Networks Hold the Key to a Decades-Old Problem About Waves

Samuel Velasco/Michael Kanyongolo/ Quanta Magazine

塞缪尔·贝拉斯科/迈克尔·坎永格洛/广达杂志

Two centuries ago, Joseph Fourier gave mathematicians a magical technique.

两个世纪前，约瑟夫·傅立叶赋予了数学家一种神奇的技巧。

He conjectured that it’s possible to write almost any function as a sum of simple waves, a trick now called the Fourier transform .

他推测几乎任何函数都可以写成简单波的和，这种技巧现在称为傅里叶变换。

These days, the Fourier transform is used to understand everything from the chemical makeup of distant stars to what’s happening far beneath the Earth’s crust .

如今，傅立叶变换被用来理解一切事物，从遥远恒星的化学组成到地壳深处正在发生的事情。

“Fourier series are everywhere in mathematics,” said Mehtaab Sawhney of Columbia University.

“傅里叶级数在数学中无处不在，”哥伦比亚大学的 Mehtaab Sawhney 说。

“It’s part of the faith of mathematicians that Fourier series are important.”

“数学家相信傅立叶级数很重要。”

Yet certain fundamental questions about the Fourier transform have remained stubbornly, and mysteriously, unanswerable.

然而，关于傅立叶变换的某些基本问题仍然顽固而神秘地无法解答。

In 1965, the mathematician Sarvadaman Chowla posed one such question.

1965 年，数学家 Sarvadaman Chowla 提出了这样一个问题。

He wanted to know how small an extremely simple type of Fourier transform — a sum of cosine waves — could get.

他想知道一种极其简单的傅立叶变换（余弦波之和）可以有多小。

His problem sounded straightforward.

他的问题听起来很简单。

But somehow, it wasn’t.

但不知何故，事实并非如此。

“The question is a bit of bait,” Sawhney said; it was designed to illuminate just how little mathematicians know.

“这个问题有点像诱饵，”索尼说。它的目的是为了说明数学家的知识是多么的少。

“Because we can’t show this, we clearly don’t understand the structure of these [sums] at all.”

“因为我们无法证明这一点，所以我们显然根本不了解这些[总和]的结构。”

For decades, mathematicians struggled with Chowla’s cosine problem.

几十年来，数学家们一直在努力解决乔拉的余弦问题。

It became a benchmark for Fourier analysis techniques, used to explore how well they could detect deeper structure in sequences of numbers.

它成为傅里叶分析技术的基准，用于探索它们如何能够检测数字序列中更深层的结构。

The results were discouraging.

结果令人沮丧。

“Progress was completely anemic,” said Tom Sanders of the University of Oxford.

“进展完全乏力，”牛津大学的汤姆·桑德斯说。

In September, that suddenly changed.

到了九月，情况突然发生了变化。

Four mathematicians — Zhihan Jin , Aleksa Milojević , István Tomon , and Shengtong Zhang — posted the first significant advance on the problem in 20 years.

四位数学家——金志瀚、阿列克萨·米洛耶维奇、伊斯特万·托蒙和张盛通——发表了 20 年来在这个问题上的第一个重大进展。

Their strategy had almost nothing to do with traditional Fourier analysis.

他们的策略几乎与传统的傅立叶分析无关。

In fact, before last summer, the foursome had never even heard of Chowla’s cosine problem.

事实上，在去年夏天之前，四人甚至从未听说过乔拉的余弦问题。

In the early 1950s, Chowla and his fellow number theorist Nesmith Ankeny wanted to use the Fourier transform to better understand patterns in sets of numbers.

20 世纪 50 年代初，Chowla 和他的数论学家同事 Nesmith Ankeny 希望使用傅里叶变换来更好地理解数字集合中的模式。

Consider the set consisting of the numbers 2, 3, and 8.

考虑由数字 2、3 和 8 组成的集合。

First, use each number in the set to define a cosine wave — 2 gives you cos(2 x ), for instance.

