Living Fully in the Math World Means Threading the Needle
Hong Wang, pictured here working on a new article at the Institute for Advanced Scientific Studies (IHES) in France, is just the third woman to receive a Fields Medal in the award’s 90-year history.
照片中的王红正在法国高级科学研究所 (IHES) 撰写一篇新文章，她是菲尔兹奖 90 年历史上第三位获得菲尔兹奖的女性。
O ne might imagine that, for a mathematician, proving a monumental theorem is a blissful experience.
人们可能会想象，对于一名数学家来说，证明一个伟大的定理是一种幸福的经历。
It wasn’t so for Hong Wang .
洪王则不然。
In February 2025, Wang and her collaborator Joshua Zahl presented a 127-page proof of a long-standing conjecture at the intersection of multiple branches of math.
2025 年 2 月，王和她的合作者 Joshua Zahl 提出了长达 127 页的证明，证明了一个长期存在的多个数学分支交叉点的猜想。
The duo had been checking their work for months and had sent it to select colleagues for review before finally mustering the courage to post it publicly.
两人几个月来一直在检查他们的工作，并将其发送给选定的同事进行审查，然后终于鼓起勇气将其公开发布。
Wang worried less that the arguments were flawed than that they might be imperfectly expressed and therefore unclear.
王担心的不是这些论点有缺陷，而是它们的表达可能不完美，因此不清楚。
No error was found, the proof’s logic was digested and streamlined, and everyone became convinced of its soundness.
没有发现任何错误，证明的逻辑被消化和简化，每个人都相信它的合理性。
Wang and Zahl had proved the three-dimensional Kakeya conjecture .
Wang和Zahl证明了三维挂谷猜想。
Hong Wang, who proved the three-dimensional Kakeya set conjecture, won the Fields Medal, making her the third woman in the award’s 90-year history to receive math’s highest honor.
证明三维挂谷集猜想的王红荣获菲尔兹奖，成为该奖90年历史上第三位获得数学最高荣誉的女性。
The proof triggered a succession of prizes for Wang, who comes from Guilin, China: the Salem Prize, the Ostrowski Prize, and the International Congress of Chinese Mathematicians’ Gold Medal of Mathematics in 2025; the Antonio Ambrosetti Medal, Sadosky Prize, Clay Research Award, and New Horizons in Mathematics Prize in the first half of 2026; and finally a 2026 Fields Medal, widely regarded as math’s highest honor, given to exceptionally accomplished mathematicians under 40.
这一证明为来自中国桂林的王先生带来了一系列奖项：塞勒姆奖、奥斯特洛夫斯基奖以及2025年国际华人数学家大会数学金奖； 2026 年上半年安东尼奥·安布罗塞蒂奖、萨多斯基奖、克莱研究奖和数学新视野奖；最后是 2026 年菲尔兹奖，该奖被广泛认为是数学的最高荣誉，颁发给 40 岁以下杰出数学家。
Wang, 35, a professor of mathematics at New York University and the Institute for Advanced Scientific Studies (IHES) in France, is just the third female winner in the Fields’ 90-year history.
35岁的王女士是纽约大学和法国高等科学研究所（IHES）的数学教授，也是菲尔兹90年历史上的第三位女性获奖者。
The attention has meant more speaking invitations and travel and interview requests that she struggles to refuse, which impinge on the time she can spend trying to understand how long, thin tubes pointing in different directions can overlap.
这种关注意味着更多她难以拒绝的演讲邀请、旅行和采访请求，这影响了她花时间尝试了解指向不同方向的细长管子可以重叠的时间。
Early morning walks help Wang handle the stress and self-doubt that come with her mathematical research.
清晨散步可以帮助王应对数学研究带来的压力和自我怀疑。
This is the issue at the heart of the “Kakeya-type problems” that are Wang’s specialty, a family of mathematical questions and conjectures that lies at the center of the Venn diagram of harmonic analysis (the study of how signals break up into constituent frequencies), geometric measure theory (pertaining to the shapes and sizes of non-smooth things), and partial differential equations (which describe systems that change in multiple ways simultaneously), with further connections to number theory, combinatorics, and beyond.
这是王的专长“挂屋型问题”的核心问题，这是一系列数学问题和猜想，位于调和分析（研究信号如何分解为组成频率）、几何测度论（与非光滑物体的形状和大小有关）和偏微分方程（描述同时以多种方式变化的系统）的维恩图的中心，并与数论、组合学等进一步联系。
“All of these connections are why this area is considered so central and why Hong is celebrated so much,” said Pablo Shmerkin , a mathematician at the University of British Columbia.
不列颠哥伦比亚大学数学家巴勃罗·什默金 (Pablo Shmerkin) 表示：“所有这些联系都是该地区被认为如此中心的原因，也是洪先生如此受人推崇的原因。”
She is always very kind and acknowledges that some of the ideas came from my work, but she was able to apply them in these fantastic ways that I couldn’t have envisioned.
她总是非常友善，并承认有些想法来自我的工作，但她能够以我无法想象的奇妙方式应用它们。
When she isn’t traveling, Wang works with the devotion and discipline of a cloistered nun or elite athlete.
