个月，并在另一家投资公司兼职。
Quantitative finance had some of the things he liked — it offered hard problems, smart colleagues, and a clear-cut sense of progress, in the form of waxing or waning account balances.
定量金融有一些他喜欢的东西——它提供了难题、聪明的同事，以及以账户余额增减的形式明确的进步感。
But after a few months it was clear to him that he preferred the open-endedness of math research.
但几个月后，他清楚地意识到，他更喜欢数学研究的开放性。
“I just like doing longer research projects.
“我只是喜欢做更长的研究项目。
They were freer,” he said.
他们更自由，”他说。
Tsimerman in his office at the IAS.
齐默尔曼在 IAS 的办公室里。
After Tsimerman’s flirtation with finance, he confronted his lingering doubts about his fitness as a researcher.
在齐默尔曼涉足金融领域之后，他对自己是否适合成为一名研究员提出了挥之不去的怀疑。
At Princeton, he had proved he could solve research problems, but many of the problems that had shaped him had come from Sarnak.
在普林斯顿大学，他证明了自己能够解决研究问题，但塑造他的许多问题都来自萨纳克。
Tsimerman wanted to prove to himself that he could set his own research agenda.
齐默曼想向自己证明他可以制定自己的研究议程。
He began by looking for places where the tools he knew gave him a competitive advantage.
他首先寻找他所知道的工具可以为他带来竞争优势的地方。
Mathematicians often talk about the value of bringing ideas from one area into another.
数学家经常谈论将一个领域的想法带到另一个领域的价值。
Tsimerman talks that way too, but with a more competitive edge.
齐默尔曼也这么说，但更具竞争优势。
To him, moving between areas of math is a way of finding problems where he has resources that established researchers in the area might not.
对他来说，在数学领域之间移动是一种发现问题的方法，他拥有该领域知名研究人员可能没有的资源。
“You want to learn expertise in one area and bring it to another area where it’s useful but not yet used,” he said.
“你想学习一个领域的专业知识，并将其带到另一个有用但尚未使用的领域，”他说。
“That’s the golden goose.”
“这就是金鹅。”
In his last year of graduate school, Tsimerman took an improv class with the Upright Citizens Brigade in New York.
在研究生院的最后一年，齐默曼参加了纽约正直公民旅的即兴表演课程。
It captivated him immediately.
这立刻就吸引了他。
“You make a choice, see what happens, react in the moment,” he said.
“你做出选择，看看会发生什么，然后立即做出反应，”他说。
“I love making decisions on the fly.”
“我喜欢即时做出决定。”
He continued with improv in Cambridge, Massachusetts, where he did postdoctoral work at Harvard University as a junior fellow, and in Toronto, where he moved in 2014 to take a job at the University of Toronto.
他在马萨诸塞州剑桥继续即兴创作，在哈佛大学担任初级研究员，并在多伦多进行博士后工作，并于 2014 年搬到多伦多大学就职。
He recently wrote a musical called The Sombrero Detective , a neo-noir detective story about a town with a severe water shortage.
他最近写了一部音乐剧《草帽侦探》，这是一个关于一个严重缺水的小镇的新黑色侦探故事。
“I follow the three-second rule in life,” he said.
“我在生活中遵循三秒规则，”他说。
“You have three seconds to do it, or just accept you’re chickening out.”
“你有三秒钟的时间去做，或者就承认你在畏缩。”
Alongside math, improv has become his most enduring interest, though he sees the two as inverses.
除了数学之外，即兴表演也成为他最持久的兴趣，尽管他认为两者是相反的。
“Improv is very much about losing control,” Tsimerman said.
“即兴表演很大程度上意味着失去控制，”齐默曼说。
“The way I think about math is, it’s really the opposite.
“我对数学的看法是，它实际上是相反的。
I’m constantly trying to gain as much control as I can.”
我一直在努力获得尽可能多的控制权。”
One possibility was to focus on Hodge theory, an area of mathematics that studies geometric spaces by translating them into analytic objects — objects built from functions, integrals, and other more flexible tools than polynomial equations.
一种可能性是关注霍奇理论，这是一个数学领域，通过将几何空间转化为分析对象来研究几何空间——由函数、积分和其他比多项式方程更灵活的工具构建的对象。
In the 1970s, a mathematician named Philip Griffiths had predicted that in some situations, even after these objects pass from the rigid geometric world into this more flexible analytic world, some underlying geometric structure should remain.
