内容来源：https://www.quantamagazine.org/graduate-student-proves-the-fractal-uncertainty-principle-20260812/

内容总结：

研究生破解分形量子不确定性原理，数学难题终获突破

近日，一位年轻的研究生成功证明了适用于分形结构的不确定性原理，这一成果被认为是数学界的一项基础性突破，为理解量子混沌现象开辟了新路径。

不确定性原理是量子力学的基石之一，它指出，我们越精确地知道一个粒子的位置，就越无法确定其动量，反之亦然。大约十年前，麻省理工学院数学家Semyon Dyatlov开始探索一个问题：在混沌系统中，量子粒子是否会被困在类似分形的复杂路径上？分形是一种无论放大多少倍都保持复杂结构的形状。要解答这个问题，需要一个能处理分形的全新不确定性原理。

2016年，Dyatlov与著名数学家Jean Bourgain合作，成功证明了一维情况下的分形不确定性原理。但当时他们在新泽西州举办的研讨会上尝试将证明推广到更高维度，却以失败告终，与会者普遍认为这几乎不可能实现。

然而，多年后，当时还是麻省理工学院博士生的Alex Cohen取得了关键突破。他的论文于2025年发表在数学领域顶级期刊《数学年刊》上，成功将分形不确定性原理推广到了所有更高维度。这一成果不仅成为Cohen的博士论文，也让他年仅25岁就获得了纽约大学的助理教授职位。普林斯顿高等研究院的Peter Sarnak评价道，这是“一个基础性成果”，对一位博士生而言是“相当了不起的成就”。

从台球到分形尘埃

不确定性原理不仅存在于量子世界。声音越短促，其音调就越难以确定；短雷达脉冲能精确定位潜艇，但需要更长的信号才能判断其运动方向。这些现象都源于同一个数学根源——傅里叶变换。该变换能将任何函数分解为一系列简单波的组合，而不确定性原理就内嵌于此。

分形不确定性原理指出，如果将一个分形频率集合的波相加，得到的曲线不可能同样是分形（即不可能布满孔洞）。反之亦然。这种分形结构在混沌系统中自然出现。例如，在无摩擦的弹珠台中，球可能永远被困在三个缓冲器之间弹跳，其每次撞击点会形成类似“康托尔集”的分布，数学上称为“分形尘埃”。

然而，分形不确定性原理揭示了量子与经典混沌的根本区别：波无法像台球那样被限制在分形路径上，它最终会泄漏出去。

攻克难题的关键

Cohen于2021年进入麻省理工学院，他很快发现挑战巨大。在二维及以上空间中，许多分形（如谢尔宾斯基地毯）包含连续直线，这会导致原原理失效。Cohen创造性地提出了“线性多孔性”这一更严格的条件来规避这些反例。

证明过程同样曲折。他借鉴了Dyatlov和Bourgain一维证明中的“阻尼函数”方法，但构造这种特殊函数在高维空间中极其困难。在Dyatlov分享的Bourgain生前未发表的笔记中，Cohen找到了关键线索——通过复分析的迂回路径来间接构造阻尼函数。这个技巧实际上源于1960年代的Beurling-Malliavin定理，但Cohen最初并不知情。他事后坦言，如果知道这个方法并非秘密，自己可能早就放弃了，是“因为不知道别人尝试过，所以充满信心”。

应用前景广阔

Cohen的结果迅速被其他数学家用于解锁新证明。2025年，哈佛大学的Elena Kim和俄克拉荷马大学的Nicholas Miller利用高维分形不确定性原理，将相关结论推广到了更高维的双曲空间。

许多数学家希望借此证明1994年由Sarnak和Zeév Rudnick提出的著名猜想：经历混沌的波不仅会完全扩散，还会在整个空间均匀分布。这一猜想若获证实，将表明混沌波的微观行为看似随机，宏观上却呈现均匀性，正如印象派画家莫奈的画作——近看笔触繁多，远看色彩均匀。

专家指出，分形不确定性原理作为一个关于傅里叶分析的基础事实，其应用远不止于量子混沌，未来在依赖信号处理的各种领域都可能产生深远影响。

中文翻译：

博士生证明分形的量子不确定原理

引言

在量子尺度上，微小粒子的行为十分奇异。造成这种现象的一个原因就是不确定原理，它指出：你对量子粒子的位置知道得越多，你能知道的粒子运动速度就越少，反之亦然。最近，这一法则得到了罕见的升级。