当她不旅行时，王会以隐居修女或精英运动员的奉献精神和纪律来工作。
“She’s crazy into math,” said Shukun Wu of Indiana University, a collaborator.
“她对数学非常着迷，”合作者、印第安纳大学的 Shukun Wu 说。
“She’s definitely one of the most dedicated people I’ve seen.” She denies herself time-consuming indulgences like reading books or having a dog, though she does make time for friends, yoga, and early-morning walks to see the dogs at Washington Square Park, which all help relieve the stress and feelings of self-doubt that, for her, can accompany mathematical research.
“她绝对是我见过的最敬业的人之一。”她否认自己喜欢读书或养狗等耗时的嗜好，尽管她确实抽出时间与朋友、瑜伽和清晨去华盛顿广场公园散步看狗，这些都有助于缓解压力和自我怀疑的感觉，对她来说，这些都伴随着数学研究。
The desire to feel confident in her knowledge and ability seems to have shaped Wang’s life.
对自己的知识和能力充满信心的渴望似乎塑造了王的生活。
It is, for example, what draws her to mathematics: to her, the least doubtful thing.
例如，这就是她对数学感兴趣的原因：对她来说，这是最不值得怀疑的事情。
Even though she never took the time to celebrate her landmark proof — nor felt any consequent ego boost — at least Wang can now say with utter certainty that tubes pointing in every direction in 3D space can’t overlap very much.
尽管她从未花时间庆祝她的里程碑式的证明——也没有感受到任何随之而来的自我提升——但至少王现在可以完全肯定地说，指向 3D 空间中各个方向的管子不能重叠太多。
Reserved at first but easily loosened up, Wang wears tortoiseshell panto glasses and rings on both index fingers, which would occasionally dart into the frame of video calls — silver on the left and gold on the right.
王一开始很拘谨，但很容易放松下来，他戴着玳瑁眼镜，食指上戴着戒指，偶尔会出现在视频通话的画面中——左边是银色，右边是金色。
From the West Coast, where she was spending a few months visiting collaborators and friends, she explained that her life started out a lot more carefree.
她在西海岸花了几个月的时间拜访合作者和朋友，她解释说，她的生活一开始无忧无虑。
In Guilin in the 1990s, she would come home from school and read (Greek myths, Harry Potter, The Lord of the Rings ), or watch television, or play table tennis or badminton.
20世纪90年代的桂林，她放学回家就会看书（希腊神话、哈利波特、指环王），或者看电视，或者打乒乓球或羽毛球。
Her parents didn’t pressure her academically, despite both being schoolteachers.
尽管她的父母都是教师，但她并没有在学业上给她施加压力。
The three of them enjoyed evening walks around Guilin, often called China’s most beautiful city, situated amid steep emerald hills with a river running through it.
他们三人喜欢在桂林周围散步，桂林通常被称为中国最美丽的城市，坐落在陡峭的翠绿山峦之中，有一条河流流经其中。
During Wang’s childhood in Guilin, China, math was just one pastime among many, but she clearly showed an early aptitude for it.
王在中国桂林度过的童年时期，数学只是众多消遣中的一种，但她很早就表现出了这方面的天赋。
Math began as one hobby among all the others.
数学最初是作为所有其他爱好之一。
Whenever she got a new textbook, she’d complete all the exercises before the semester started.
每当她拿到新课本时，她都会在开学前完成所有练习。
She’d get more math books from the bookstore and solve all their problems as well.
她会从书店买更多数学书并解决他们所有的问题。
She remembers one brainteaser at the end of a chapter that asked: How should you plant seven trees to get the greatest number of rows of three trees?
她记得一章末尾的一个脑筋急转弯问：你应该如何种植七棵树才能获得最多三棵树的行数？
Like the Kakeya-type problems she would study later, the puzzle is about “incidence geometry,” or how objects overlap — in this case, lines (rows of trees) and points (trees).
就像她后来研究的挂谷类型问题一样，这个谜题是关于“重合几何”，或者物体如何重叠——在这种情况下，是线（树行）和点（树）。
Wang quickly figured out that you should plant the trees in the shape of an equilateral triangle, with a tree at each corner, a tree at the midpoint of each side, and the seventh tree in the center.
王很快就想到，应该把树种成等边三角形，每个角各有一棵树，每条边的中点各有一棵树，第七棵树在中间。
This forms six rows of trees: three along the sides and three through the middle.
这样就形成了六排树：三排在两侧，三排在中间。
She got the answer faster than her father, a math teacher.
她比她的数学老师父亲更快得到答案。
She was 6.
她六岁。
Wang came to appreciate math for its permanence.
王开始欣赏数学的永恒性。
History is forever being revised; learning English was frustrating because it seemed to have as many exceptions as rules.
历史永远被修正；学习英语是令人沮丧的，因为它似乎有与规则一样多的例外。
Only theorems seemed reliable.
只有定理似乎是可靠的。
“No one can come and say, ‘Oh, actually, this is not true,’” she said.
“没有人可以说，‘哦，实际上，这不是真的，’”她说。
I sent her an email and said, ‘Wow.’
