Graduate Student Proves a Quantum Uncertainty Principle for Fractals

Ada Zejun Shen/ Quanta Magazine

沉泽君/广达杂志

A t the quantum scale, tiny particles behave in bizarre ways.

在量子尺度上，微小粒子的行为方式很奇怪。

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.

造成这种情况的原因之一是不确定性原理，即您对量子粒子的位置了解得越多，您对其移动速度的了解就越少，反之亦然。

Recently, this rule got a rare upgrade.

最近，这一规则得到了罕见的升级。

The new uncertainty principle relates to fractals, shapes that remain equally complex no matter how much you zoom in on them.

新的不确定性原理与分形有关，无论你放大多少形状，它们仍然同样复杂。

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.

大约十年前，麻省理工学院的数学家谢苗·迪亚特洛夫（Semyon Dyatlov）正在研究量子粒子在置于相同的混沌环境中时是否与普通粒子表现不同。

Sometimes, an object moving chaotically can become trapped into following a fractal-like path forever.

有时，混乱移动的物体可能会永远陷入分形路径中。

Could quantum particles do the same?

量子粒子能做同样的事吗？

The uncertainty principle tells us 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.

不确定性原理告诉我们，你对量子粒子的位置了解得越多，你对它移动速度的了解就越少，反之亦然。

Recently, this rule got a rare upgrade.

最近，这一规则得到了罕见的升级。

Quantum particles tend to spread out like waves, which blurs their exact location.

量子粒子往往像波一样扩散，这模糊了它们的确切位置。

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.

为了弄清楚量子粒子是否过于模糊而无法走上这些复杂的被困路径，迪亚特洛夫需要一种新的不确定性原理——一种可以解决分形问题的原理。

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.

2016 年，凭借著名数学家 Jean Bourgain（一位在这项工作完成后不久去世）的关键思想，Dyatlov 证明了一维分形（看起来像锯齿状线）的分形不确定性原理。

These lines can represent the paths taken by objects moving in two dimensions, like balls traveling around a billiard table.

这些线可以代表二维运动物体所采取的路径，就像球绕着台球桌移动一样。

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.

那年秋天，迪亚特洛夫和布尔根在新泽西州召集了来自世界各地的数学家参加了一个研讨会，希望将证明扩展到更高的维度。

An extended proof could be used to study the three-dimensional world and would become a universal mathematical tool in its own right.

扩展证明可用于研究三维世界，并且其本身将成为一种通用的数学工具。

But the task proved too difficult.

但事实证明这项任务太艰巨了。

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.

到研讨会结束时，“没有人真正相信这是可以做到的”，其中一位与会者、索邦大学的数学家弗雷德里克·诺德 (Frédéric Naud) 说。

Alex Cohen’s proof of the fractal uncertainty principle has already proven influential across math.

亚历克斯·科恩（Alex Cohen）对分形不确定性原理的证明已经证明对数学界具有影响力。

It wasn’t until years later that Alex Cohen , while a doctoral student at MIT, finally made a breakthrough.

直到多年后，麻省理工学院的博士生亚历克斯·科恩终于取得了突破。

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.

在 2025 年发表在被广泛认为是该领域顶级期刊的《数学年鉴》上的一篇论文中，他将分形不确定性原理扩展到了所有更高维度。

The result became Cohen’s thesis and earned him an assistant professorship at New York University at the age of 25.

这一成果成为科恩的论文，并让他在 25 岁时获得了纽约大学助理教授的职位。

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.”

高级研究所的彼得·萨纳克（Peter Sarnak）说，分形不确定性原理是“一个基础性的结果”——“对于他的论文中的一个人来说，这是一个相当了不起的成就。”

Already, this principle has revealed a new deep way that quantum particles differ from classical ones.

这一原理已经揭示了量子粒子与经典粒子不同的新的深层方式。

Though the uncertainty principle may seem strange in the context of particles, it can also crop up in less mysterious forms.

尽管不确定性原理在粒子的背景下可能看起来很奇怪，但它也可能以不那么神秘的形式出现。

The briefer a sound, the less sure you can be about the tones that make it up.

声音越短，你对构成它的音调就越不确定。

A short radar pulse can accurately locate a submarine, but it takes a longer signal to determine where it’s moving.

短雷达脉冲可以准确定位潜艇，但需要较长的信号才能确定其移动位置。

All these uncertainty principles, including the quantum one, arise from the same mathematical source.

所有这些不确定性原理，包括量子原理，都源于相同的数学来源。

This deeper mathematical uncertainty principle applies broadly to any function — or any curve, roughly speaking, no matter how bumpy and wild it looks.