新的不确定原理与分形有关——所谓分形，是指无论你放大多少倍，其复杂程度都不变的形状。

大约十年前，麻省理工学院的数学家谢苗·贾特洛夫一直在研究：当量子粒子和普通粒子被置于同样的混沌情境中时，它们的行为是否会有所不同。有时，一个做混沌运动的物体会被永久地困在一条类似分形的路径上。量子粒子也能做到同样的事情吗？

量子粒子倾向于像波一样扩散，这使它们的确切位置变得模糊。为了弄清楚量子粒子是否会因过度模糊而无法沿这些复杂曲折的被困路径运动，贾特洛夫需要一个全新的不确定原理——一个能够处理分形问题的不确定原理。

2016年，凭借让·布尔甘（一位著名数学家，在完成这项工作后不久便去世了）提出的关键思路，贾特洛夫证明了一维分形的不确定原理。一维分形看起来像是参差不齐的线条，这些线条可以表示物体在二维空间中运动的轨迹，比如在台球桌上滚动的球。那年秋天，贾特洛夫和布尔甘召集了来自世界各地的数学家齐聚新泽西州，举办了一场研讨会，希望将这一证明推广到更高维度。扩展后的证明可用于研究三维世界，而且本身也将成为一项具有普适性的数学工具。

但这项任务过于困难。研讨会结束时，“没有人真正相信这件事能做成，”与会者之一、索邦大学的数学家弗雷德里克·诺德说。

直到多年以后，亚历克斯·科恩——当时还是麻省理工学院的博士生——终于取得了突破。在一篇发表于2025年《数学年刊》（被广泛认为是该领域顶级期刊）的论文中，他将分形不确定原理推广到了所有更高维度。这项成果成为科恩的博士论文，也为他赢得了纽约大学助理教授的职位，年仅25岁。

分形不确定原理是“一个奠基性的结果”，高等研究院的彼得·萨纳克说——“对一个用来做博士论文的人来说，这是相当了不起的成就。”

这一原理已经揭示出量子粒子与经典粒子之间一种新的深刻差异。

弹球奇才

尽管不确定原理在粒子语境下看起来很奇怪，但它也可以以不那么神秘的形态出现。声音越短促，你对构成它的音调就越不确定。短促的雷达脉冲可以精确定位潜艇，但要想确定潜艇的移动方向，则需要更长的信号。

所有这些不确定原理，包括量子力学中的那一个，都源自同一个数学源头。这个更深层的数学不确定原理广泛适用于任何函数——或者粗略地说，任何曲线——无论它看起来多么崎岖不平、多么狂野。它来自19世纪发明的一个方程，叫作傅里叶变换。傅里叶变换以法国人约瑟夫·傅里叶命名，它将任何函数分解为一组简单波的集合，每个波具有不同的频率或音调。将这些简单波加在一起，就能得到原来的函数。

不确定原理就内置其中。一个无限延伸的简单正弦波只有一个确定的频率，因此它的傅里叶变换是一个单独的尖峰。但一声军鼓的“啪”响，在空气中传播时看起来像一个脉冲，它的傅里叶变换则具有分布很广的频率——这是一种听不出音高的声音。

一般来说，一个函数越窄，它的傅里叶变换就必须越分散，反之亦然。换句话说，你对一个函数在空间中尖峰的位置越确定，你对它的频率或随时间变化的速率就越不确定。

在量子力学中，粒子在数学上用波来描述，波峰出现在最可能找到粒子的地方。该波的傅里叶变换描述了粒子的运动。这就是为什么量子粒子的位置和动量无法同时被精确知道：它们通过傅里叶变换联系在一起。

但是，如果对一个看起来像分形的函数进行傅里叶变换，会怎样呢？

要理解这个问题，不妨想象一种简单的分形：从一条线段开始，把它切成三段，然后删除中间那段。重复这个过程：把剩下的每一段切成三段，移除中间段，如此反复。最终得到的集合叫作康托尔集。这个集合具有数学家所说的“多孔”特征——它到处都是空隙，像海绵一样，在每一个尺度上都是如此。