我给她发了一封电子邮件，说：“哇。”
Both parents wanted her to live as normal a life as possible, Wang recalled.
王回忆说，父母都希望她尽可能过正常的生活。
She just wanted to learn as much as possible, so she skipped a couple of grades.
她只是想尽可能多地学习，所以她跳了几个年级。
Toward the end of middle school, her grades started to matter for getting into a good high school in Guilin.
初中快结束时，她的成绩开始成为进入桂林一所好高中的重要因素。
Wang set her sights on studying math at Peking University in Beijing, one of the most prestigious universities in China.
王女士的目标是在中国最负盛名的大学之一北京大学学习数学。
According to her, Peking reserved a single spot in its math program for a student from her province in 2007, with admission based on a standardized test.
据她介绍，2007年，北京大学在数学课程中为一名来自她所在省份的学生预留了一个名额，并根据标准化考试录取。
She did not receive the highest score.
她没有获得最高分。
But she did well enough to attend the university as an earth sciences major and hoped to transfer to math once there.
但她表现出色，进入大学主修地球科学，并希望在那里转入数学专业。
Her geophysics professor encouraged her, affirming the importance of math to science, for instance in the way seismic waves are used to map Earth’s interior (math that, in fact, relates to the Kakeya problem, though she didn’t know it at the time).
她的地球物理学教授鼓励她，肯定数学对科学的重要性，例如用地震波绘制地球内部地图的方式（事实上，数学与挂谷问题有关，尽管她当时并不知道）。
She worked hard and was eventually allowed to switch majors.
她努力学习，最终被允许转专业。
She recalled an offhand remark by her father that analysis, the type of math concerned with evolving quantities, limits, measures, and approximations, is much easier than algebra, with its symbolic equations and exact solutions.
她回忆起父亲随口说过的一句话，即分析这种涉及不断变化的数量、极限、度量和近似值的数学类型，比代数容易得多，因为代数有符号方程和精确解。
Wang rebelled by focusing on algebra.
王通过专注于代数来反叛。
A detour into architecture while studying in France was short-lived.
在法国学习期间绕道进入建筑界的时间是短暂的。
The tangible goals of math drew Wang back to analysis.
数学的具体目标将王拉回分析领域。
She didn’t have top grades and doubted she would get into graduate school.
她成绩不佳，怀疑自己能否进入研究生院。
Then representatives from the École Polytechnique near Paris, one of France’s prestigious, highly selective grandes écoles, held exams in Beijing for its postbaccalaureate program.
随后，巴黎附近的巴黎综合理工学院（法国著名的、严格筛选的高等院校之一）的代表在北京举行了学士学位后课程的考试。
In 2011 she moved to France.
2011年，她移居法国。
All her courses were taught in French, which she didn’t speak.
她所有的课程都是用法语授课，但她不会说法语。
Fortunately, she already knew much of the math in her first-semester classes.
幸运的是，她在第一学期的课程中已经了解了很多数学知识。
She shared meals with her French-speaking classmates, quietly listening, politely asking for clarification, gradually understanding more.
她与讲法语的同学一起吃饭，静静地听，礼貌地询问，渐渐了解更多。
She took classes in analysis and found that it came more naturally to her than algebra.
她参加了分析课程，发现这对她来说比代数更自然。
She was faring well in her classes, but she didn’t think she was good enough to do mathematical research.
她在课堂上表现很好，但她认为自己不足以进行数学研究。
For a semester, she switched to architecture and interned at a Paris firm.
一个学期，她转而学习建筑，并在巴黎的一家公司实习。
“I didn’t have much stress,” she said.
“我没有太大压力，”她说。
“But I didn’t know the goal.” So she recommitted to math.
“但我不知道目标。”于是她重新致力于数学。
“I decided not to worry about whether I would be good or not and just work on understanding better,” she said.
“我决定不再担心自己是否表现出色，而只是努力更好地理解，”她说。
In 2013, during a research internship at the Massachusetts Institute of Technology, Wang attended a seminar by an analyst named Larry Guth .
2013年，在麻省理工学院进行研究实习期间，王参加了一位名叫拉里·古斯(Larry Guth)的分析师举办的研讨会。
Guth discussed his recent proof, with the mathematician Nets Katz of Rice University, of the Erdős distinct distances problem, which asks: For any finite set of points on a plane, how many distances are there between pairs of points?
Guth 与莱斯大学的数学家 Nets Katz 讨论了他最近关于 Erdő 不同距离问题的证明，该问题提出：对于平面上的任何有限点集，点对之间有多少距离？
Guth explained the problem and his solution so clearly that Wang felt she could grasp part of the proof well, and this motivated her.
古斯把问题和他的解决方案解释得非常清楚，王觉得自己能很好地掌握部分证明，这激励了她。
She learned that it was closely related to many other problems of a similar character, all with incidence geometry at their heart.
她了解到，它与许多其他具有类似特征的问题密切相关，所有问题都以入射几何为核心。
She was admitted to the doctoral program at MIT the next year and chose Guth as her adviser.