这种更深层次的数学不确定性原理广泛适用于任何函数——或者任何曲线，粗略地说，无论它看起来多么崎岖和狂野。

It comes from an equation invented in the 19th century called the Fourier transform .

它来自 19 世纪发明的一个方程，称为傅里叶变换。

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.

傅里叶变换以法国人约瑟夫·傅里叶的名字命名，可将任何函数分解为一组简单的波，每个波具有不同的频率或音调。

Add those simple waves together, and you’ll get back your original function.

将这些简单的波浪加在一起，你就会恢复原来的功能。

The uncertainty principle comes built in.

不确定性原理是内置的。

A simple sine wave, which spreads infinitely throughout space, has one definite frequency, so its Fourier transform is a single peak.

一个简单的正弦波在空间中无限传播，具有一个确定的频率，因此它的傅里叶变换是单峰。

But the snap of a snare drum looks like a pulse as it travels through the air.

但小军鼓在空气中传播时发出的声音看起来就像是脉冲。

Its Fourier transform has a wide spread of frequencies — a sound with no discernible pitch.

它的傅里叶变换具有很宽的频率范围——一种没有明显音调的声音。

In general, the narrower a function, the more spread out its Fourier transform must be, and vice versa.

一般来说，函数越窄，其傅里叶变换必须越分散，反之亦然。

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.

或者，换句话说，你对一个函数在空间中的峰值位置越确定，你对其频率或它随时间的变化方式就越不确定。

In quantum mechanics, a particle is described mathematically as a wave, with peaks in places where it’s most likely to be found.

在量子力学中，粒子在数学上被描述为波，在最有可能发现它的地方有波峰。

The Fourier transform of that wave describes the particle’s motion.

该波的傅立叶变换描述了粒子的运动。

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.

这就是为什么不能同时精确地知道量子粒子的位置和动量：它们通过傅里叶变换相关联。

But what if you take the Fourier transform of a function that looks like a fractal?

但是，如果对看起来像分形的函数进行傅立叶变换会怎样呢？

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.

为了理解这个问题，想象一种简单的分形会有所帮助：从一条线开始，将其切成三段。

Then delete the center segment.

然后删除中心部分。

Repeat the process: Cut each remaining segment into three, remove the center, and so on.

重复此过程：将剩余的每个片段切成三部分，去除中心，依此类推。

The result is a fractal set called the Cantor set.

结果是一个称为康托集的分形集。

The set has the feature of being what mathematicians call porous — it’s full of holes, like a sponge, at every scale.

该集合具有数学家所说的多孔的特征——它在各个尺度上都充满了孔，就像海绵一样。

Now imagine that this Cantor set lives on the x -axis, and at each point on this set is a peak representing a frequency.

现在想象一下这个康托集位于 x 轴上，并且该集上的每个点都有一个代表频率的峰值。

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.

分形不确定性原理表明，如果将类似分形的频率集的波加在一起，所得曲线就不可能看起来像分形——它不可能是多孔的。

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.

反之亦然：如果对类分形函数进行傅立叶变换，则结果不可能是类分形频率集。

Such fractal-like functions might sound hard to come by.

这种类似分形的函数听起来可能很难实现。

But they pop up when mathematicians study what happens to quantum particles — or waves more generally — in chaotic situations.

但当数学家研究量子粒子（或更普遍的波）在混沌情况下发生的情况时，它们就会突然出现。

Like a ball bouncing around a pinball machine, an object experiencing chaos will often travel erratically around the entire space.

就像弹球机周围弹跳的球一样，经历混乱的物体通常会在整个空间中不稳定地移动。

“If you look at your path, it’s going to look like a random scribble,” said Elena Kim , a mathematician at Harvard University.

哈佛大学数学家埃琳娜·金 (Elena Kim) 表示：“如果你仔细观察你的路径，它看起来就像是随机的涂鸦。”

But in rare instances, an object can find stability in the chaos.

但在极少数情况下，物体可以在混乱中找到稳定。

In a flat pinball machine, a ball could stay trapped bouncing between three bumpers forever.

在平板弹球机中，球可能永远被困在三个缓冲器之间弹跳。

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.

球不会重复完全相同的路径，但它会被限制在保险杠之间，每次从略有不同的位置弹起。

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.

如果你标记球撞击每个保险杠的每个点，你会发现这些标记组成了类似于康托集的东西。

This collection of marks is also called fractal dust.

这种标记的集合也称为分形灰尘。

Most balls that hit the bumpers will fly off.