首先，使用集合中的每个数字来定义余弦波 - 例如，2 给出 cos(2 x )。

Then add up all your waves to get cos(2 x ) + cos(3 x ) + cos(8 x ).

然后将所有波相加得到 cos(2 x ) + cos(3 x ) + cos(8 x )。

This is just another way of writing your original set as a Fourier series.

这只是将原始集合编写为傅里叶级数的另一种方式。

The series is highly structured: All the waves are cosines, and because there are no numbers in front of any of the cosines, all the waves are the same size.

该级数是高度结构化的：所有波都是余弦，并且由于任何余弦前面没有数字，所以所有波的大小相同。

“It’s the simplest possible type of Fourier series you can have,” said Benjamin Bedert of the University of Cambridge.

剑桥大学的本杰明·贝德特 (Benjamin Bedert) 表示：“这是最简单的傅立叶级数类型。”

“And in general, we know quite a lot about Fourier series.”

“总的来说，我们对傅立叶级数了解很多。”

The new wave defined by cos(2 x ) + cos(3 x ) + cos(8 x ) has peaks and valleys that reveal interesting properties of the original set of numbers.

由 cos(2 x ) + cos(3 x ) + cos(8 x ) 定义的新波具有峰值和谷值，揭示了原始数字集的有趣属性。

So Ankeny and Chowla sought to test how much they really understood about such a series.

因此，安克尼和乔拉试图测试他们对这样一个系列的真正理解程度。

They wondered: For any set of N integers, what is the lowest value that the sum will ever take?

他们想知道：对于任何 N 个整数的集合，其总和的最小值是多少？

It’s easy to figure out the sum’s maximum.

很容易算出总和的最大值。

When x is zero, any cosine wave hits its maximum at 1.

当 x 为零时，任何余弦波在 1 处达到最大值。

So our sum of three cosine waves gives us 1 + 1 + 1, or 3.

所以三个余弦波的和就是 1 + 1 + 1，即 3。

Similarly, a sum of 10 million cosine waves has a maximum of 10 million.

同样，1000万个余弦波的总和最多为1000万个。

For any set of N integers, the maximum is simply N .

对于任何 N 个整数集合，最大值就是 N 。

Yet understanding the cosine sum’s minimum is surprisingly difficult.

然而，理解余弦和的最小值却出奇地困难。

While the different waves all hit their maximum simultaneously at least once (when x is zero), this is not true for the minimum.

虽然不同的波至少同时达到最大值一次（当 x 为零时），但对于最小值则不然。

Perhaps the lowest points of the different waves will still align enough to produce a very low sum.

也许不同波浪的最低点仍会足够对齐以产生非常低的总和。

Or perhaps the waves will interfere with each other so that it becomes impossible for the sum to get too low.

或者也许这些波会互相干扰，这样总和就不可能太低。

Zhihan Jin (left), Aleksa Milojević (top right), and István Tomon set out to solve a major problem in graph theory.

金志瀚（左）、Aleksa Milojević（右上）和 István Tomon 着手解决图论中的一个重大问题。

In doing so, they unwittingly came up with a new approach to a seemingly unrelated problem on the Fourier transform.

在此过程中，他们无意中提出了一种新方法来解决傅里叶变换上看似不相关的问题。

From left: Courtesy of Zhihan Jin; Archives of the Mathematisches Forschungsinstitut Oberwolfach; Livia Tomon-Horvath

左起：金志瀚提供； Oberwolfach 数学研究所档案馆；利维娅·托蒙-霍瓦斯

In 1952, Ankeny and Chowla conjectured that just as the maximum gets higher and higher as the number of integers in the original set gets bigger, the minimum should get lower and lower .

1952年，Ankeny和Chowla推测，正如随着原始集合中整数的数量变大，最大值会越来越高，最小值也应该越来越低。

This was proved several years later — prompting Chowla to sharpen the question in 1965.