次年，她考入麻省理工学院攻读博士学位，并选择古斯作为她的导师。
The “Kakeya-type problems” Wang focuses on are connected to harmonic analysis, geometric measure theory, and partial differential equations, as well as number theory, combinatorics, and more.
王重点关注的“挂谷型问题”与调和分析、几何测度论、偏微分方程以及数论、组合学等相关。
Later her specialty would be the Kakeya-type problems.
后来她的专长是挂谷类问题。
The most famous problem in this family and the one most of the others depend on is the Kakeya set conjecture.
这个家族中最著名的问题，也是其他大多数问题所依赖的问题，就是挂谷集合猜想。
It’s the modern version of a question asked in 1917 by the Tohoku University mathematician Sōichi Kakeya, who wondered about the smallest area you can sweep out with a needle as you rotate it in every direction on a plane.
这是东北大学数学家挂谷宗一 (Sōichi Kakeya) 在 1917 年提出的一个问题的现代版本，他想知道当你在平面上向各个方向旋转一根针时，你可以扫出的最小面积。
Simply turning the needle around its midpoint sweeps out a disk.
只需围绕中点旋转针即可扫出圆盘。
Kakeya realized that by shifting the needle as you turn it, as if making a three-point turn in a car, you can keep it within half the disk’s area, sweeping out a shape called a deltoid.
挂谷意识到，通过在转动时移动指针，就像在汽车中进行三点转弯一样，可以将其保持在圆盘面积的一半之内，从而扫出一种称为三角肌的形状。
A couple of years later, the Russian and later Cambridge-based mathematician Abram Besicovitch showed that you can do dramatically better.
几年后，俄罗斯数学家艾布拉姆·贝西科维奇 (Abram Besicovitch) 证明，你可以做得更好。
You can point the needle in all directions while covering no area at all — at least with an abstracted version of the problem.
你可以将针指向各个方向，同时不覆盖任何区域——至少对于问题的抽象版本来说是这样。
In this case there’s no sweeping needle, but an infinite set of line segments that point in all directions on the plane — one for each possible way the needle could be pointing — called a Kakeya set.
在这种情况下，没有扫过的针，而是一组无限的线段，它们指向平面上的所有方向 - 一个线段代表针可能指向的每种可能的方向 - 称为挂谷集。
Without changing the direction of any line segment, you can move them around to pack them into a smaller and smaller area.
在不改变任何线段方向的情况下，您可以移动它们以将它们打包到越来越小的区域中。
Besicovitch found that these line segments can, in fact, be packed into zero area.
贝西科维奇发现这些线段实际上可以被填充到零区域中。
How?
如何？
An equilateral triangle is another shape, like a deltoid but with a larger area, that can contain a Kakeya set.
等边三角形是另一种形状，类似于三角肌，但面积更大，可以包含挂谷组。
Imagine cutting that triangle straight down the middle and shifting (without rotating) one of the halves so that it lies on top of the other as much as possible.
想象一下，将三角形直接从中间切开，然后移动（不旋转）其中的一半，使其尽可能位于另一半的顶部。
Now cut each half of the original triangle in half and shift the halves again to maximize the overlap.
现在将原始三角形的每一半切成两半，然后再次移动两半以最大化重叠。
You’ve again squeezed the line segments into a smaller area.
您再次将线段压缩到更小的区域中。
Repeat this procedure ad infinitum, sliding the line segments back and forth using a geometric procedure called a Pál joint, and then rotate the triangle so you can repeat the procedure from each corner.
无限重复此过程，使用称为 Pál 关节的几何过程来回滑动线段，然后旋转三角形，以便可以从每个角重复该过程。
Put the results together and you will have accounted for line segments pointing in every direction, squeezed into zero area.
将结果放在一起，您将计算出指向各个方向的线段，这些线段被压缩到零区域。
Besicovitch showed that this works in higher dimensions too.
贝西科维奇表明这在更高维度上也有效。
Kakeya sets consisting of line segments pointing in all directions in 3D space can have zero volume.
由指向 3D 空间中所有方向的线段组成的挂屋集可以具有零体积。
But despite lacking area or volume, these strange, spiky sets still occupy space.
尽管缺乏面积或体积，这些奇怪的、尖刺的集合仍然占据着空间。
The line segments in the Kakeya set were not somehow squeezed together into a single, one-dimensional line; they still point in all directions.
挂谷集中的线段并没有以某种方式被挤在一起形成一条单一的、一维的线；而是以某种方式被压缩在一起。它们仍然指向各个方向。
The question is: How much space does a Kakeya set occupy?
问题是：挂屋套装占用多少空间？
Another way to put that question, mathematically, is: How many dimensions does it exist in?
从数学上来说，提出这个问题的另一种方式是：它存在于多少个维度？
An operational way to determine an object’s dimensionality is to imagine covering it with tiny boxes of uniform size and counting how many boxes it takes.
确定物体维度的一种操作方法是想象用大小均匀的小盒子覆盖它并计算它需要多少个盒子。
The number depends on how small the boxes are, but an object’s dimensionality relates to how that number changes as the size of the box changes.