大多数撞击保险杠的球都会飞走。

“This is the dust that’s left,” said Maciej Zworski of the University of California, Berkeley.

“这就是留下的灰尘，”加州大学伯克利分校的 Maciej Zworski 说。

Elena Kim, a mathematician at Harvard University, has used the fractal uncertainty principle to understand chaotic systems in unconventional spaces.

哈佛大学数学家埃琳娜·金 (Elena Kim) 使用分形不确定性原理来理解非常规空间中的混沌系统。

Unlike a pinball, however, a wave cannot be confined to a fractal-like path, according to the fractal uncertainty principle.

然而，与弹球不同的是，根据分形不确定性原理，波不能被限制在类似分形的路径上。

If you tried to trap a wave between three bumpers, it would leak out and escape.

如果你试图将波浪限制在三个保险杠之间，它就会泄漏并逃逸。

“So there’s something different about quantum and classical [chaos],” Dyatlov said.

“所以量子和经典[混沌]有一些不同，”迪亚特洛夫说。

“And one ingredient that you can try to use for that would be uncertainty principles.”

“你可以尝试使用的一个要素是不确定性原理。”

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.

科恩于 2021 年来到麻省理工学院，他自称是“年轻​​、精力充沛的调和分析师”——调和分析是一个致力于研究函数及其傅立叶变换的数学领域。

As a doctoral student, he sat in Dyatlov’s talks about the fractal uncertainty principle.

作为一名博士生，他参加了迪亚特洛夫关于分形不确定性原理的演讲。

“He would end every talk being like, ‘We don’t have a higher-dimensional fractal uncertainty principle.

“他每次谈话结束时都会说，‘我们没有更高维的分形不确定性原理。

I wish we had that!’” Cohen said.

我希望我们能拥有它！’”科恩说。

“I was like, OK, this seems like a fun thing to work on.”

“我当时想，好吧，这看起来是一件有趣的事情。”

His enthusiasm was soon checked.

他的热情很快就受到了抑制。

For one thing, mathematicians already knew of many situations where the fractal uncertainty principle would fail in higher dimensions.

一方面，数学家已经知道分形不确定性原理在更高维度中会失效的许多情况。

Cohen’s first challenge was to come up with a clean way to avoid these cases.

科恩面临的第一个挑战是想出一种干净的方法来避免这些情况。

The typical requirement of being a fractal is to be porous — meaning that there are holes everywhere you look.

分形的典型要求是多孔的——这意味着你看到的任何地方都有洞。

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.

称为 Sierpiński 地毯的二维分形就是一个例子；它是通过将一个正方形分成较小的正方形网格并重复删除中间的正方形来构建的。

But while this shape has many holes, it’s also possible to draw a straight line that is fully contained within it.

但是，虽然这个形状有很多孔，但也可以画一条完全包含在其中的直线。

Lines like these spell trouble for the fractal uncertainty principle.

像这样的线条给分形不确定性原理带来了麻烦。

The Fourier transform of a vertical line returns a horizontal line, and vice versa.

垂直线的傅里叶变换返回水平线，反之亦然。

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.

在二维或更多维度中，这两条线都算作分形——这是因为一条线不占据任何面积，并且周围的大部分空间都是空的，这满足了有很多洞的条件。

So any fractal that contains uninterrupted lines breaks the fractal uncertainty principle.

因此，任何包含不间断线的分形都打破了分形不确定性原理。

Cohen needed a new, more stringent kind of porosity.

科恩需要一种新的、更严格的孔隙率。

He came up with what he called “line porosity” — any line you draw on the fractal should encounter many holes.

他提出了所谓的“线孔隙率”——你在分形上绘制的任何线都应该遇到很多孔。

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.

他的证明排除了任何不具备此条件的分形，包括典型的谢尔宾斯基地毯，但具有更多空白空间的改良谢尔宾斯基地毯满足了该规则。

With his assumptions in place, Cohen moved on to the actual proof.

有了假设，科恩就开始进行实际证明。

He quickly realized that this was unlike any Fourier-related problem he had worked on before.

他很快意识到这与他之前研究过的任何与傅立叶相关的问题都不同。

“I tried using all my tools to prove the fractal uncertainty principle, and none of them even came remotely close to working,” he said.

“我尝试使用我所有的工具来证明分形不确定性原理，但没有一个工具能够发挥作用，”他说。

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.

由于感到陷入困境，他回到了迪亚特洛夫和布尔干在一维原理上的证明，并试图准确地理解它是如何运作的。

Dyatlov and Bourgain used an uncommon method in their proof.

Dyatlov 和 Bourgain 在证明中使用了一种不常见的方法。

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.