几年后这一点得到了证明，促使 Chowla 在 1965 年进一步尖锐化了这个问题。

He wanted to know exactly how fast the minimum drops as N grows.

他想确切地知道随着 N 的增长，最小值下降的速度有多快。

He knew of sets of N integers whose cosine sum had a minimum value around −$latex \sqrt{\textit{N}}$.

他知道 N 个整数的集合，其余弦和的最小值约为 −$latex \sqrt{\textit{N}}$。

Every other set he could think of dipped even lower, leading him to conjecture that for any set of N positive integers, the minimum of the corresponding cosine sum must be below −$latex \sqrt{\textit{N}}$.

他能想到的所有其他集合都下降得更低，这导致他推测对于任何 N 个正整数集合，相应余弦和的最小值必须低于 -$latex \sqrt{\textit{N}}$。

Over the ensuing decades, a few mathematicians chipped away at the problem.

在接下来的几十年里，一些数学家逐渐解决了这个问题。

But by the mid-2000s, there was still a massive gulf between what they were able to prove and what Chowla had predicted.

但到了 2000 年代中期，他们能够证明的内容与 Chowla 的预测之间仍然存在巨大差距。

According to the latest bound , proved in 2004 by Imre Ruzsa of the Alfréd Rényi Institute of Mathematics in Hungary, a sum of 10 20 cosines — that’s a 1 with 20 zeros after it, about the number of molecules in a cubic inch of air — must have a minimum value smaller than about −7.

根据匈牙利 Alfred Rényi 数学研究所的 Imre Ruzsa 于 2004 年证明的最新界限 ，10 20 个余弦之和（即一个 1 后面有 20 个零，大约是一立方英寸空气中的分子数量）的最小值必须小于约 -7。

By comparison, Chowla had predicted that the minimum would have to dip below −10 10 .

相比之下，Chowla 预测最小值必须降至 -10 10 以下。

And yet, for the past 20 years, Ruzsa’s result has represented the pinnacle of progress on Chowla’s cosine problem.

然而，在过去的 20 年里，Ruzsa 的结果代表了 Chowla 余弦问题进展的顶峰。

Then an entirely unrelated research program finally broke the barrier.

然后，一个完全不相关的研究项目终于打破了障碍。

The program dealt with networks of nodes and edges called graphs.

该程序处理称为图的节点和边的网络。

Last summer, two sets of graph theorists — Jin, Milojević, and Tomon in Europe, and Zhang at Stanford University — were enthusiastically making progress on one of graph theory’s most central questions.

去年夏天，两组图论学家——欧洲的 Jin、Milojević 和 Tomon，以及斯坦福大学的张——正在热情地在图论最核心的问题之一上取得进展。

The “MaxCut” problem is about the optimal way to cut a graph into two parts so that there are as many edges as possible connecting the parts.

“MaxCut”问题是关于将图切割成两部分的最佳方法，以便有尽可能多的边连接这些部分。

It’s a basic question about the structure of a graph, with real-world applications: A graph’s MaxCut might represent an efficient circuit design, for instance, or the lowest-energy state of a system of particles.

这是一个关于图结构和实际应用的基本问题：例如，图的 MaxCut 可能代表有效的电路设计，或者粒子系统的最低能量状态。

Mark Belan/Samuel Velasco/ Quanta Magazine

马克·贝兰/塞缪尔·贝拉斯科/广达杂志

But there’s no one-size-fits-all approach to finding a graph’s MaxCut, at least at the moment.

但至少目前还没有一种通用的方法来查找图的 MaxCut。

(It’s what’s known as an NP-hard problem.) And so mathematicians instead attempt to estimate the MaxCut for specific classes of graphs.

（这就是所谓的 NP 难问题。）因此数学家转而尝试估计特定类别图的 MaxCut。

In 2003, Benjamin Sudakov , a mathematician at the Swiss Federal Institute of Technology Zurich who would later mentor Jin, Milojević, and Tomon, posed a conjecture with three colleagues about the MaxCut of a particular kind of graph .