现在想象这个康托尔集位于x轴上，这个集合上的每个点都有一个代表某个频率的尖峰。

分形不确定原理指出：如果你将这一组类分形频率的波相加，得到的曲线不可能也看起来像分形——它不可能是多孔的。反过来也成立：如果你对类分形函数做傅里叶变换，结果不可能是一个类分形的频率集合。

这类类分形函数听起来可能很难遇到。但当数学家研究量子粒子——更一般地说，是波——在混沌情境下的行为时，它们就会出现。

就像弹球机中弹来弹去的球一样，经历混沌运动的物体通常会在整个空间中杂乱无章地运动。“如果你看看自己的轨迹，它看起来就像一个随机的涂鸦，”哈佛大学的数学家埃琳娜·金说。

但在罕见的情况下，物体可以在混沌中找到稳定。在平面的弹球机中，一个球可以永远被困在三个缓冲器之间弹来弹去。球不会重复完全相同的路径，但它会被限制在缓冲器之间，每次都撞在略微不同的位置。如果你标出球撞击每个缓冲器的位置，你会发现这些标记构成了类似康托尔集的东西。

这组标记也被称为“分形尘埃”。大多数撞到缓冲器的球都会飞走。“这就是留下的尘埃，”加州大学伯克利分校的马切伊·兹沃尔斯基说。

然而，与弹球不同，根据分形不确定原理，波无法被限制在类分形的路径上。如果你试图把波困在三个缓冲器之间，它会泄漏出来并逃逸。

“所以量子混沌和经典混沌之间存在某种差异，”贾特洛夫说。“你可以尝试用来解释这一点的一个要素就是不确定原理。”

未完成的知识

科恩于2021年来到麻省理工学院，他自称是一个“年轻而充满活力的调和分析学者”——调和分析是数学中专门研究函数及其傅里叶变换的分支领域。作为博士生，他参加了贾特洛夫关于分形不确定原理的讲座。“他每次讲座结尾都会说：‘我们没有高维分形不确定原理。我希望我们有！’”科恩说。“我当时想，好吧，这似乎是个有意思的研究方向。”

他的热情很快遇到了挑战。首先，数学家们已经知道在很多情况下，分形不确定原理在高维中会失效。科恩的第一个挑战是想出一种干净的方法来避开这些情况。

成为分形的典型要求是多孔性——也就是说，你到处看都能看到空隙。二维分形“谢尔宾斯基地毯”就是一个例子；它的构造方法是将一个正方形分成若干小方格，然后反复移除中间的那个方格。但是，虽然这种形状有很多空隙，你仍然可以画出一条完全包含在其中的直线。

这样的直线会给分形不确定原理带来麻烦。垂直线的傅里叶变换会得到一条水平线，反之亦然。在二维或更高维度中，这两条线都算作分形——因为一条线不占据面积，使周围大部分空间保持空白，这满足“有很多空隙”的条件。因此，任何包含不间断直线的分形都会打破分形不确定原理。

科恩需要一种新的、更严格的多孔性定义。他提出了所谓的“线性多孔性”——你在分形上画出的任何直线都应该遇到许多空隙。他的证明排除了所有不满足这一条件的分形，包括典型的谢尔宾斯基地毯，但一个经过修改、拥有更多空白空间的谢尔宾斯基地毯则满足这一规则。

在确定假设之后，科恩开始了实际的证明。他很快意识到，这与他之前处理过的任何傅里叶相关问题都不同。“我试着用我所有的工具去证明分形不确定原理，但没有任何一个工具能接近成功，”他说。

感到困顿的他回到贾特洛夫和布尔甘的一维证明，试图彻底理解它的运作方式。

贾特洛夫和布尔甘在证明中使用了一种不常见的方法。他们一次从类分形函数中隔离一个尖峰，并证明该尖峰的傅里叶变换会扩散开来。对每个尖峰都这样做，并考虑所有傅里叶变换如何叠加，他们证明了总的傅里叶变换不可能有足够多的零点来形成分形——不会有足够的空隙。

要隔离每个尖峰，需要构造一个非常特殊的函数，当它与原来的类分形函数相乘时，只会提取出那个尖峰，而在其他所有地方都接近零。这叫作阻尼函数，它必须量身定做才能发挥作用。“这是一个很难构造的东西，”科恩说。但他知道，如果他能在高维中构造出来，就能解开整个证明。

