The Year in Mathematics
Video : 2025 marked a historic year in mathematics.
视频：2025 年对于数学来说是历史性的一年。
Researchers solved a major case of Hilbert’s ambitious sixth problem, proved a sweeping new theorem about hyperbolic surfaces, and settled the longstanding three-dimensional Kakeya conjecture.
研究人员解决了希尔伯特雄心勃勃的第六个问题的一个重大案例，证明了一个关于双曲曲面的全新定理，并解决了长期存在的三维挂谷猜想。
M athematics is, at its core, an art.
数学的核心是一门艺术。
Like painters, musicians or writers, mathematicians create and explore new worlds.
就像画家、音乐家或作家一样，数学家创造和探索新世界。
They test, and then push past, the limits of their imagination.
他们测试并突破自己想象力的极限。
They engage with thousands of years of history, with concepts and tastes and fashions that are constantly in flux.
他们涉足数千年的历史，观念、品味和时尚不断变化。
This artistic pursuit of beauty, truth and meaning is what every Quanta math story is about, to some extent.
在某种程度上，这种对美、真理和意义的艺术追求是每个广达数学故事的主题。
This was on full display in one of my favorite articles of the year, an account by Kevin Hartnett of how a mathematician named Hannah Cairo solved an important problem in the field of harmonic analysis — at just 17 years old.
这一点在我今年最喜欢的一篇文章中得到了充分展示，凯文·哈特尼特 (Kevin Hartnett) 的一篇文章讲述了一位名叫汉娜·开罗 (Hannah Cairo) 的数学家如何解决调和分析领域的一个重要问题——当时他只有 17 岁。
Cairo grew up in the Bahamas, where she was homeschooled, learning math by watching Khan Academy videos and consuming everything else she could find online.
开罗在巴哈马长大，在那里她在家接受教育，通过观看可汗学院的视频学习数学，并消耗她在网上能找到的所有其他东西。
She found the homeschooling experience overwhelmingly lonely and confining.
她发现在家上学的经历极其孤独和受限。
“There was this inescapable sameness,” she told Hartnett.
“这是不可避免的相似性，”她告诉哈内特。
“No matter what I did, I was in the same place doing mostly the same things.
“无论我做什么，我都会在同一个地方做几乎相同的事情。
I was very isolated, and nothing I could do could really change that.”
我非常孤立，我无能为力改变这一点。”
Except studying math.
除了学数学。
Math gave her the escape she needed, an entire universe to roam — in Cairo’s words, a “world of ideas that I can explore on my own.” It’s impossible not to see math as art here: a way of experimenting with new ideas, of grappling with a world that doesn’t always make sense.
数学给了她所需的逃避，一个可以漫游的整个宇宙——用开罗的话说，一个“我可以自己探索的思想世界”。在这里，不可能不将数学视为艺术：一种尝试新想法、应对一个并不总是有意义的世界的方式。
And, crucially, of questioning assumptions.
而且，最重要的是，质疑假设。
As a teenager, Cairo moved to California, where she took graduate-level classes at the University of California, Berkeley and encountered a 40-year-old conjecture about the behavior of functions.
十几岁的时候，Cairo 搬到了加利福尼亚州，在那里她在加州大学伯克利分校攻读研究生课程，并遇到了一个 40 年前关于函数行为的猜想。
After several months of persistent work, she constructed a counterexample to the conjecture that more seasoned mathematicians had missed.
经过几个月的坚持不懈的工作，她构建了一个反例来反驳更多经验丰富的数学家错过的猜想。
Just as art is so frequently inextricable from the artist who makes it, Cairo was uniquely positioned to formulate a fresh perspective on the functions she was studying, which allowed her to show that they can behave in more counterintuitive ways than mathematicians had imagined.
正如艺术常常与创作它的艺术家密不可分一样，开罗处于独特的地位，能够对她正在研究的函数提出新的视角，这使她能够证明函数的行为方式比数学家想象的更违反直觉。
That’s often what success in math is all about.
这通常就是数学成功的全部意义所在。
M ath is also beautiful and strange.
数学也是美丽而奇怪的。
I remember when I first heard about the solution to the “ten martini problem” — a result about how the energy levels of electrons can form a well-known fractal pattern called the Cantor set — I was floored.
我记得当我第一次听说“十马提尼问题”的解决方案时——这个结果是关于电子的能级如何形成一种众所周知的分形图案，称为康托集——我感到很震惊。
It brought to mind Eugene Wigner’s famous essay on the “ unreasonable effectiveness of mathematics ,” which examines how abstract math often mysteriously provides the perfect language for understanding the natural world.
它让人想起尤金·维格纳（Eugene Wigner）关于“数学不合理的有效性”的著名文章，该文章探讨了抽象数学如何常常神秘地为理解自然世界提供完美的语言。
For the Cantor set to rear its head in solutions to Schrödinger’s equation in quantum physics, for it to be able to give insights into how electrons in a crystal behave when placed near a magnet — seriously?