在 20 世纪 70 年代，一位名叫菲利普·格里菲斯 (Philip Griffiths) 的数学家曾预测，在某些情况下，即使这些物体从严格的几何世界进入这个更灵活的分析世界之后，一些潜在的几何结构也应该保留下来。
The problem, known as Griffiths’ conjecture, had the same flavor as André-Oort, where a loose analytic object, when viewed the right way, should turn out to have hidden algebraic structure.
这个被称为格里菲斯猜想的问题与安德烈-奥尔特问题具有相同的风格，即一个松散的分析对象，如果以正确的方式观察，应该具有隐藏的代数结构。
The work also brought him closer to Benjamin Bakker , a friend from graduate school.
这项工作也让他与研究生院的朋友本杰明·巴克 (Benjamin Bakker) 走得更近了。
“In grad school I had the feeling we were on pretty different paths math-wise,” said Bakker, now a professor at the University of Illinois, Chicago.
“在研究生院时，我感觉我们在数学方面走的是截然不同的道路，”现任芝加哥伊利诺伊大学教授的巴克说。
“He was more on the number-theory side, and I was more on the geometry side of things.
“他更多地关注数论方面，而我更多地关注几何方面。
But of course you can draw a line from any one point in math to any other point, so that’s all an illusion.”
但当然，你可以从数学中的任何一点到任何其他点画一条线，所以这都是一种幻觉。”
Working with Bakker and Yohan Brunebarbe, Tsimerman brought tools he had learned from André-Oort into Hodge theory.
齐默曼与巴克和约翰·布鲁内巴贝合作，将他从安德烈·奥尔特那里学到的工具带入了霍奇理论。
In 2018, they proved Griffiths’ conjecture and developed a framework called o-minimal GAGA, which gave mathematicians a new way to show that some analytic spaces have hidden algebraic structure.
2018年，他们证明了格里菲斯猜想，并开发了一个名为o-minimal GAGA的框架，该框架为数学家提供了一种新的方法来证明某些解析空间具有隐藏的代数结构。
After years of demonstrating his ability as a problem solver, Tsimerman had now shown that he could build new mathematical machinery as well.
经过多年证明自己作为问题解决者的能力，齐默曼现在也证明了他也可以建造新的数学机器。
“We like in math to say people are problem solvers and theory builders,” Sarnak said.
“我们喜欢在数学中说人是问题解决者和理论构建者，”萨纳克说。
“Jacob is a problem solver and a theory builder; I say that in a very strong way.”
“雅各布是一位问题解决者和理论构建者；我以非常强烈的方式这么说。”
By the early 2020s, Tsimerman had established himself as one of the best mathematicians in the world.
到 2020 年代初，齐默曼已成为世界上最好的数学家之一。
The doubts that had followed him to Princeton and for years after had fallen away, and he began racking up results at a rapid clip.
伴随他来到普林斯顿以及此后多年的疑虑已经消失，他开始迅速取得成果。
In 2021, Tsimerman, Pila, and Ananth Shankar posted a proof of the full André-Oort conjecture , completing the long process that had begun when Tsimerman first took up the problem as a graduate student.
2021 年，Tsimerman、Pila 和 Ananth Shankar 发布了完整的安德烈-奥尔特猜想的证明，完成了 Tsimerman 作为研究生首次解决这个问题时开始的漫长过程。
By then he had also delved deeper into Hodge theory, where the machinery he’d helped develop kept producing new results.
那时，他还深入研究了霍奇理论，他帮助开发的机器不断产生新的结果。
In 2024, Tsimerman, Bakker, and two collaborators proved a major new theorem about certain spaces arising in Hodge theory.
2024 年，齐默曼、巴克和两位合作者证明了霍奇理论中关于某些空间的一个重要新定理。
The results brought prizes.
结果带来了奖品。
Tsimerman won the 2022 New Horizons in Mathematics award and the 2023 Ostrowski Prize.
齐默曼荣获 2022 年数学新视野奖和 2023 年奥斯特罗斯基奖。
By the time he entered his mid-30s and last cycle of eligibility for the Fields Medal, winning it no longer seemed a remote possibility.