2003 年，瑞士苏黎世联邦理工学院的数学家本杰明·苏达科夫（Benjamin Sudakov）（后来成为 Jin、Milojević 和 Tomon 的导师）与三位同事提出了关于特定类型图的 MaxCut 的猜想。

This graph had no cliques — clusters of nodes that are all connected to one another.

该图没有派系——所有节点都相互连接。

A cluster of interconnected nodes forms a clique.

互连节点的集群形成一个派系。

This graph has a five-node clique, highlighted in red.

该图有一个五节点团，以红色突出显示。

Last July, more than two decades later, Zhang proved a new bound on the MaxCut for such graphs.

二十多年后的去年七月，张证明了此类图的 MaxCut 的新界限。

A few days later, Jin, Milojević, and Tomon improved on his result .

几天后，Jin、Milojević 和 Tomon 的成绩有所提高。

To do this, the researchers investigated important quantities called eigenvalues.

为此，研究人员研究了称为特征值的重要量。

Eigenvalues provide information about a graph’s structure.

特征值提供有关图结构的信息。

For example, the largest eigenvalue counts the number of edges in the graph; the second-largest measures the graph’s connectivity.

例如，最大特征值统计图中的边数；第二大衡量图表的连通性。

Jin, Milojević, Tomon, and Zhang focused on the negative eigenvalues, building on a recent line of research that had linked them to a graph’s MaxCut.

Jin、Milojević、Tomon 和Zhang 重点研究了负特征值，他们基于最近的一系列研究，将负特征值与图的 MaxCut 联系起来。

Their analysis of these eigenvalues ultimately allowed them to prove their new results.

他们对这些特征值的分析最终使他们能够证明他们的新结果。

The mathematicians decided to combine their separate results into a joint paper.

数学家们决定将他们各自的结果合并成一篇联合论文。

But before they could finish, they received an unexpected email about Chowla’s cosine problem.

但在他们完成之前，他们意外地收到了一封关于 Chowla 余弦问题的电子邮件。

The email was from Ilya Shkredov , a mathematician at Purdue University in Indiana.

这封电子邮件来自印第安纳州普渡大学的数学家伊利亚·什克雷多夫 (Ilya Shkredov)。

Shkredov, a number theorist, pointed out that Chowla’s cosine problem could be reformulated in terms of graphs.

数论学家什克雷多夫指出，乔拉的余弦问题可以用图表重新表述。

Not the general kinds of graphs that the team was studying, but a special type of graph invented in 1878 by the mathematician Arthur Cayley.

该团队正在研究的不是一般类型的图表，而是数学家 Arthur Cayley 于 1878 年发明的一种特殊类型的图表。

To build a Cayley graph, imagine you’re once again working with the set {2, 3, 8}.

要构建凯莱图，假设您再次使用集合 {2, 3, 8}。

Start with a bunch of nodes — it doesn’t really matter how many, so long as the number of nodes is larger than the biggest integer in the set.

从一堆节点开始——有多少节点并不重要，只要节点的数量大于集合中的最大整数即可。

(Depending on how you want to use the graph, you might also add some extra constraints on this number; in this case, it also has to be prime.) Next, arrange the nodes in a circle and label each one with an integer.

（根据您想要如何使用该图，您还可以对此数字添加一些额外的约束；在这种情况下，它也必须是质数。）接下来，将节点排列在一个圆圈中，并用一个整数标记每个节点。

Then place an edge between two nodes if the difference between them is in the original set.

如果两个节点之间的差异在原始集合中，则在两个节点之间放置一条边。

So the nodes labeled 1 and 3 will be connected by an edge, because they differ by 2, and 2 is in the set {2, 3, 8}.

因此标记为 1 和 3 的节点将通过一条边连接，因为它们相差 2，并且 2 在集合 {2, 3, 8} 中。

By the 1970s, mathematicians had figured out that embedded within the structure of Cayley graphs is information about the Fourier series from Chowla’s problem.