康托尔将在量子物理中薛定谔方程的解中崭露头角，以便能够深入了解晶体中的电子在靠近磁铁时的行为——认真的吗？
In a fascinating article, Lyndie Chiou and Quanta staff writer Joe Howlett explore this problem , which was famously so difficult to solve that a mathematician offered 10 martinis to whoever could figure it out.
在一篇引人入胜的文章中，Lyndie Chiou 和 Quanta 特约撰稿人 Joe Howlett 探讨了这个问题，众所周知，这个问题很难解决，以至于一位数学家向任何能解决这个问题的人提供了 10 杯马提尼酒。
The problem was originally solved in 2004, but in a way that one of the proof’s authors, Svetlana Jitomirskaya, found unsatisfying.
该问题最初于 2004 年得到解决，但证明的作者之一 Svetlana Jitomirskaya 认为解决方式并不令人满意。
The proof “was a patchwork quilt, each square stitched out of distinct arguments,” Chiou and Howlett write.
Chiou 和 Howlett 写道，这个证明“就像一床拼凑而成的被子，每个方格都是由不同的论点缝制而成的”。
And it couldn’t be applied to solve the problem in more general and realistic settings — prompting Jitomirskaya to return to it 20 years later.
而且它无法应用于解决更普遍、更现实的环境中的问题——这促使吉托米尔斯卡娅在 20 年后再次回到这个问题上。
She and her colleagues have now produced a new, more powerful proof of this strange connection between number theory and quantum physics, cementing it as something deep and true.
她和她的同事现在提出了一个新的、更强有力的证据，证明了数论和量子物理学之间的这种奇怪联系，巩固了它的深刻和真实。
Along the way, Chiou and Howlett take you on a journey involving graphs that resemble butterfly wings, a calculator named Rumpelstilzchen, and Douglas Hofstadter’s delightful book Gödel, Escher, Bach .
一路上，Chiou 和 Howlett 将带您踏上一段旅程，其中涉及类似蝴蝶翅膀的图形、名为 Rumpelstilzchen 的计算器，以及道格拉斯·霍夫施塔特 (Douglas Hofstadter) 的令人愉快的书《哥德尔、埃舍尔、巴赫》。
Kristina Armitage/ Quanta Magazine ; sources (from left): Fondation L’Oréal For Women in Science, Jan Vondrák, P.
克里斯蒂娜·阿米蒂奇/广达杂志；资料来源（左起）：欧莱雅科学女性基金会，Jan Vondrák, P.
Imbert/Collège de France
安贝尔/法兰西学院
M ath doesn’t happen in a vacuum.
数学不会在真空中发生。
It’s influenced by philosophies of thought and, ultimately, people.
它受到思想哲学的影响，并最终受到人的影响。
Sometimes revolutionaries come on the scene, guiding how mathematicians think about a particular field for generations.
有时，革命者会出现，指导几代数学家如何思考某个特定领域。
One of those revolutionaries was Maryam Mirzakhani.
玛丽亚姆·米尔扎哈尼（Maryam Mirzakhani）就是其中一位革命者。
As a graduate student, she transformed the field of hyperbolic geometry, developing groundbreaking techniques for understanding mind-bending surfaces that appear throughout math and physics.
作为一名研究生，她改变了双曲几何领域，开发了突破性的技术来理解数学和物理中出现的令人费解的表面。
She became the first woman to win a Fields Medal , in part for this work.
她成为第一位获得菲尔兹奖的女性，部分原因是这项工作。
But she died at age 40, before she could fully explore the ramifications of these discoveries.
但她在 40 岁时就去世了，当时她还没来得及充分探索这些发现的影响。
Earlier this year, Howlett examined her legacy — and how two other mathematicians, Nalini Anantharaman and Laura Monk, have picked up where she left off , aiming to better understand the world of hyperbolic surfaces.
今年早些时候，豪利特研究了她的遗产，以及另外两位数学家 Nalini Anantharaman 和 Laura Monk 如何从她离开的地方继续，旨在更好地理解双曲曲面的世界。
Howlett expertly weaves together the stories of these three women and the research that unites them across time and space.
豪利特熟练地将这三位女性的故事以及跨越时空将她们联系在一起的研究编织在一起。
It’s also worth noting that Mirzakhani had a deep passion for literature, at one point hoping to be a writer.
还值得注意的是，米尔扎哈尼对文学有着浓厚的热情，一度希望成为一名作家。
Anantharaman trained as a classical pianist and seriously considered pursuing a career in music instead of math.
阿南塔拉曼接受过古典钢琴家训练，并认真考虑过从事音乐而不是数学事业。
And throughout this latest project, Monk acted as an archaeologist of sorts, excavating all of Mirzakhani’s papers to develop an intimate understanding of her work — and, through her work, of her.