这个数字取决于盒子的大小，但对象的维度与该数字随着盒子大小的变化而变化有关。
If you want to cover a 1D object such as a line segment, the number of boxes you’ll need is proportional to 1/ r , where r is the width of the box.
如果要覆盖一维对象（例如线段），则需要的框数量与 1/ r 成正比，其中 r 是框的宽度。
If you want to cover a 2D object, the number you’ll need is proportional to 1/ r 2 .
如果你想覆盖一个 2D 对象，你需要的数字与 1/ r 2 成正比。
For a 3D object, it’s 1/ r 3 , and so on.
对于 3D 对象，它是 1/ r 3 ，依此类推。
That exponent is the object’s dimension.
该指数就是物体的尺寸。
However, that exponent is not always equivalent to the apparent number of dimensions of the space the object exists in.
然而，该指数并不总是等于物体所在空间的表观维数。
Take the case of fractals, or objects that have ever more detailed structure as you zoom in, such as the Koch curve or Koch snowflake.
以分形为例，或者当放大时具有更详细结构的对象，例如科赫曲线或科赫雪花。
In these cases, the smaller the boxes are, the more structure there is for them to line up along, so they scale differently.
在这些情况下，盒子越小，它们排列的结构就越多，因此它们的缩放比例不同。
The exponent of r in the scaling law is a fraction, which indicates how much new structure is revealed as you zoom in.
缩放定律中 r 的指数是一个分数，它表示放大时会显示出多少新结构。
Fractals such as the Koch curve or snowflake, with its infinitely detailed edge, are somewhere in between integer dimensions.
科赫曲线或雪花等分形具有无限详细的边缘，介于整数维度之间。
The crux of the Kakeya set conjecture is whether these sets occupy all dimensions of the space they point in every direction of: that is, whether a Kakeya set of line segments pointing in every direction on a plane is itself 2D, whether a Kakeya set pointing in every direction in a volume of space is itself 3D, and so on.
挂谷集猜想的关键在于这些集合是否占据了它们指向各个方向的空间的所有维度：也就是说，指向平面上每个方向的线段的挂谷集本身是否是 2D，指向空间体积中每个方向的挂谷集本身是否是 3D，等等。
The sets behave, in this specific way, like solid objects, even though, as Besicovitch showed, they can have no area or volume.
在这种特定的方式下，这些集合的行为就像固体一样，尽管如贝西科维奇所示，它们可以没有面积或体积。
It’s not clear who first posed this conjecture or when, but it existed in 1971, when the British mathematician Roy Davies proved that it is true in the 2D case.
目前尚不清楚是谁或何时首先提出这个猜想，但它存在于 1971 年，当时英国数学家 Roy Davies 证明了它在 2D 情况下是正确的。
That it holds in three dimensions became known as the 3D Kakeya set conjecture.
它在三维空间中成立，被称为 3D Kakeya 集合猜想。
If it were to be disproved, this would mean the line segments of a Kakeya set, despite pointing in all directions, can be so cleverly bundled that they behave more like fractals than solid objects — that they have lower dimension than the 3D space they point in.
如果它被反驳，这将意味着挂屋集的线段尽管指向所有方向，但可以如此巧妙地捆绑在一起，以至于它们的行为更像分形而不是实体——它们的维度比它们指向的 3D 空间更低。
Because the Kakeya set is made up of line segments, mathematicians have reimagined the boxes used to measure dimensionality as being fused together into long, thin tubes.
由于挂屋集是由线段组成的，数学家们将用于测量维度的盒子重新想象为融合在一起形成长而细的管子。
The question is how the number of tubes you need to cover the set scales with the thickness of the tubes.
问题是覆盖设定范围所需的管子数量如何与管子的厚度成比例。
The Kakeya set conjecture says the scaling is like that of any other 3D object.
Kakeya 猜想认为缩放比例与任何其他 3D 对象的缩放比例相同。
If instead Kakeya sets can occupy a fraction of the full dimensionality of the space, then more detail — more line segments clustered in spaces inside — would reveal itself at smaller scales.
相反，如果挂谷集可以占据整个空间维度的一小部分，那么更多的细节——更多的线段聚集在内部的空间中——就会在更小的尺度上显现出来。
Wang speaks with mathematician Frank Merle over tea.
王与数学家弗兰克·梅尔喝茶交谈。
Prior to the proof of the 3D Kakeya set conjecture, her reputation had been growing through a series of collaborations with blockbuster results.
在3D挂谷集合猜想被证明之前，她的声誉已经通过一系列轰动一时的合作而声名鹊起。
Interest in the 3D Kakeya conjecture rose in the 1970s when the Princeton mathematician Charles Fefferman established that it has a close connection to a powerful phenomenon called Fourier transformation, where any signal or function can be decomposed into sine waves of different frequencies and directions.
20 世纪 70 年代，普林斯顿大学数学家查尔斯·费弗曼 (Charles Fefferman) 证实 3D Kakeya 猜想与一种名为傅里叶变换的强大现象密切相关，在傅里叶变换中，任何信号或函数都可以分解为不同频率和方向的正弦波，人们对 3D Kakeya 猜想的兴趣随之兴起。
A major open question in Fourier analysis (also called harmonic analysis) is what happens when the waves all have the same wavelength but are oriented in different directions.