科恩就构造这个特殊函数的计划咨询了贾特洛夫。在布尔甘于2018年底去世之前，他也曾在这个问题上苦苦思索，并将未发表的笔记分享给了贾特洛夫。现在，贾特洛夫把它们分享给了科恩。“布尔甘是一位传奇式的分析学家，”科恩说。阅读这份笔记感觉像是“从他那里接收了这份未完成的知识”。

笔记里恰好包含了科恩需要的线索。“这让我叹为观止，”科恩说。“它真的为我解锁了这个问题。”

在读到布尔甘的笔记之前，科恩对于如何构造阻尼函数有几个想法，但它们高度复杂且精确，就像一块砖一块砖地设计建房的图纸。笔记揭示了一种意想不到的做法。它涉及绕道进入复分析——对包含负1的平方根的虚数函数的研究。这条迂回路线让科恩构建了一个更加灵活的对象，然后可以用它来间接构造阻尼函数。

有了这一洞见，科恩接下来需要找到一种方法，创造出这个灵活对象的恰到好处的版本，以产生一个合适的阻尼函数。“要构造一个具有如此特定性质的东西是极其不平凡的，非常微妙，”耶鲁大学的威廉·施拉格说，科恩在本科期间曾跟随他学习。“在二维中，没有人知道怎么做，亚历克斯提出了一个精彩的构造。”

当科恩于2023年5月将证明发布到网上时，数学界为之震惊。

“他的论文非常漂亮，给人留下了极其深刻的印象，”施拉格说。

后来，科恩发现，布尔甘笔记中向他揭示的技巧其实并非秘密。这个方法出自20世纪60年代一个著名的定理，叫作伯林-马利亚万定理。“我以为我掌握了特殊的内幕知识，”科恩说。“后来我发现这个领域的人早就知道这个策略。”

如果他知道自己所谓的内幕消息其实不是秘密，科恩可能早就放弃了。“我想我之所以很有信心，是因为我不知道别人已经试过这条路，”他说。

迷宫般的混沌

科恩分享了他的结果之后不久，其他数学家就开始利用它来解锁关于波在混沌情境中行为的新证明。

在自然界中，混沌出现在湍急的水流和天气等系统中——在这些系统中，开始时距离很近的物体很快就会走向截然不同的位置。这些系统过于复杂，无法用数学来描述。因此，寻求研究混沌的数学家们往往会求助于一种本身就内置了混沌的奇特空间，叫作双曲空间。

在双曲空间中，平行线会急剧发散，沿着它们的路径延伸会彼此越来越远。（这与球面正好相反，在球面上平行线会汇聚。）这意味着物体之间的微小分离在后续可能会变成巨大的差距——这正是混沌的标志性特征。

“当然，这看起来和天气预报毫无关系，”贾特洛夫说。但双曲空间为数学家提供了一个简单的混沌系统来探索。“我们研究我们能处理的东西，我们隔离现象。这在数学中是常见的做法。”

2017年，贾特洛夫和北京清华大学的金龙使用一维分形不确定原理证明：你永远无法将波困在双曲曲面上；它会一直扩散，直到触及每一个角落。

为此，他们设想了一个波永远不会进入的区域，即使给了波无限的时间去扩散也不会进入。当他们移除所有进入该区域的轨迹后，剩下的正是弹球示例中出现的那种分形尘埃。由于分形不确定原理禁止波被限制在分形上，这样的区域就不可能存在——波必须扩散到各处。

2025年，金与俄克拉荷马大学的尼古拉斯·米勒合作，利用科恩的高维分形不确定原理，将结果推广到了更高维的双曲空间。“这是高维版本迄今最令人瞩目的应用，”萨纳克说。

许多数学家希望能证明，经历混沌的波不仅会完全扩散，而且会完全均匀地扩散到整个空间。这正是萨纳克和泽夫·鲁德尼克1994年提出的一个著名猜想的内容，他们对该问题与数论的联系感兴趣。“这是一个巨大的开放猜想，”金说。“这是一个非常困难的问题，很多人都在乎它。”

证明这个猜想将表明，经历混沌的波在微小尺度上的运动看起来出人意料且随机，但在宏观层面却是简单的。

“就像那些莫奈的画，”贾特洛夫说。“你走近看，画面上有很多微观特征，有很多笔触。”但如果你退后一步眯起眼睛，“它看起来就是均匀的颜色。”这与经典混沌不同，在经典混沌中，物体会陷入复杂而精细的路径中。

