内容来源：https://www.quantamagazine.org/the-year-in-mathematics-20251218/

内容总结：

年度数学研究回顾：从少年天才到未解之谜，数学疆域持续拓展

过去一年，数学领域见证了多项突破性进展与动人故事，展现了这门学科作为创造性艺术与探索工具的无限魅力。

少年解谜：17岁攻克数十年难题
来自巴哈马的17岁少女汉娜·开罗，在家庭自学期间通过在线资源深入数学世界。移居加州后，她在加州大学伯克利分校攻读研究生课程，并成功解决了一个悬置40年的函数猜想，提出了资深数学家未曾发现的反例。她的经历凸显了数学作为“可独自探索的思想宇宙”如何为孤独的学习者打开广阔天地。

理论交融：“十杯马提尼”问题揭示物理深层次结构
数学家们通过数论工具，最终严格证明了量子物理中电子能级可形成著名的康托尔集分形结构，解决了长期悬而未决的“十杯马提尼”难题。该证明不仅完善了2004年的原有工作，更深刻揭示了数论与量子物理之间“不可思议”的内在联系。

薪火相传：已故天才数学家思想重获新生
已故菲尔兹奖得主玛丽亚姆·米尔扎哈尼在双曲几何领域的开创性工作，正由后续数学家继承与发展。两位女性数学家纳里尼·阿南塔拉曼和劳拉·蒙克深入研读其遗留手稿，持续推进该领域研究，体现了数学思想跨越时空的传承力量。

基础探索：理性与混沌的永恒之问
数学家在无穷概念的探索中不断挑战认知边界。新近提出的两种新型无穷概念行为反常，暗示数学宇宙可能比想象中更为混沌莫测。与此同时，在数论基础领域，数学家发展出新工具，成功证明了一系列重要数字的无理性，增进了对“数轴”这一基本景观的理解。

几何趣闻：“不可穿越”的立体与“单面稳定”四面体
研究人员首次发现了一个具有90个顶点、152个面的凸多面体无法让自身的复制品穿过，结束了长达数个世纪的“鲁珀特性质”搜索。此外，数学家构造出仅能稳定立于单一面的四面体，再次证明即使对看似简单的几何对象，人类认知仍存未知。

这些进展共同表明，数学既是严谨的逻辑体系，也是充满想象力与人文温度的创造性探索，其边界仍在不断重塑与扩展之中。

中文翻译：

卡洛斯·阿罗霍为《量子杂志》撰稿
17岁的汉娜·开罗解开了一道重大数学谜题
数学本质上是一门艺术。如同画家、音乐家或作家，数学家也在创造并探索新世界。他们不断试探想象力的边界，然后将其向前推进。他们与数千年的历史对话，与不断流变的概念、品味和风尚互动。
在某种程度上，对美、真理与意义的艺术性追求，正是《量子》每一则数学故事的核心。这一点在我今年最喜爱的文章之一中展现得淋漓尽致——凯文·哈特尼特撰文讲述了数学家汉娜·开罗如何在年仅17岁时，解决了调和分析领域的一个重要难题。
开罗在巴哈马长大，在家自学数学，主要通过观看可汗学院的视频，并在网上搜寻一切能找到的学习资源。她感到在家学习的经历异常孤独且束缚。“那种千篇一律的感觉无法摆脱，”她告诉哈特尼特，“无论我做什么，我都在同一个地方，做着几乎相同的事情。我非常孤立，而且我无法真正改变这种状况。”
除了学习数学。数学给了她所需的逃离，一个可以自由漫游的完整宇宙——用开罗的话说，一个“我可以独自探索的思想世界”。在此，你很难不将数学视为艺术：一种实验新思想、应对一个并非总是合乎逻辑的世界的方式。
而且，关键在于质疑假设。十几岁时，开罗移居加利福尼亚，在加州大学伯克利分校攻读研究生课程，并遇到了一个关于函数行为的、存在了40年的猜想。经过数月的持续努力，她构建了一个反例，推翻了那些经验更丰富的数学家们未曾察觉的猜想。正如艺术品常常与其创作者密不可分一样，开罗因其独特经历，对她所研究的函数提出了全新的见解，从而证明这些函数的行为方式比数学家们想象的更为反直觉。
这往往是数学成功的关键所在。

魏安进为《量子杂志》撰稿
“十杯马提尼”证明：数论如何解释量子分形
数学也兼具美感与奇异。我记得第一次听说“十杯马提尼问题”的解决方案时——该结果涉及电子能级如何形成著名的康托尔集分形图案——我震惊不已。这让我想起了尤金·维格纳那篇关于“数学在自然科学中不可思议的有效性”的著名文章，探讨了抽象的数学如何常常神秘地为理解自然界提供完美的语言。康托尔集竟然出现在量子物理学中薛定谔方程的解里，还能帮助理解晶体中的电子在磁铁附近的行为——这真的可能吗？
在一篇引人入胜的文章中，林迪·邱和《量子》特约撰稿人乔·豪利特探讨了这个问题。该问题曾因极其难解而闻名，以至于一位数学家悬赏十杯马提尼，奖励能解决它的人。该问题最初于2004年得到解决，但证明方式连其作者之一斯韦特兰娜·吉托米尔斯基娅都感到不满意。邱和豪利特写道，该证明“像一块拼凑的被子，每一块都由不同的论证缝合而成”，并且无法应用于更普遍和现实的场景中——这促使吉托米尔斯基娅在20年后重新研究它。如今，她和同事们提出了一个新的、更有力的证明，揭示了数论与量子物理学之间这种奇特的联系，并确立了其深刻性与真实性。
在此过程中，邱和豪利特带领读者踏上了一段旅程，涉及形如蝴蝶翅膀的图形、一个名为“侏儒怪”的计算器，以及道格拉斯·霍夫施塔特那本引人入胜的著作《哥德尔、埃舍尔、巴赫》。