当他进入 30 多岁、也是获得菲尔兹奖资格的最后一个周期时，赢得菲尔兹奖似乎不再是遥不可及的事情。
Realizing that, he abandoned his vow not to become obsessed with the honor and made it his explicit goal.
意识到这一点后，他放弃了不沉迷于荣誉的誓言，并将其作为自己的明确目标。
“I worked hard and sacrificed enjoyment to try and win it,” he said.
“我努力工作并牺牲了享受来尝试赢得胜利，”他说。
The sacrifice was less about working longer hours and more about changing what he chose to work on.
这种牺牲不是为了延长工作时间，而是为了改变他选择的工作内容。
Until then, Tsimerman had preferred to work on fun projects that caught his interest — what he describes as “cool little things here and there.” But in the last few years, he decided to commit himself to projects with the kind of depth and impact he thought would matter most to mathematicians judging his body of work, including the committee that chooses the Fields medalists.
在那之前，齐默曼更喜欢从事引起他兴趣的有趣项目——他将其描述为“到处都是很酷的小事情”。但在过去的几年里，他决定致力于一些具有深度和影响力的项目，他认为这些项目对于评判他的工作的数学家来说最重要，包括选择菲尔兹奖得主的委员会。
“I stopped doing fun side projects and buckled down on projects that were high-return,” he said.
“我不再做有趣的副业项目，而是专注于高回报的项目，”他说。
Tsimerman has stopped taking students because of the uncertainty surrounding AI’s effects on math.
由于人工智能对数学影响的不确定性，齐默曼已经停止招收学生。
One of those projects was a paper with Arul Shankar , a mathematician at Toronto, on “secondary main terms for quartics.” Tsimerman had wanted to finish it for a long time; however, the project required working through some dense technical details that he’d been avoiding.
其中一个项目是与多伦多数学家 Arul Shankar 合作撰写的一篇关于“四次次主要术语”的论文。齐默尔曼很早就想完成它了。然而，该项目需要解决一些他一直回避的密集技术细节。
But with the Fields Medal in view, he and Shankar spent several months pushing through the calculations, posting the result in August 2025.
但考虑到获得菲尔兹奖，他和香卡花了几个月的时间进行计算，并于 2025 年 8 月公布了结果。
From 2024 to 2025, Tsimerman posted 11 papers to the preprint site arxiv.org across a range of fields.
从 2024 年到 2025 年，Tsimerman 在预印本网站 arxiv.org 上发布了 11 篇涉及多个领域的论文。
Then, late one Friday in January 2026, he received an email from Hiraku Nakajima, the president of the International Mathematical Union, asking to schedule a time to speak with him.
然后，2026 年 1 月的一个星期五晚些时候，他收到了国际数学联盟主席 Hiraku Nakajima 的一封电子邮件，要求安排与他交谈的时间。
Tsimerman woke his wife to tell her that he thought he had won the Fields Medal, then spent the weekend waiting nervously to find out.
齐默曼叫醒了他的妻子，告诉她他认为自己赢得了菲尔兹奖，然后整个周末都在紧张地等待着结果。
When the call finally came through the following Monday, Tsimerman, who was at the Institute for Advanced Study (IAS) in Princeton for the year, went outside to absorb the good news.
当下周一终于接到电话时，当时在普林斯顿高等研究院 (IAS) 呆了一年的齐默曼走到外面去了解这个好消息。
“I walked in the IAS woods for a while to work off some residual nervous energy and take it in,” he said.
“我在 IAS 树林里走了一会儿，以释放一些残余的神经能量并吸收它，”他说。
“It was nice.”
“这很好。”
As before, success finds Tsimerman at another crossroads.
和以前一样，齐默曼面临着成功的另一个十字路口。
He has received the Fields Medal at a moment when the future of mathematics feels more uncertain than ever.
在数学的未来比以往任何时候都更加不确定的时刻，他获得了菲尔兹奖。
As recently as 2025, artificial intelligence was still struggling to solve high-school-level competition problems.
就在2025年，人工智能还在努力解决高中水平的竞赛问题。
But early in 2026, AI systems began proving increasingly sophisticated results.
但 2026 年初，人工智能系统开始证明其结果越来越复杂。
The advance has culminated, for now, in OpenAI’s disproof in May 2026 of the unit distance conjecture, a result widely viewed as good enough to be published in the top journals in mathematics.