分形不确定原理的影响不会止步于量子混沌。傅里叶分析的工具被广泛应用于数学的许多领域，并且是所有依赖信号处理的行业的基础。“这是关于傅里叶分析的一个基础性事实，”萨纳克说。“我们还没有看到它的全部应用。”

英文来源：

Graduate Student Proves a Quantum Uncertainty Principle for Fractals
Introduction
At the quantum scale, tiny particles behave in bizarre ways. One reason for this is the uncertainty principle, which says that the more you know about where a quantum particle is, the less you can know about how fast it’s moving, and vice versa. Recently, this rule got a rare upgrade.
The new uncertainty principle relates to fractals, shapes that remain equally complex no matter how much you zoom in on them.
Around a decade ago, Semyon Dyatlov, a mathematician at the Massachusetts Institute of Technology, was studying whether quantum particles behave differently than ordinary particles when put into the same chaotic situations. Sometimes, an object moving chaotically can become trapped into following a fractal-like path forever. Could quantum particles do the same?
Quantum particles tend to spread out like waves, which blurs their exact location. To figure out whether quantum particles blur too much to take on these intricate trapped paths, Dyatlov needed a new uncertainty principle — one that could tackle fractals.
In 2016, with key ideas from Jean Bourgain — a renowned mathematician who died shortly after this work — Dyatlov proved the fractal uncertainty principle for one-dimensional fractals, which look like jagged lines. These lines can represent the paths taken by objects moving in two dimensions, like balls traveling around a billiard table. That fall, Dyatlov and Bourgain gathered mathematicians from around the world in New Jersey for a workshop, hoping to extend the proof to higher dimensions. An extended proof could be used to study the three-dimensional world and would become a universal mathematical tool in its own right.
But the task proved too difficult. By the end of the workshop, “nobody really believed that it could be done,” said one of the attendees, Frédéric Naud, a mathematician from Sorbonne University.
It wasn’t until years later that Alex Cohen, while a doctoral student at MIT, finally made a breakthrough. In a paper published in 2025 in the Annals of Mathematics, widely considered to be the field’s top journal, he extended the fractal uncertainty principle to all higher dimensions. The result became Cohen’s thesis and earned him an assistant professorship at New York University at the age of 25.
The fractal uncertainty principle is “a foundational result,” said Peter Sarnak of the Institute for Advanced Study — “a pretty remarkable achievement for a guy in his thesis.”
Already, this principle has revealed a new deep way that quantum particles differ from classical ones.
Pinball Wizard
Though the uncertainty principle may seem strange in the context of particles, it can also crop up in less mysterious forms. The briefer a sound, the less sure you can be about the tones that make it up. A short radar pulse can accurately locate a submarine, but it takes a longer signal to determine where it’s moving.
All these uncertainty principles, including the quantum one, arise from the same mathematical source. This deeper mathematical uncertainty principle applies broadly to any function — or any curve, roughly speaking, no matter how bumpy and wild it looks. It comes from an equation invented in the 19th century called the Fourier transform. Named for the Frenchman Joseph Fourier, the Fourier transform decomposes any function into a set of simple waves, each with a different frequency, or tone. Add those simple waves together, and you’ll get back your original function.
The uncertainty principle comes built in. A simple sine wave, which spreads infinitely throughout space, has one definite frequency, so its Fourier transform is a single peak. But the snap of a snare drum looks like a pulse as it travels through the air. Its Fourier transform has a wide spread of frequencies — a sound with no discernible pitch.
In general, the narrower a function, the more spread out its Fourier transform must be, and vice versa. Or, in other words, the more certain you are about where a function peaks in space, the less certain you can be about its frequency, or how it’s changing in time.
In quantum mechanics, a particle is described mathematically as a wave, with peaks in places where it’s most likely to be found. The Fourier transform of that wave describes the particle’s motion. This is why a quantum particle’s position and momentum can’t both be precisely known at once: They are related by a Fourier transform.
But what if you take the Fourier transform of a function that looks like a fractal?
To make sense of the question, it helps to imagine a simple kind of fractal: Start with a line and cut it into three segments. Then delete the center segment. Repeat the process: Cut each remaining segment into three, remove the center, and so on. The result is a fractal set called the Cantor set. The set has the feature of being what mathematicians call porous — it’s full of holes, like a sponge, at every scale.
Now imagine that this Cantor set lives on the x-axis, and at each point on this set is a peak representing a frequency.