它涉及一次从类似分形的函数中分离出一个峰值，并显示该峰值的傅里叶变换会展开。

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.

对所有峰值执行此操作，并考虑傅里叶变换如何加在一起，他们证明总傅里叶变换永远不可能等于零，足以形成分形 - 不会有足够的孔。

Jean Bourgain, seen here in 2012, was an author on more than 500 papers spanning much of mathematics.

Jean Bourgain（2012 年出现在此处）是 500 多篇涉及数学领域论文的作者。

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.

隔离每个峰值需要构建一个非常具体的函数，当该函数乘以原始的分形函数时，将只提取峰值，而其他地方都接近于零。

This is called a damping function, and it needs to be perfectly tailor-made to work.

这称为阻尼功能，需要完美定制才能发挥作用。

“This is a challenging thing to construct,” Cohen said.

“这是一个具有挑战性的事情，”科恩说。

But he knew that if he could do it in higher dimensions, he could unlock the entire proof.

但他知道，如果他能在更高的维度上做到这一点，他就能解开整个证明。

Cohen consulted Dyatlov about his plan to construct this special function.

科恩向迪亚特洛夫咨询了建造这一特殊功能的计划。

Before Bourgain died in late 2018, he too struggled with this problem, and he shared his unpublished notes with Dyatlov.

布尔甘于 2018 年底去世之前，他也曾与这个问题作过斗争，他与迪亚特洛夫分享了他未发表的笔记。

Now, Dyatlov shared them with Cohen.

现在，迪亚特洛夫与科恩分享了它们。

“Bourgain was a legendary analyst,” Cohen said.

“布尔根是一位传奇的分析师，”科恩说。

Reading the note felt like “receiving this unfinished knowledge from him.”

读这张纸条感觉就像“从他那里得到了这些未完成的知识”。

The notes contained exactly the hint Cohen needed.

这些笔记正是科恩所需要的提示。

“It just blew my mind,” Cohen said.

“这让我大吃一惊，”科恩说。

“It really unlocked the problem for me.”

“这确实为我解开了这个问题。”

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.

在阅读布尔根的笔记之前，科恩对如何构建阻尼功能有一些想法，但它们非常复杂和精确，就像一块砖建造房屋的设计一样。

The note revealed an unexpected way to do it.

这张纸条揭示了一种意想不到的方法。

It involved taking a detour into complex analysis — the study of functions of imaginary numbers, which include the square root of negative 1.

它涉及到复杂分析——虚数函数的研究，其中包括负 1 的平方根。

This detour allowed Cohen to build a much more flexible object, which could then be used to construct the damping function indirectly.

这种绕行使得科恩能够构建一个更加灵活的对象，然后可以使用它来间接构造阻尼函数。

Semyon Dyatlov, a mathematician at the Massachusetts Institute of Technology, proved the one-dimensional fractal uncertainty principle in 2016.

麻省理工学院数学家谢苗·迪亚特洛夫 (Semyon Dyatlov) 于 2016 年证明了一维分形不确定性原理。

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.

有了这种洞察力，科恩需要找到一种方法来创建这个柔性物体的正确版本，以产生适当的阻尼功能。

“To construct something like this that has very specific properties is highly nontrivial.

“构建这样具有非常特定属性的东西是非常重要的。

It’s delicate,” said Wilhelm Schlag of Yale University, with whom Cohen studied as an undergraduate.

这很微妙，”耶鲁大学的威廉·施拉格（Wilhelm Schlag）说道，科恩本科时就师从该大学。

“In two dimensions, nobody knew how to do that, and Alex came up with a brilliant construction of such a thing.”

“在二维空间中，没有人知道如何做到这一点，而亚历克斯想出了一个出色的构建这样的东西。”

Cohen stunned the math world when he posted the proof online in May 2023.

2023 年 5 月，科恩在网上发布了这一证明，震惊了数学界。

“His paper is very beautiful, and it made a huge impression,” Schlag said.

“他的论文非常漂亮，给人留下了深刻的印象，”施拉格说。

Later, Cohen found out that the trick revealed to him in Bourgain’s note wasn’t actually a secret.

后来，科恩发现布尔根在笔记中向他透露的伎俩实际上并不是秘密。

The method came from a well-known theorem from the 1960s called the Beurling-Malliavin theorem.

该方法源自 20 世纪 60 年代著名的 Beurling-Malliavin 定理。

“I thought that I had this special inside knowledge,” Cohen said.

“我以为我拥有这种特殊的内幕知识，”科恩说。

“I found out later that everyone in the field already knew about this strategy.”