目前，这一进展已达到顶峰，OpenAI 于 2026 年 5 月反驳了单位距离猜想，这一结果被广泛认为足以发表在数学顶级期刊上。
The rapid improvement in AI math capabilities has occurred so suddenly that mathematicians do not know what to make of it.
人工智能数学能力的快速提升来得如此突然，以至于数学家们不知道该如何理解它。
Some remain skeptical that AI is actually useful for serious research.
一些人仍然怀疑人工智能对于严肃的研究是否真的有用。
Others, including Terry Tao, have begun to imagine it as a powerful assistant , able to extend what human mathematicians can do.
包括陶哲轩在内的其他人已经开始将其想象为一个强大的助手，能够扩展人类数学家的能力。
Tsimerman has become one of the highest-profile voices for a third, more disruptive take.
齐默曼已成为第三次更具颠覆性的行动中最引人注目的声音之一。
As he looks at the high-slope trajectory AI is on, he thinks the future of the field is in jeopardy.
当他看到人工智能所走的高坡轨迹时，他认为该领域的未来正处于危险之中。
“I think AI will be better than mathematicians at doing math within two years,” he said.
“我认为两年内人工智能将比数学家做得更好，”他说。
That belief has already changed how he spends his time.
这种信念已经改变了他度过时间的方式。
Tsimerman has stopped taking graduate students, because he worries that starting a conventional research project now could leave a student preparing for a mathematical career that may not exist.
齐默曼已经停止招收研究生，因为他担心现在开始一个传统的研究项目可能会让学生为可能不存在的数学职业做好准备。
“I don’t know what to do with a grad student in math who’s not very AI motivated,” he said.
“我不知道该如何对待一个对人工智能不太感兴趣的数学研究生，”他说。
Tsimerman now spends less time on classical math research.
齐默曼现在花在经典数学研究上的时间更少了。
Rather than organize math conferences, he is coordinating events that he hopes will prepare mathematicians for the changes ahead.
他没有组织数学会议，而是协调一些活动，希望这些活动能让数学家为未来的变化做好准备。
He has also shifted his research focus toward AI safety.
他还将研究重点转向人工智能安全。
Tsimerman thinks mathematicians have a role to play in understanding how systems of AI agents behave, and in deriving proofs that ensure that these complex systems won’t act in unintended ways.
Tsimerman 认为数学家可以在理解人工智能代理系统的行为方式以及得出证明以确保这些复杂系统不会以意想不到的方式行事方面发挥作用。
“The risks are super high, the stakes are super high, so we need a very high level of assurance,” he said.
“风险非常高，利害关系非常高，所以我们需要非常高水平的保证，”他说。
Societal impacts aside, AI’s march also has very personal implications for Tsimerman.
除了社会影响之外，人工智能的进步对齐默尔曼也有非常个人的影响。
He has always loved mathematics as a problem-solving activity, where the point is to reach the answer first.
他一直喜欢数学作为一种解决问题的活动，其重点是首先找到答案。
If machines become better problem solvers than any human, he is not sure the work will hold enough meaning to keep him or other mathematicians engaged.
如果机器比任何人类都能更好地解决问题，他不确定这项工作是否具有足够的意义来让他或其他数学家参与其中。
In early May 2026, making his debut on the social media platform X, he made this point bluntly .
2026年5月上旬，他在社交媒体平台X上首次亮相时，直言不讳地表达了这一点。
“There are many people whose primary enjoyment of math comes through problem solving in one of its incarnations,” he wrote.
他写道：“很多人对数学的主要兴趣来自于解决数学的某一形式中的问题。”
“If that disappears, that is not a trivial issue and many of them might not want to do it anymore.”
“如果这种情况消失，那就不是一个小问题，他们中的许多人可能不想再这样做了。”
Correction: July 23, 2026 The original version of this article stated that Arul Shankar was a co-author on both the André-Oort and secondary terms for quartics papers.
更正：2026 年 7 月 23 日本文的原始版本指出 Arul Shankar 是四次论文的 André-Oort 和次要术语的合著者。
In fact, Tsimerman’s collaborator on the on the André-Oort paper was Ananth Shankar.
事实上，齐默曼在安德烈-奥尔特论文中的合作者是阿南斯·香卡 (Ananth Shankar)。
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