The fractal uncertainty principle says that if you add together waves of that fractal-like set of frequencies, the resulting curve cannot also look like a fractal — it cannot be porous. The reverse also holds: If you take the Fourier transform of a fractal-like function, the result cannot be a fractal-like set of frequencies.
Such fractal-like functions might sound hard to come by. But they pop up when mathematicians study what happens to quantum particles — or waves more generally — in chaotic situations.
Like a ball bouncing around a pinball machine, an object experiencing chaos will often travel erratically around the entire space. “If you look at your path, it’s going to look like a random scribble,” said Elena Kim, a mathematician at Harvard University.
But in rare instances, an object can find stability in the chaos. In a flat pinball machine, a ball could stay trapped bouncing between three bumpers forever. The ball wouldn’t repeat the exact same path, but it would stay confined between the bumpers, bouncing off a slightly different spot each time. And if you marked each spot where the ball hits each bumper, you’d find that the marks make up something like the Cantor set.
This collection of marks is also called fractal dust. Most balls that hit the bumpers will fly off. “This is the dust that’s left,” said Maciej Zworski of the University of California, Berkeley.
Unlike a pinball, however, a wave cannot be confined to a fractal-like path, according to the fractal uncertainty principle. If you tried to trap a wave between three bumpers, it would leak out and escape.
“So there’s something different about quantum and classical [chaos],” Dyatlov said. “And one ingredient that you can try to use for that would be uncertainty principles.”
Unfinished Knowledge
Cohen arrived at MIT in 2021 a self-described “young, energetic harmonic analyst” — harmonic analysis being a field of mathematics dedicated to studying functions and their Fourier transforms. As a doctoral student, he sat in Dyatlov’s talks about the fractal uncertainty principle. “He would end every talk being like, ‘We don’t have a higher-dimensional fractal uncertainty principle. I wish we had that!’” Cohen said. “I was like, OK, this seems like a fun thing to work on.”
His enthusiasm was soon checked. For one thing, mathematicians already knew of many situations where the fractal uncertainty principle would fail in higher dimensions. Cohen’s first challenge was to come up with a clean way to avoid these cases.
The typical requirement of being a fractal is to be porous — meaning that there are holes everywhere you look. The two-dimensional fractal called the Sierpiński carpet is an example; it’s built by dividing a square into a grid of smaller squares and removing the middle square, repeatedly. But while this shape has many holes, it’s also possible to draw a straight line that is fully contained within it.
Lines like these spell trouble for the fractal uncertainty principle. The Fourier transform of a vertical line returns a horizontal line, and vice versa. In two or more dimensions, both of these lines count as fractals — that’s because a line takes up no area and leaves most of the surrounding space empty, which satisfies the condition of having many holes. So any fractal that contains uninterrupted lines breaks the fractal uncertainty principle.
Cohen needed a new, more stringent kind of porosity. He came up with what he called “line porosity” — any line you draw on the fractal should encounter many holes. His proof excludes any fractals that don’t have this condition, including the typical Sierpiński carpet, but a modified Sierpiński carpet with much more empty space satisfies the rule.
With his assumptions in place, Cohen moved on to the actual proof. He quickly realized that this was unlike any Fourier-related problem he had worked on before. “I tried using all my tools to prove the fractal uncertainty principle, and none of them even came remotely close to working,” he said.
Feeling stuck, he went back to Dyatlov and Bourgain’s proof of the principle in one dimension and sought to understand exactly how it worked.
Dyatlov and Bourgain used an uncommon method in their proof. It involved isolating one peak from a fractal-like function at a time and showing that the Fourier transform of that peak would spread out. Doing this for all peaks, and considering how the Fourier transforms would add together, they proved that the total Fourier transform could never equal zero often enough to form a fractal — there wouldn’t be enough holes.
Isolating each peak required constructing a very specific function that, when multiplied by the original fractal-like function, would pull out just the peak and be close to zero everywhere else. This is called a damping function, and it needs to be perfectly tailor-made to work. “This is a challenging thing to construct,” Cohen said. But he knew that if he could do it in higher dimensions, he could unlock the entire proof.