“后来我发现这个领域的每个人都已经知道这个策略了。”

Had he known that his insider tip was no secret, Cohen might have given up too soon.

如果科恩知道他的内幕消息并不是秘密，他可能会很快放弃。

“I think I had a lot of confidence because I didn’t know other people had tried it,” he said.

“我认为我很有信心，因为我不知道其他人已经尝试过，”他说。

Soon after Cohen shared his result, other mathematicians started using it to unlock new proofs about how waves behave in chaotic situations.

科恩分享他的结果后不久，其他数学家开始使用它来解锁有关波在混沌情况下如何表现的新证据。

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.

在自然界中，混乱出现在湍流水和天气等系统中——开始时靠近的物体很快就会出现在截然不同的地方。

These systems are too complex to describe mathematically.

这些系统太复杂，无法用数学方法描述。

Instead, mathematicians seeking to study chaos often turn to an odd kind of space that has chaos built in, called hyperbolic space.

相反，寻求研究混沌的数学家经常转向一种内置混沌的奇怪空间，称为双曲空间。

In hyperbolic space, parallel lines diverge dramatically, getting farther from each other as you follow their paths.

在双曲空间中，平行线会急剧发散，当您沿着平行线的路径行驶时，它们之间的距离会越来越远。

(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.

（它与球体相反，球体中平行线会聚。）这意味着物体之间的微小间隔可能会在直线上变得很大——这是混乱的迹象。

“Of course, this looks nothing like predicting weather,” Dyatlov said.

“当然，这看起来与预测天气完全不同，”迪亚特洛夫说。

But hyperbolic spaces offer mathematicians a simple chaotic system to explore.

但双曲空间为数学家提供了一个简单的混沌系统来探索。

“We study what we can handle, and we isolate phenomena.

“我们研究我们可以处理的问题，并隔离现象。

That’s a common thing to do in math.”

这在数学中是很常见的事情。”

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.

2017年，北京清华大学的Dyatlov和Long Jin使用一维分形不确定性原理证明了永远不可能在双曲面上捕获波；它总是会散开，直到触及每个角落。

To do so, they imagined a region on the surface that a wave never enters, even after having infinite time to spread out.

为此，他们想象了表面上的一个区域，即使在有无限的时间传播之后，波也永远不会进入。

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.

当他们移除进入该区域的所有轨迹时，剩下的就是弹球示例中出现的相同类型的分形灰尘。

Since the fractal uncertainty principle forbids a wave from being trapped on a fractal, no such region can exist — the wave must spread everywhere.

由于分形不确定性原理禁止波被困在分形上，因此不可能存在这样的区域——波必须传播到各处。

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.

2025 年，Kim 与俄克拉荷马大学的 Nicholas Miller 一起利用科恩的高维分形不确定性原理将结果扩展到更高维度的双曲空间。

“That’s the most spectacular application of the higher-dimensional one so far,” Sarnak said.

“这是迄今为止高维应用最引人注目的应用，”萨纳克说。

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.

许多数学家希望证明经历混沌的波不仅完全扩散，而且在整个空间上精确均匀地扩散。

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.

这是 Sarnak 和 Zeév Rudnick 1994 年提出的一个著名猜想的表述，他们对该问题与数论的联系很感兴趣。

“This is a huge open conjecture,” Kim said.

“这是一个巨大的公开猜想，”金说。

“It’s a really difficult problem that a lot of people care about.”

“这确实是一个很多人关心的难题。”

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.

证明这一猜想将表明，经历混沌的波的运动方式在小尺度上看起来令人惊讶和随机，但在宏观层面上却很简单。

“It’s like those Monet paintings,” Dyatlov said.

“这就像莫奈的画作，”迪亚特洛夫说。

“When you go very close, they have a lot of microscopic features.

“当你走得很近时，它们有很多微观特征。

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.

有很多笔触。”但如果你退后一步眯起眼睛，“它看起来只是颜色均匀。”这与经典的混沌不同，在经典的混沌中，对象可能会陷入复杂而详细的路径中。

The impact of the fractal uncertainty principle won’t stop with quantum chaos.

分形不确定性原理的影响不会随着量子混沌而停止。

The tools of Fourier analysis are used in many fields of mathematics and form the backbone of any industry that relies on signal processing.

傅里叶分析工具用于许多数学领域，并构成任何依赖信号处理的行业的支柱。

“It’s a foundational fact about Fourier analysis,” Sarnak said.

“这是傅里叶分析的基本事实，”萨纳克说。

“We haven’t seen all the applications yet.”

“我们还没有看到所有的申请。”

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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。