到了 20 世纪 70 年代，数学家们发现凯莱图的结构中嵌入了 Chowla 问题中的傅立叶级数信息。

A Cayley graph’s eigenvalues, it turns out, correspond exactly to different values that the cosine sum can have.

事实证明，凯莱图的特征值与余弦和可以具有的不同值完全对应。

The smallest eigenvalue therefore tells you how low the cosine sum can get.

因此，最小特征值告诉您余弦和可以达到多低。

“It’s a well-known thing,” Milojević said.

“这是众所周知的事情，”米洛耶维奇说。

“The connection is very classical.”

“这种联系非常经典。”

It allowed mathematicians to reframe the problem.

它使数学家能够重新构建问题。

If they could show that the smallest eigenvalue of a Cayley graph gets very small, it would mean that the cosine sum has to get very small as well — precisely what Chowla’s cosine problem is all about.

如果他们能够证明凯莱图的最小特征值变得非常小，那就意味着余弦和也必须变得非常小——这正是乔拉余弦问题的全部内容。

But no one could figure out how to exploit that connection.

但没有人能弄清楚如何利用这种联系。

“You try to hit a nail with a hammer only once you have a hammer,” Sudakov said.

“只有当你拥有锤子时，你才会尝试用锤子敲钉子，”苏达科夫说。

Mathematicians didn’t have a way of analyzing the lowest eigenvalue accurately enough to find out what they wanted to know about the minimum of the cosine sum.

数学家没有足够准确地分析最低特征值的方法来找出他们想要了解的余弦和最小值的信息。

Shengtong Zhang made major progress on the famous “MaxCut” problem, a fundamental question about graphs that has a host of practical applications.

张盛通在著名的“MaxCut”问题上取得了重大进展，这是一个具有大量实际应用的图的基本问题。

But in their work on the MaxCut of graphs, Jin, Milojević, Tomon, and Zhang had unwittingly produced a hammer.

但在有关图表 MaxCut 的工作中，Jin、Milojević、Tomon 和Zhang 无意中制作了一把锤子。

While studying how the eigenvalues of a graph relate to its structure, they’d discovered that any graph that doesn’t have a low eigenvalue must be dominated by cliques.

在研究图的特征值与其结构的关系时，他们发现任何不具有低特征值的图一定是由派系主导的。

Shkredov, reading their proof, realized that this meant that the team had actually reframed Chowla’s cosine problem once more: There was no longer any need to analyze the eigenvalue directly.

Shkredov 阅读了他们的证明，意识到这意味着团队实际上再次重新构建了 Chowla 的余弦问题：不再需要直接分析特征值。

Instead, they just had to prove that their Cayley graphs didn’t have any large cliques.

相反，他们只需要证明他们的凯莱图没有任何大的派系。

That would imply that the graphs each had a very low eigenvalue, finally enabling them to exploit the link between Chowla’s conjecture and graph theory.

这意味着每个图都有一个非常低的特征值，最终使他们能够利用乔拉猜想和图论之间的联系。

From then on, “I think the main obstacle was believing we can do it,” Tomon said.

从那时起，“我认为主要的障碍是相信我们能做到，”托蒙说。

When Shkredov sent his email, the mathematicians were all on vacation.

当什克雷多夫发送电子邮件时，数学家们都在度假。

But Tomon, who was visiting his home city of Budapest, found time to toy with the Cayley graph.

但是正在访问他的家乡布达佩斯的托蒙抽出了时间摆弄凯利图。

After a bit of thinking, “it just clicked,” he said.

经过一番思考后，他说：“就这样了。”

To see how Tomon’s idea works, let’s go back to our Cayley graph for the set {2, 3, 8}.

为了了解 Tomon 的想法是如何运作的，让我们回到集合 {2, 3, 8} 的凯莱图。

Remember that proving Chowla’s conjecture means showing that the graph’s smallest eigenvalue gets very low.