Cohen consulted Dyatlov about his plan to construct this special function. Before Bourgain died in late 2018, he too struggled with this problem, and he shared his unpublished notes with Dyatlov. Now, Dyatlov shared them with Cohen. “Bourgain was a legendary analyst,” Cohen said. Reading the note felt like “receiving this unfinished knowledge from him.”
The notes contained exactly the hint Cohen needed. “It just blew my mind,” Cohen said. “It really unlocked the problem for me.”
Before reading Bourgain’s note, Cohen had a few ideas for how to construct the damping function, but they were highly complicated and precise, like the designs for building a house brick by brick. The note revealed an unexpected way to do it. It involved taking a detour into complex analysis — the study of functions of imaginary numbers, which include the square root of negative 1. This detour allowed Cohen to build a much more flexible object, which could then be used to construct the damping function indirectly.
Armed with this insight, Cohen then needed to find a way to create just the right version of this flexible object to produce a proper damping function. “To construct something like this that has very specific properties is highly nontrivial. It’s delicate,” said Wilhelm Schlag of Yale University, with whom Cohen studied as an undergraduate. “In two dimensions, nobody knew how to do that, and Alex came up with a brilliant construction of such a thing.”
Cohen stunned the math world when he posted the proof online in May 2023.
“His paper is very beautiful, and it made a huge impression,” Schlag said.
Later, Cohen found out that the trick revealed to him in Bourgain’s note wasn’t actually a secret. The method came from a well-known theorem from the 1960s called the Beurling-Malliavin theorem. “I thought that I had this special inside knowledge,” Cohen said. “I found out later that everyone in the field already knew about this strategy.”
Had he known that his insider tip was no secret, Cohen might have given up too soon. “I think I had a lot of confidence because I didn’t know other people had tried it,” he said.
Funhouse Chaos
Soon after Cohen shared his result, other mathematicians started using it to unlock new proofs about how waves behave in chaotic situations.
In nature, chaos appears in systems like turbulent water and the weather — situations where objects that start close together quickly end up in drastically different places. These systems are too complex to describe mathematically. Instead, mathematicians seeking to study chaos often turn to an odd kind of space that has chaos built in, called hyperbolic space.
In hyperbolic space, parallel lines diverge dramatically, getting farther from each other as you follow their paths. (It’s the opposite of a sphere, where parallel lines converge.) This means that small separations between objects can become huge down the line — the telltale sign of chaos.
“Of course, this looks nothing like predicting weather,” Dyatlov said. But hyperbolic spaces offer mathematicians a simple chaotic system to explore. “We study what we can handle, and we isolate phenomena. That’s a common thing to do in math.”
In 2017, Dyatlov and Long Jin from Tsinghua University in Beijing used the one-dimensional fractal uncertainty principle to prove that you can never trap a wave on a hyperbolic surface; it will always spread out until it touches every corner.
To do so, they imagined a region on the surface that a wave never enters, even after having infinite time to spread out. When they removed all the trajectories that entered that region, what remained was the same sort of fractal dust that appeared in the pinball example. Since the fractal uncertainty principle forbids a wave from being trapped on a fractal, no such region can exist — the wave must spread everywhere.
In 2025, Kim, along with Nicholas Miller of the University of Oklahoma, used Cohen’s higher-dimensional fractal uncertainty principle to extend the results to hyperbolic spaces of higher dimensions. “That’s the most spectacular application of the higher-dimensional one so far,” Sarnak said.
Many mathematicians are hoping to prove that waves experiencing chaos not only spread out completely but also spread out exactly evenly over the entire space. This is the statement of a famous 1994 conjecture by Sarnak and Zeév Rudnick, who were interested in the problem’s connections to number theory. “This is a huge open conjecture,” Kim said. “It’s a really difficult problem that a lot of people care about.”
Proving this conjecture would show that waves experiencing chaos move in ways that look surprising and random at small scales but are simple at the macroscopic level.
“It’s like those Monet paintings,” Dyatlov said. “When you go very close, they have a lot of microscopic features. There are a lot of brushstrokes.” But if you stand back and squint, “it just looks uniformly colored.” This is different from classical chaos, where objects can get stuck in complicated and detailed paths.
The impact of the fractal uncertainty principle won’t stop with quantum chaos. The tools of Fourier analysis are used in many fields of mathematics and form the backbone of any industry that relies on signal processing. “It’s a foundational fact about Fourier analysis,” Sarnak said. “We haven’t seen all the applications yet.”