傅里叶分析（也称为谐波分析）中的一个主要悬而未决的问题是，当波都具有相同的波长但方向不同时会发生什么。
What happens to the combined signal where they interact?
它们相互作用的组合信号会发生什么？
The Fourier restriction conjecture says that in such cases, the combined signal must be very spatially spread out and diffuse.
傅里叶限制猜想表明，在这种情况下，组合信号必须在空间上非常分散和扩散。
Because the component frequencies are all traveling in slightly different directions, they can’t overlap and constructively interfere enough to form a highly concentrated wave or signal, a consideration relevant to signal processing and imaging.
由于分量频率的传播方向略有不同，因此它们无法重叠和相长干扰足以形成高度集中的波或信号，这是与信号处理和成像相关的考虑因素。
“You can imagine that each wave lives on one of the long thin tubes,” Wang said.
“你可以想象每个波都生活在一根细长的管子上，”王说。
Will they overlap a lot, concentrating a signal, or not?
它们是否会重叠很多，从而集中信号？
A couple of years after arriving at MIT, Wang was chipping away at relatives of the Kakeya problem alongside Guth and others.
到达麻省理工学院几年后，王正与古斯等人一起解决挂谷问题的相关问题。
Within a year, they’d solved a special case of the Falconer distance problem, a harder variant of the problem Guth solved with Katz.
一年之内，他们解决了福尔科纳距离问题的一个特例，这是古斯和卡茨解决的问题的一个更难的变体。
Wang also made progress on the Fourier restriction problem by framing it in terms of incidence geometry patterns.
王还在傅里叶限制问题上取得了进展，通过将其定义为入射几何模式。
“It’s pretty rare that anyone could say anything new about this problem because it’s so difficult, so it was surprising for a Ph.D. student [to do so],” said Jonathan Hickman , a mathematician at the University of Edinburgh.
爱丁堡大学数学家乔纳森·希克曼 (Jonathan Hickman) 表示：“很少有人能够对这个问题提出任何新的见解，因为它太难了，因此对于一名博士生来说（这样做）感到惊讶。”
“I sent her an email and said, ‘Wow.’”
“我给她发了一封电子邮件并说，‘哇。’”
“I decided not to worry about whether I would be good or not and just work on understanding better,” Wang said.
“我决定不再担心自己是否优秀，而只是努力更好地理解，”王说。
Shmerkin, from the University of British Columbia, who had also worked on the Falconer distance problem, noticed Wang’s talent while visiting Guth at MIT.
来自不列颠哥伦比亚大学的什默金也曾研究过福尔科纳距离问题，他在麻省理工学院拜访古斯时注意到了王的才华。
She struck him as creative, and as a sponge for new techniques.
她给他的印象是富有创造力，并且是新技术的海绵。
She seemed to understand incidence geometry deeply even though she had only recently started working on it.
尽管她最近才开始研究入射几何，但她似乎对入射几何有着深入的了解。
The research area was already replete with new tools, and Wang soon imported more tools from geometric measure theory that she had learned during a collaboration with Shmerkin on the Falconer distance set problem.
该研究领域已经充满了新工具，王很快从几何测度理论中引入了更多工具，这些工具是她在与 Shmerkin 合作解决 Falconer 距离集问题时学到的。
She was greatly inspired by Shmerkin’s “multiscale analysis” approach, a way of comparing what happens at different scales in order to arrive at contradictions.
她受到什默金的“多尺度分析”方法的极大启发，这是一种比较不同尺度下发生的情况以得出矛盾的方法。
“She is always very kind and acknowledges that some of the ideas came from my work, but she was able to apply them in these fantastic ways that I couldn’t have envisioned,” Shmerkin said.
“她总是非常友善，并承认有些想法来自我的工作，但她能够以我无法想象的奇妙方式应用它们，”施默金说。
Wang’s reputation grew in 2019 when she, Guth, and Ruixiang Zhang of the University of California, Berkeley proved the 2D case of the local smoothing conjecture , one of the holy grails of harmonic analysis, which says that the solution to the wave equation can’t concentrate energy in a small region for an extended period.
2019年，王的名气越来越大，当时她、古斯和加州大学伯克利分校的张瑞祥证明了局部平滑猜想的二维情况，这是调和分析的圣杯之一，它表明波动方程的解不能在较长时间内将能量集中在一个小区域。
The same year, her doctorate complete, she moved to Princeton, New Jersey, for a postdoc at the Institute for Advanced Study.
同年，完成博士学位后，她搬到新泽西州普林斯顿高级研究所攻读博士后。
She worried that she couldn’t succeed without Guth, but gradually she began new collaborations and made headway on more Kakeya-type problems.
她担心没有古斯就无法取得成功，但她逐渐开始了新的合作，并在更多挂谷类型的问题上取得了进展。
Hiring committees took notice; she got a faculty job at the University of California, Los Angeles in 2021, and in 2023 she became an associate professor at NYU.