克里斯蒂娜·阿米蒂奇/《量子杂志》；图片来源（从左至右）：欧莱雅-联合国教科文组织“世界杰出女科学家成就奖”基金会、扬·冯德拉科、P. 安贝尔/法兰西公学院
数学天才早逝多年后，她的思想重获新生
数学并非在真空中产生。它受思想哲学的影响，并最终与人息息相关。有时，革命者会出现，引导数学家们思考特定领域数代之久。
玛丽亚姆·米尔扎哈尼便是这样的革命者之一。在读研究生期间，她革新了双曲几何领域，开创了突破性的技术，用于理解数学和物理学中无处不在的复杂曲面。她因此成为首位获得菲尔兹奖的女性。但她在40岁时去世，未能充分探索这些发现的深远影响。
今年早些时候，豪利特审视了她的遗产，以及另外两位数学家——娜丽妮·阿南塔拉曼和劳拉·蒙克——如何接续她的工作，旨在更好地理解双曲曲面的世界。豪利特巧妙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。
值得一提的是，米尔扎哈尼对文学有着深厚的热情，一度希望成为一名作家。阿南塔拉曼曾接受古典钢琴家训练，并认真考虑过从事音乐而非数学事业。而在此最新项目中，蒙克扮演了类似考古学家的角色，仔细研读米尔扎哈尼的所有论文，以深入理解她的工作——并通过她的工作，理解她这个人。

魏安进/《量子杂志》
数学主要是混沌还是有序？
我想不出比“无穷”更迷人、更浪漫的数学概念了。
自19世纪70年代以来，数学家们就知道，说得委婉些，无穷确实非常奇特。首先，它有许多不同的形状和大小。整数集（0, 1, 2, 3…）与分数集大小相同，但小于实数集。除了这些较为熟悉的无穷类型之外，还有一系列更大的无穷（拥有诸如“强”和“超紧”等有趣名称），几乎无法描述。
20世纪30年代，库尔特·哥德尔证明，数学宇宙从根本上是无法完全被认知的。其中有些部分我们永远无法触及：大量真实的陈述无法被证明。但数学家们能多接近理解它呢？不同类型的无穷为他们提供了一种测试自身极限的方式，并用以判断数学宇宙是井然有序（因而或多或少可以理解），还是无可救药地混沌。
格雷戈里·巴伯报道了一个数学家团队最近如何发明了两种新型的无穷，据称它们的行为方式出人意料。这项研究计划比数学的其他部分更具实验性和争议性：“如果数学是一幅由传统共识缝合而成的织锦，那么无穷的更高层次就是其破损的边缘，”巴伯写道。但如果这些数学家是对的，那就意味着数学宇宙充满了我们甚至未曾瞥见的各种奥秘与怪物。

塞缪尔·贝拉斯科/《量子杂志》
有理还是无理？这个基础数学问题花了几十年才找到答案。
我最喜欢的一些故事，揭示了我们对数学最基本构成要素仍知之甚少。例如，数学家们知道大多数数是无理数，意味着它们不能写成两个整数的分数形式。但要为特定数证明这一点极其困难。例如，证明数e是无理数花了几十年，证明π是无理数则用了一个多世纪。而数学家们尚未证明π + e是无理数。
此类无理数证明一直很罕见——并且，据《量子》长期撰稿人埃里卡·克拉尔赖希所述，有时还颇具戏剧性。当一位数学家宣布他证明了某个特定数是无理数时，“讲座很快陷入一片混乱，”她写道，“数学家们对他的断言报以哄堂大笑，隔着房间呼唤朋友，还扔起了纸飞机。”（我参加过的数学会议从未出现过如此混乱的场面。）
克拉尔赖希解释了数学家们最近如何开发出新的重要技术，使他们能够证明一系列重要数字的无理性。“在迷雾中摸索了这么多年之后，”她写道，“数学家们终于开始在他们最基本的领域之一——数轴上——清晰地辨认出一系列地标。”