请记住，证明 Chowla 猜想意味着表明图的最小特征值变得非常低。

So first assume the opposite: that none of the eigenvalues are low.

因此，首先假设相反的情况：没有一个特征值是低的。

You’ll want to show that this assumption will eventually lead to a contradiction.

你需要证明这个假设最终会导致矛盾。

Based on the team’s work on MaxCut, if the Cayley graph has no small eigenvalues, then it must have a large clique — say, five nodes that are all connected to each other.

根据团队在 MaxCut 上的工作，如果凯莱图没有小的特征值，那么它一定有一个大的派系——比如，五个节点都相互连接。

This, in turn, means that if you take any two of those nodes, the difference between their integer labels is 2, 3, or 8.

反过来，这意味着如果您采用这些节点中的任意两个，它们的整数标签之间的差为 2、3 或 8。

Using a completely different set of techniques, Benjamin Bedert inched even closer to resolving Chowla’s cosine problem.

本杰明·贝德特 (Benjamin Bedert) 使用一套完全不同的技术，距离解决乔拉的余弦问题又近了一步。

But now add 1 to each node to get a new set of five nodes.

但现在向每个节点添加 1，以获得一组新的五个节点。

They’ll differ by the same amounts as the first set, meaning that they, too, will form a clique.

它们的差异与第一组相同，这意味着它们也将形成一个派系。

Keep going, and you’ll generate more and more cliques.

继续下去，你会产生越来越多的派系。

But there’s a problem: Cliques have lots of edges, while the Cayley graph, based on how it’s defined, has relatively few edges, which obey a very particular structure.

但有一个问题：派系有很多边，而凯莱图根据其定义方式，边相对较少，遵循非常特殊的结构。

Eventually, you’ll get so many cliques that you’ll have generated more edges than the Cayley graph can hold.

最终，你会得到如此多的派系，以至于生成的边数超出了凯莱图所能容纳的数量。

This means that the earlier assumption that there was a large clique must have been false.

这意味着之前关于存在一个大派系的假设肯定是错误的。

Which, in turn, means that the smallest eigenvalue had to be low.

反过来，这意味着最小特征值必须很低。

Once Tomon figured this out, the rest of the proof came together relatively easily.

一旦托蒙弄清楚了这一点，其余的证明就相对容易地组合在一起。

In September, he, Jin, Milojević, and Zhang posted their joint paper online.

九月，他、金、米洛耶维奇和张在网上发布了他们的联合论文。

It mainly focused on how to analyze the lowest eigenvalues of graphs — work that, for one thing, allowed them to strengthen the bounds they’d found a few months earlier on the MaxCut of graphs without cliques.

它主要关注如何分析图的最低特征值——一方面，这项工作使他们能够加强几个月前在没有派系的图的 MaxCut 上发现的界限。

But their headline result was about Chowla’s cosine problem.

但他们的主要结果是关于乔拉的余弦问题。

They’d proved that for any set of N integers, the corresponding cosine sum attains a value lower than − N 1/10 .

他们证明，对于任何 N 个整数集合，相应的余弦和的值都低于 - N 1/10 。

For any realistic value of N , − N 1/10 doesn’t differ too much from Ruzsa’s decades-old bound.

对于 N 的任何实际值，−N 1/10 与 Ruzsa 几十年前的界限没有太大区别。

But for huge values of N , like 10 20 , the difference starts to be noticeable: Jin, Milojević, Tomon, and Zhang show that a sum of 10 20 cosines slides below −100, in comparison to Ruzsa’s bound of −7.

但对于 N 的巨大值，例如 10 20 ，差异开始变得明显：Jin、Milojević、Tomon 和Zhang 表明，与 Ruzsa 的界限 -7 相比，10 20 余弦之和低于 -100。

“For me, it’s very surprising,” Sudakov said.

“对我来说，这非常令人惊讶，”苏达科夫说。

The group started with a result about graphs, and out of nowhere, they gained fresh insight on a seemingly unrelated problem.