招聘委员会注意到了这一点； 2021年，她在加州大学洛杉矶分校获得教职，2023年成为纽约大学副教授。
A month after getting there, she produced another blockbuster result : Along with Kevin Ren , she proved the 2D Furstenberg set conjecture, which estimates the dimensionality of a set of points on the plane that contains a subset of lines pointing in every direction.
到达那里一个月后，她得出了另一个重磅成果：与 Kevin Ren 一起证明了二维弗斯滕伯格集合猜想，该猜想估计了平面上一组点的维数，这些点包含指向各个方向的线的子集。
She’s crazy into math.
她对数学非常着迷。
She’s definitely one of the most dedicated people I’ve seen.
她绝对是我见过的最敬业的人之一。
Despite these successes, Wang still felt like an underdog.
尽管取得了这些成功，王仍然感觉自己是一个失败者。
“Each time it just felt lucky,” she said of her various proofs.
“每次都感觉很幸运，”她谈到她的各种证明时说道。
It was at the Institute for Advanced Study during the pandemic that Wang had decided to turn her attention to the 3D Kakeya set conjecture.
正是在疫情期间的高等研究院，王决定将注意力转向3D挂谷集猜想。
The problem had resisted many challengers over the decades; proofs had even been announced and then shown to be wrong.
几十年来，这个问题一直困扰着许多挑战者。证据甚至已经公布，然后被证明是错误的。
“I never figured out the Kakeya conjecture, but there have been four or five times I thought I might have,” Guth said.
“我从未弄清楚挂谷猜想，但有四五次我认为我可能明白了，”古斯说。
Each time, he spotted his error before going public.
每次他都会在公开之前发现自己的错误。
Wang read a 2014 blog post by the UCLA mathematician Terence Tao laying out a proof strategy that Tao and Katz had developed but never pursued.
Wang 阅读了加州大学洛杉矶分校 (UCLA) 数学家特伦斯·陶 2014 年发表的一篇博客文章，其中列出了陶和卡茨已经开发但从未采用的证明策略。
The approach seemed promising, so she contacted Joshua Zahl, then at the University of British Columbia, another mathematician who had studied aspects of the Kakeya problem in his thesis.
这种方法似乎很有前途，所以她联系了当时在不列颠哥伦比亚大学的约书亚·扎尔(Joshua Zahl)，他是另一位数学家，他在论文中研究了挂谷问题的各个方面。
They decided to investigate what Tao and Katz had described.
他们决定调查陶和卡茨的描述。
The strategy, which wound up requiring an enormous arsenal of techniques, was to show that any hypothetical counterexample to the conjecture — a potential way of squeezing line segments pointing in every direction in 3D space into fewer than three dimensions — must have such a rigid and efficient structure that it would contradict theorems governing the interplay between addition and multiplication.
该策略最终需要大量的技术，旨在证明该猜想的任何假设反例（一种将 3D 空间中指向每个方向的线段压缩到少于三个维度的潜在方法）必须具有如此严格和有效的结构，以至于它与控制加法和乘法之间相互作用的定理相矛盾。
This general approach of positing a counterexample that disproves some conjecture, then twisting it into such knots that you conclude it can’t exist after all (leaving the conjecture as the only option), is a common proof strategy.
这种通用的方法是提出一个反例来反驳某些猜想，然后将其扭曲成这样的结，让你得出结论它根本不存在（将猜想作为唯一的选择），这是一种常见的证明策略。
The renowned proof of the 3D Kakeya set conjecture, with Joshua Zahl, ran to 127 pages and required a range of cleverly applied mathematical tools.
Joshua Zahl 对 3D Kakeya 集合猜想的著名证明长达 127 页，需要一系列巧妙应用的数学工具。
“You have doubt,” Wang said.
王说：“你有疑问。”
“But also the argument feels natural.”
“但这种争论也很自然。”
In this case, Wang and Zahl focused on what are called “sticky” Kakeya sets, because line segments in these sets with similar orientations tend to stick together in space.
在这种情况下，Wang 和 Zahl 专注于所谓的“粘性”Kakeya 集合，因为这些集合中具有相似方向的线段往往在空间中粘在一起。
These line segments bundle closely enough that thicker tubes could efficiently hold them, so the set as a whole could potentially exhibit fractal scaling, rather than behaving like a 3D object.
这些线段紧密地捆绑在一起，较粗的管子可以有效地容纳它们，因此整个集合可能会表现出分形缩放，而不是表现得像 3D 对象。
They first proved that sticky sets aren’t counterexamples; they are 3D after all.
他们首先证明粘性集不是反例；毕竟它们是 3D 的。
But there was more to do.
但还有更多事情要做。
“It wasn’t clear to people whether the sticky case truly was the main enemy,” said Zahl, who has the clean-cut look and comportment of a fighter pilot.
“人们不清楚这个粘糊糊的盒子是否真的是主要敌人，”扎尔说，他有着战斗机飞行员的干净外观和举止。
So next they assumed a more general counterexample: a Kakeya set with some dimensionality less than three (say, 3 − x ).