在这个最新的项目中，蒙克扮演了某种考古学家的角色，挖掘了米尔扎哈尼的所有论文，以深入了解她的工作——并通过她的工作，深入了解她。
I ’m not sure I can think of a mathematical concept more alluring or romantic than infinity.
我不确定我能想到一个比无穷大更诱人或更浪漫的数学概念。
Mathematicians have known since the 1870s that infinity is, to put it mildly, really weird.
自 1870 年代以来，数学家们就知道无穷大，说得客气一点，真的很奇怪。
For one thing, it comes in many different shapes and sizes.
一方面，它有许多不同的形状和尺寸。
The set of whole numbers (0, 1, 2, 3…) is the same size as the set of fractions, but smaller than the set of real numbers.
整数集（0、1、2、3…）与分数集大小相同，但小于实数集。
Beyond these more familiar types of infinity, there’s a menagerie of larger infinities (with fun names like “strong” and “supercompact”) that are nearly impossible to describe.
除了这些更熟悉的无穷大类型之外，还有一系列几乎无法描述的更大的无穷大（具有有趣的名称，如“强”和“超紧凑”）。
In the 1930s, Kurt Gödel proved that the mathematical universe is fundamentally unknowable in its entirety.
20 世纪 30 年代，库尔特·哥德尔 (Kurt Gödel) 证明，数学宇宙从根本上来说是完全不可知的。
There are parts of it that we can never access: reams of true statements that can’t be proved.
其中有些部分我们永远无法访问：大量无法证明的真实陈述。
But how close can mathematicians get to understanding it?
但数学家对它的理解能有多接近呢？
The different types of infinities give them a way to test their limits, and to decide whether the mathematical universe is nicely ordered, and therefore something they can more or less comprehend, or hopelessly chaotic.
不同类型的无穷大为他们提供了一种测试自身极限的方法，并确定数学宇宙是否有序，因此他们或多或少可以理解，或者是无可救药的混乱。
Gregory Barber reported on how a team of mathematicians recently invented two new types of infinity that, they claim, don’t behave the way you’d expect.
格雷戈里·巴伯 (Gregory Barber) 报道了一组数学家最近如何发明了两种新型无穷大，他们声称，这两种无穷大的表现方式与您预期的不同。
This research program is more experimental and controversial than the rest of mathematics: “If mathematics is a tapestry sewn together by traditional assumptions that everyone agrees on, the higher reaches of the infinite are its tattered fringes,” Barber writes.
这个研究项目比其他数学领域更具实验性和争议性：“如果数学是一张由每个人都同意的传统假设缝制而成的挂毯，那么无限的更高境界就是它破烂的边缘，”巴伯写道。
But if these mathematicians are right, it suggests that the mathematical universe is full of all sorts of mysteries and monsters we haven’t so much as caught a glimpse of.
但如果这些数学家是对的，那就表明数学宇宙充满了我们从未见过的各种奥秘和怪物。
Samuel Velasco/ Quanta Magazine
塞缪尔·贝拉斯科/广达杂志
S ome of my favorite stories are those that reveal how much we still don’t know about math’s most basic building blocks.
我最喜欢的一些故事揭示了我们对数学最基本的组成部分仍然知之甚少。
For instance, mathematicians know that most numbers are irrational, meaning that they can’t be written as a fraction of two whole numbers.
例如，数学家知道大多数数字都是无理数，这意味着它们不能写成两个整数的分数。
But it’s exceedingly hard to prove this for specific numbers.
但用具体数字来证明这一点非常困难。
It took decades, for instance, to definitively show that the number e is irrational, and over a century to do the same for π.
例如，花了几十年的时间才明确证明数字 e 是无理数，而对 π 也花了一个多世纪的时间才证明它是无理数。
And mathematicians have yet to prove that π + e is irrational.
数学家还没有证明 π + e 是无理数。
Such irrationality proofs have been rare — and at times, according to longtime Quanta contributor Erica Klarreich, dramatic.
这种非理性的证明很少见，而且根据广达的长期撰稿人艾丽卡·克拉瑞奇 (Erica Klarreich) 的说法，有时甚至是戏剧性的。
When one mathematician announced his proof of a particular number’s irrationality, “the lecture quickly descended into pandemonium,” she writes.
当一位数学家宣布证明某个数字的非理性时，“讲座很快陷入混乱，”她写道。
“Mathematicians greeted his assertions with hoots of laughter, called out to friends across the room, and threw paper airplanes.” (None of the math conferences I’ve attended have featured such chaos.)
“数学家们用大笑声回应他的主张，向房间另一边的朋友们喊话，并扔纸飞机。” （我参加过的数学会议都没有出现过如此混乱的情况。）
Klarreich explains how mathematicians recently developed new, important techniques that allowed them to prove irrationality for a slew of important numbers .