该小组从一个关于图表的结果开始，突然间，他们对一个看似不相关的问题获得了新的见解。

Just two days after the researchers posted their paper, Bedert, the Cambridge mathematician, posted his own advance on the problem, using a more traditional approach from Fourier analysis.

就在研究人员发表论文两天后，剑桥数学家贝德特发表了自己在这个问题上的进展，使用了更传统的傅里叶分析方法。

His result edges out the team’s bound by a hair: It says that for any set of N integers, the cosine sum attains a value less than − N 1/7 .

他的结果超出了团队的极限：它表明对于任何 N 个整数集合，余弦和的值都小于 − N 1/7 。

For 10 20 , this lowers the minimum that Jin, Milojević, Tomon, and Zhang identified from −100 to around −720.

对于 10 20 ，这将 Jin、Milojević、Tomon 和Zhang 确定的最小值从 -100 降低到 -720 左右。

But what mathematicians find most noteworthy is that both of these results mark the first time that a proven estimate has the same form as Chowla’s conjectured bound.

但数学家发现最值得注意的是，这两个结果标志着第一次经过验证的估计与乔拉猜想的界限具有相同的形式。

That is, the new bounds, like Chowla’s, can be written as a power of N .

也就是说，新的界限，就像 Chowla 的界限一样，可以写成 N 的幂。

(Chowla’s bound of −$latex \sqrt{\textit{N}}$ is equivalent to − N 1/2 .) Ruzsa’s previous estimate cannot be written in this form.

（Chowla 的 −$latex \sqrt{\textit{N}}$ 的界限等于 − N 1/2 。）Ruzsa 之前的估计不能写成这种形式。

The fog surrounding the Fourier transform is still dense.

傅里叶变换周围的迷雾仍然很浓。

But these new techniques are a little better at seeing through it.

但这些新技术更能看穿它。

Though neither proof has fully bridged the gap to prove Chowla’s conjecture, mathematicians are excited.

尽管这两个证明都没有完全弥合证明乔拉猜想的差距，但数学家们还是很兴奋。

For now, “it is a little bit, I think, like the moon landing or the 4-minute mile,” Sanders said.

桑德斯说，就目前而言，“我认为这有点像登月或 4 分钟一英里”。

“It’s not clear ahead of time what this is going to open up.”

“目前尚不清楚这会带来什么。”

The role that graphs played in the story is particularly intriguing.

图表在故事中扮演的角色特别有趣。

This isn’t the first time that graph theory and Fourier analysis have met.

这并不是图论和傅立叶分析第一次相遇。

But so far, the links between the two fields have been one-offs.

但到目前为止，这两个领域之间的联系还只是一次性的。

Now, Jin hopes that the specific connection between Chowla’s cosine problem and MaxCut hints at something broader.

现在，Jin 希望 Chowla 的余弦问题和 MaxCut 之间的具体联系暗示了更广泛的东西。

“Whatever is predicted in the Chowla problem, that phenomenon is more general,” he said.

“无论对乔拉问题的预测如何，这种现象都更为普遍，”他说。

“It works in graphs.”

“它适用于图表。”

“We now have more problems that are in the same spheres of influence,” Sawhney said.

“我们现在在同一影响范围内遇到了更多问题，”索尼说。

“Knowing that things are living in the same world is very useful information.

“知道事物生活在同一个世界是非常有用的信息。

It’s very powerful.”

它非常强大。”

Correction: January 29, 2026 An earlier version of the text implied that Benjamin Sudakov was the sole author of a 2003 conjecture about the MaxCut of certain graphs.

更正：2026 年 1 月 29 日文本的早期版本暗示本杰明·苏达科夫 (Benjamin Sudakov) 是 2003 年有关某些图的 MaxCut 猜想的唯一作者。

He was in fact one of four authors.

他实际上是四位作者之一。

The text was also revised to clarify that not all Cayley graphs have a prime number of nodes.

文本还进行了修改，以澄清并非所有凯莱图都具有素数节点。

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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。