接下来，他们假设了一个更一般的反例：一个维数小于三（例如 3 − x ）的挂谷集。
By analyzing how many tubes intersect some midsize spherical region in the set, they could show that it must be either sticky (proved to be 3D) or not sticky, which, they could show, meant it must be higher-dimensional than had been assumed.
通过分析有多少管子与集合中的某个中等大小的球形区域相交，他们可以表明它必须是粘性的（被证明是 3D 的）或非粘性的，他们可以表明，这意味着它必须比假设的维度更高。
“Then we win either way,” Wang said.
“那么无论怎样我们都会赢，”王说。
They could repeat this argument for smaller and smaller values of x , until ultimately they could conclude that every Kakeya set must be 3D.
他们可以对越来越小的 x 值重复这个论证，直到最终他们得出结论：每个 Kakeya 集都必须是 3D 的。
In early 2024, four years after they began their attempt, the proof’s complicated inductive logic seemed to be working out, though neither Wang nor Zahl let themselves draw that conclusion.
2024 年初，即他们开始尝试四年后，证明的复杂归纳逻辑似乎正在发挥作用，尽管王和扎尔都没有让自己得出这个结论。
“This is a difficult problem, and who am I to solve it?” Wang said.
“这是一个难题，我是谁来解决它呢？”王说。
“You have doubt.
“你有疑问。
But also the argument feels natural.” She could visualize the logic in her mind’s eye: tubes at one scale turning into planks on larger scales, then expanding into slabs that fill the entire volume.
但这种争论也感觉很自然。”她可以在脑海中想象出其中的逻辑：一种尺寸的管子变成更大尺寸的木板，然后扩展到填充整个体积的板。
The way the shapes evolved into each other showed her that Kakeya sets are inevitably 3D.
形状相互演变的方式向她表明，挂屋套装不可避免地是 3D 的。
But it would take another year for Wang and Zahl to publicly announce the proof.
但王和扎尔又过了一年才公开宣布证据。
As the proof was examined, studied, and confirmed, it shifted the landscape in Wang’s corner of math.
随着这个证明的检验、研究和证实，它改变了王数学领域的面貌。
“It’s a bit strange in the community because the holy grail has been achieved in some sense,” Shmerkin said.
“这在社区里有点奇怪，因为从某种意义上说，圣杯已经实现了，”施默金说。
The Kakeya problem “is not solved in every dimension, but a big part of the holy grail has been achieved.
挂谷问题“并未在各个方面得到解决，但圣杯的很大一部分已经实现。
So we are all asking ourselves, what do we do next?
所以我们都在问自己，下一步我们该做什么？
And of course, all of these techniques have opened the way to do many nice things.”
当然，所有这些技术都为做许多好事开辟了道路。”
Even prior to her Fields Medal win, Wang was recognized on the street in China.
甚至在获得菲尔兹奖之前，王在中国的大街上就被认出了。
The award means her profile — and the travel and distractions that come with it — will grow.
该奖项意味着她的形象——以及随之而来的旅行和干扰——将会增长。
The 3D versions of other, even harder problems remain open, such as the 3D restriction and local smoothing conjectures.
其他更难问题的 3D 版本仍然悬而未决，例如 3D 限制和局部平滑猜想。
Another obvious target for Wang and her community is the 4D Kakeya conjecture, if not a general proof in all dimensions, though no one knows what that would look like yet.
Wang 和她的社区的另一个明显目标是 4D Kakeya 猜想，即使不是所有维度的通用证明，尽管还没人知道它会是什么样子。
Wang said that at the moment she doesn’t want to get too invested in solving any particular problem.
王说，目前她不想在解决任何特定问题上投入太多。
Her star has risen in China to the extent that she now gets recognized in public.
她在中国的明星地位不断上升，现在她在公众面前得到了认可。
The Fields Medal will make her a person of distinction there, and in the worldwide math community.
菲尔兹奖将使她成为当地乃至全球数学界的杰出人物。
She worries about falling behind, and still believes that she must put in more hours than her colleagues to be as good as they are.
她担心自己会落后，但仍然相信自己必须比同事投入更多的时间才能达到和他们一样的水平。
When challenged on this — everyone considers her extremely talented — she seemed genuinely surprised and pleased.
当在这个问题上受到挑战时——每个人都认为她非常有才华——她似乎真的感到惊讶和高兴。
With more Kakeya-type problems in view, Wang is excited for what is to come.
考虑到更多挂谷型问题，王对即将发生的事情感到兴奋。
Another hard problem, reserved for the future, is how to feel confident as a mathematician while holding space that doesn’t center around math.
另一个为未来保留的难题是，如何在保留不以数学为中心的空间的同时，作为一名数学家感到自信。
Wang is encouraged that Guth, her adviser, finds time to read for pleasure.
王女士的导师古斯（Guth）抽出时间阅读来消遣，这让王女士深受鼓舞。
“I always wish that I could come back to that time when I just feel like I had infinite time to read,” she said.
“我总是希望能回到那个时候，那时我觉得我有无限的时间读书，”她说。
“I think that time was really nice.”
“我觉得那段时光真的很美好。”
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