Klarreich 解释了数学家最近如何开发出新的重要技术，使他们能够证明大量重要数字的非理性。
“After so many years spent peering through the fog,” she writes, “mathematicians are finally starting to clearly discern an array of landmarks in one of their most fundamental landscapes — the number line.”
她写道：“经过这么多年的迷雾观察，数学家们终于开始清楚地辨别出他们最基本的领域之一——数轴中的一系列地标。”
S peaking of not knowing something basic: I learned from another article Klarreich wrote this year that the vast majority of convex polyhedra (shapes that have flat sides and no indentations, such as the cube, tetrahedron and dodecahedron) have a really strange property.
说到不了解一些基本知识：我从 Klarreich 今年写的另一篇文章中了解到，绝大多数凸多面体（具有平坦侧面且没有凹痕的形状，例如立方体、四面体和十二面体）都具有非常奇怪的特性。
If you take such a polyhedron, it’s possible to bore a straight tunnel through it so that another, identical copy of the polyhedron can pass through.
如果你采用这样的多面体，就可以在其中钻出一条笔直的隧道，以便另一个相同的多面体副本可以通过。
That might seem really counterintuitive, but check out the video and graphics in the article to see how it happens.
这可能看起来确实违反直觉，但请查看文章中的视频和图形，了解它是如何发生的。
Mathematicians have spent centuries looking for an example of a convex polyhedron that does not have this so-called Rupert property.
数学家花了几个世纪的时间寻找不具有所谓的鲁珀特性质的凸多面体的例子。
This year, they finally found one: a shape with 90 vertices and 152 faces that its discoverers dubbed the Noperthedron .
今年，他们终于找到了一个：具有 90 个顶点和 152 个面的形状，其发现者将其称为 Noperthedron。
And in other “simple shapes can do weird things” news from 2025, Elise Cutts tells the story of how a group of mathematicians finally built a tetrahedron that can only rest on one of its four triangular sides.
在 2025 年的其他“简单形状可以做奇怪的事情”新闻中，Elise Cutts 讲述了一群数学家如何最终构建出一个只能依靠其四个三角形边之一的四面体的故事。
If you try to place it on any of its other sides, it will flip to its stable side.
如果您尝试将其放在任何其他侧面，它会翻转到其稳定的一侧。
“I didn’t expect more work to come out on tetrahedra,” one researcher told Cutts.
“我没想到会有更多关于四面体的研究成果，”一位研究人员告诉卡茨。
But that’s the thing about math: There’s always more to learn, even about things we think we fully understand.
但这就是数学的特点：总是有更多东西需要学习，即使是我们认为自己完全理解的东西。
This video showcases the direct relationship between art and math.
该视频展示了艺术与数学之间的直接关系。
It explains the group theory behind the artist Sol LeWitt’s Variations of Incomplete Open Cubes , a series of 122 wire-frame cubes that have one or more edges removed.
它解释了艺术家 Sol LeWitt 的《不完整开放立方体变体》背后的群论，该作品由一系列 122 个线框立方体组成，其中一个或多个边缘已被移除。
But it also does a great job of describing mathematical thinking and exploration more generally — and what the process looks like for developing an intuition into an informal proof.
但它也很好地描述了更普遍的数学思维和探索——以及将直觉发展为非正式证明的过程。
Anyone who does math, or follows math news, is no stranger to arxiv.org , an online repository of research papers (called preprints) that have not yet been peer-reviewed.
任何从事数学研究或关注数学新闻的人对 arxiv.org 都不陌生，这是一个尚未经过同行评审的研究论文（称为预印本）的在线存储库。
Writer Sheon Han gives a compelling account of how arxiv.org revolutionized mathematics, and of the man behind that revolution, the physicist Paul Ginsparg.
作家 Sheon Han 令人信服地讲述了 arxiv.org 如何彻底改变了数学，以及这场革命背后的人物、物理学家 Paul Ginsparg。
In 1952, just a couple of years before his death, the mathematician Alan Turing (today considered the founder of modern computer science) made a surprising foray into biology.
1952 年，就在他去世前几年，数学家艾伦·图灵（今天被认为是现代计算机科学的创始人）令人惊讶地涉足生物学。
He proposed a mathematical model to explain the formation of a zebra’s stripes, a leopard’s spots and other patterns.
他提出了一个数学模型来解释斑马条纹、豹子斑点和其他图案的形成。
The writer Rivka Galchen explains recent research that extends Turing’s model, which now allows scientists to describe other kinds of pigmentation patterns that appear throughout the natural world.
作家 Rivka Galchen 解释了最近扩展图灵模型的研究，该模型现在允许科学家描述自然世界中出现的其他种类的色素沉着模式。
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《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。
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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。