首个被发现无法穿过自身的形状
说到对基础事物的未知：我从克拉尔赖希今年写的另一篇文章中了解到，绝大多数凸多面体（具有平坦侧面且无凹陷的形状，如立方体、四面体和十二面体）都有一个非常奇特的性质。如果你取这样一个多面体，可以在其上钻一个笔直的隧道，使得另一个完全相同的该多面体副本能够穿过。这听起来可能非常反直觉，但请查看文章中的视频和图形以了解其原理。
几个世纪以来，数学家们一直在寻找一个不具有这种所谓“鲁珀特性”的凸多面体例子。今年，他们终于找到了一个：一个拥有90个顶点和152个面的形状，其发现者称之为“诺珀特面体”。
在2025年其他关于“简单形状能做奇怪事情”的新闻中，埃莉斯·卡茨讲述了一群数学家如何最终构建出一个只能以其四个三角形面之一稳定放置的四面体。如果你试图将它放在其他任何面上，它都会翻转到那个稳定的面。“我没想到四面体方面还会有更多工作出现，”一位研究人员告诉卡茨。但这就是数学的特点：即使是我们认为已经完全理解的事物，也总有更多东西需要学习。

英文来源：

Carlos Arrojo for Quanta Magazine
At 17, Hannah Cairo Solved a Major Math Mystery
Mathematics is, at its core, an art. Like painters, musicians or writers, mathematicians create and explore new worlds. They test, and then push past, the limits of their imagination. They engage with thousands of years of history, with concepts and tastes and fashions that are constantly in flux.
This artistic pursuit of beauty, truth and meaning is what every Quanta math story is about, to some extent. 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.
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. She found the homeschooling experience overwhelmingly lonely and confining. “There was this inescapable sameness,” she told Hartnett. “No matter what I did, I was in the same place doing mostly the same things. I was very isolated, and nothing I could do could really change that.”
Except studying math. 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.
And, crucially, of questioning assumptions. 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. After several months of persistent work, she constructed a counterexample to the conjecture that more seasoned mathematicians had missed. 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.
That’s often what success in math is all about.
Wei-An Jin for Quanta Magazine
‘Ten Martini’ Proof Uses Number Theory To Explain Quantum Fractals
Math is also beautiful and strange. 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. 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. 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?
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. The problem was originally solved in 2004, but in a way that one of the proof’s authors, Svetlana Jitomirskaya, found unsatisfying. The proof “was a patchwork quilt, each square stitched out of distinct arguments,” Chiou and Howlett write. 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. 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.
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.
Kristina Armitage/Quanta Magazine; sources (from left): Fondation L’Oréal For Women in Science, Jan Vondrák, P. Imbert/Collège de France
Years After the Early Death of a Math Genius, Her Ideas Gain New Life
Math doesn’t happen in a vacuum. It’s influenced by philosophies of thought and, ultimately, people. Sometimes revolutionaries come on the scene, guiding how mathematicians think about a particular field for generations.
One of those revolutionaries was Maryam Mirzakhani. 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. She became the first woman to win a Fields Medal, in part for this work. But she died at age 40, before she could fully explore the ramifications of these discoveries.
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. Howlett expertly weaves together the stories of these three women and the research that unites them across time and space.
It’s also worth noting that Mirzakhani had a deep passion for literature, at one point hoping to be a writer. Anantharaman trained as a classical pianist and seriously considered pursuing a career in music instead of math. 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.
Wei-An Jin/Quanta Magazine
Is Mathematics Mostly Chaos or Mostly Order?
I’m not sure I can think of a mathematical concept more alluring or romantic than infinity.
Mathematicians have known since the 1870s that infinity is, to put it mildly, really weird. For one thing, it comes in many different shapes and sizes. 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. 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.
In the 1930s, Kurt Gödel proved that the mathematical universe is fundamentally unknowable in its entirety. There are parts of it that we can never access: reams of true statements that can’t be proved. But how close can mathematicians get to understanding it? 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.
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. 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. 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.
Samuel Velasco/Quanta Magazine
Rational or Not? This Basic Math Question Took Decades To Answer.
Some of my favorite stories are those that reveal how much we still don’t know about math’s most basic building blocks. For instance, mathematicians know that most numbers are irrational, meaning that they can’t be written as a fraction of two whole numbers. But it’s exceedingly hard to prove this for specific numbers. It took decades, for instance, to definitively show that the number e is irrational, and over a century to do the same for π. And mathematicians have yet to prove that π + e is irrational.
Such irrationality proofs have been rare — and at times, according to longtime Quanta contributor Erica Klarreich, dramatic. When one mathematician announced his proof of a particular number’s irrationality, “the lecture quickly descended into pandemonium,” she writes. “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.)
Klarreich explains how mathematicians recently developed new, important techniques that allowed them to prove irrationality for a slew of important numbers. “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.”
First Shape Found That Can’t Pass Through Itself
Speaking 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. 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. That might seem really counterintuitive, but check out the video and graphics in the article to see how it happens.
Mathematicians have spent centuries looking for an example of a convex polyhedron that does not have this so-called Rupert property. This year, they finally found one: a shape with 90 vertices and 152 faces that its discoverers dubbed the 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. If you try to place it on any of its other sides, it will flip to its stable side. “I didn’t expect more work to come out on tetrahedra,” one researcher told Cutts. But that’s the thing about math: There’s always more to learn, even about things we think we fully understand.