What Is Math’s Mysterious Langlands Program Really About?
Samuel Velasco/ Quanta Magazine
塞缪尔·贝拉斯科/广达杂志
T he mathematical universe is boundless and heterogeneous, encompassing, over here, truths about numbers and equations; there, the logic of shapes and spaces; and over there, the study of change and probability.
数学宇宙是无边无际的、异质的，在这里包含了关于数字和方程的真理；在那里，形状和空间的逻辑；在那里，研究变化和概率。
Its domains have grown organically over centuries, each centered around its own objects, methods, and questions.
它的领域经过几个世纪的有机发展，每个领域都围绕着自己的对象、方法和问题。
That’s why it’s so surprising when direct connections between different areas of math are discovered, like wormholes linking distant galaxies.
这就是为什么当发现不同数学领域之间的直接联系时会如此令人惊讶，就像连接遥远星系的虫洞一样。
The wormholes are known as Langlands correspondences, and the decades-long effort by hundreds of mathematicians to extend and exploit these correspondences is the Langlands program .
虫洞被称为朗兰兹对应关系，数百名数学家数十年来为扩展和利用这些对应关系所做的努力就是朗兰兹计划。
Because it suggests an underlying unity to the universe of mathematical truths, the Langlands program has been called a “grand unified theory of mathematics.”
由于朗兰兹纲领提出了数学真理宇宙的根本统一性，因此它被称为“数学大统一理论”。
Yet even most mathematicians don’t quite know what to make of it.
然而，即使是大多数数学家也不太清楚如何理解它。
I asked several experts connected to the Langlands program whether they thought the average attendee at the recent International Congress of Mathematicians would have a decent understanding of what it is, or if they would have no idea.
我询问了几位与朗兰兹计划相关的专家，他们是否认为参加最近举行的国际数学家大会的普通与会者会对朗兰兹计划有一个很好的理解，或者他们是否不知道。
“I would think more the latter than the former,” said David Ben-Zvi , a mathematician at the University of Texas, Austin who studies the geometric Langlands correspondence , echoing the typical response.
“我更倾向于后者，”德克萨斯大学奥斯汀分校研究几何朗兰兹对应的数学家戴维·本-兹维（David Ben-Zvi）说，这呼应了典型的回应。
“Everyone will have heard of it, certainly.”
“当然，每个人都会听说过它。”
I n philosophy, “qualia” refers to the subjective qualities of our experience: what it’s like for Alice to see blue or for Bob to feel delighted.
在哲学中，“感受性”指的是我们体验的主观品质：爱丽丝看到蓝色或鲍勃感到高兴是什么感觉。
Qualia are “the ways things seem to us,” as the late philosopher Daniel Dennett put it.
正如已故哲学家丹尼尔·丹尼特所说，感受性是“事物在我们看来的方式”。
In these essays, our columnists follow their curiosity, and explore important but not necessarily answerable scientific questions.
在这些文章中，我们的专栏作家追随他们的好奇心，探索重要但不一定可以回答的科学问题。
Even among experts, descriptions of the Langlands program sound nothing alike.
即使在专家中，对朗兰兹计划的描述听起来也毫无相似之处。
One said it’s all about “unexpected symmetries.” Another defined it as “bridges between two areas of mathematics.” Someone else said it’s “the best vision we have to understand non-abelian versions of Fourier theory” — a statement I’ll unpack later, because it might be the deepest explanation available so far.
有人说这都是关于“意想不到的对称性”。另一位将其定义为“两个数学领域之间的桥梁”。其他人说这是“我们理解傅里叶理论的非阿贝尔版本的最佳愿景”——我稍后会解释这一说法，因为它可能是迄今为止最深刻的解释。
References were made to the parable of the blind men and the elephant .
参考了盲人和大象的寓言。
The program has so many aspects and corners and consequences that it can be hard to interpret.
该计划有很多方面、角落和后果，以至于很难解释。
Mathematicians shy away from interpretation by nature, anyway, since anything they say will be unproven.
无论如何，数学家本质上回避解释，因为他们所说的任何事情都未经证实。
Another challenge is that though the Langlands program is sweeping and unifying, the mathematical correspondences themselves are excruciatingly specific and esoteric.
另一个挑战是，尽管朗兰兹纲领具有广泛性和统一性，但数学对应关系本身却极其具体和深奥。
Here is my attempt, as someone who has covered math and physics for years, to unpack Langlands from its origins to its current form as a mathematical project with no precedent.
作为一个多年来研究数学和物理的人，我试图将朗兰兹从其起源解释为目前的形式，作为一个史无前例的数学项目。
I’ll also explore what this set of connections means.
我还将探讨这组连接的含义。
It’s not easy, because mathematicians have yet to mine the deepest meaning of the Langlands program — an obscure message that seemingly pertains not only to the mathematical universe, but also to the physical one.
这并不容易，因为数学家尚未挖掘朗兰兹纲领的最深层含义——这是一个晦涩的信息，似乎不仅与数学宇宙有关，而且与物理宇宙有关。
Robert Langlands , a Canadian mathematician now in his 80s who occupies Albert Einstein’s former office at the Institute for Advanced Study in Princeton, New Jersey, launched the program that bears his name in 1967.
现年 80 多岁的加拿大数学家罗伯特·朗兰兹 (Robert Langlands) 曾在新泽西州普林斯顿高等研究院担任阿尔伯特·爱因斯坦 (Albert Einstein) 的办公室，他于 1967 年启动了以他的名字命名的项目。
In a letter to a colleague, he picked up on a connection between far-flung mathematical realms: number theory and harmonic analysis (the study of signals and waves).
在给同事的一封信中，他谈到了遥远的数学领域之间的联系：数论和调和分析（信号和波的研究）。
From there, Langlands conjectured the existence of a family of correspondences between objects, symmetries, and properties.
从那里，朗兰兹推测物体、对称性和属性之间存在一系列对应关系。
To understand that original Langlands wormhole, we can start with a collection of numbers called $latex \mathbb{Q}$.
为了理解原始的朗兰兹虫洞，我们可以从一组名为 $latex \mathbb{Q}$ 的数字开始。
Specifically, $latex \mathbb{Q}$ is the set of all rational numbers, meaning those expressible as fractions, such as 1, or $latex \frac{5}{16}$, or $latex – 5.3277$.
具体来说，$latex \mathbb{Q}$ 是所有有理数的集合，意味着可以用分数表示的数，例如 1、$latex \frac{5}{16}$ 或 $latex – 5.3277$。
These form a special kind of set called a number field, because adding, subtracting, multiplying, or dividing rationals (except for dividing by zero) yields a number that is also in the set.
它们形成了一种特殊的集合，称为数域，因为有理数的加、减、乘或除（除了除零之外）都会产生一个也在该集合中的数字。
In 1967, Robert Langlands revealed the first surprising correspondence of the sprawling mathematical endeavor that now bears his name.
1967 年，罗伯特·朗兰兹 (Robert Langlands) 揭示了与现在以他的名字命名的庞大数学事业的第一个令人惊讶的对应关系。
Dan Komoda/Institute for Advanced Study
丹·科莫达/高等研究院
But things change when you introduce a polynomial equation, such as $latex x^{2}\kern0.5pt -2=0$.
但当您引入多项式方程时，情况就会发生变化，例如 $latex x^{2}\kern0.5pt -2=0$。
Even though these equations exclusively feature rational numbers in their terms, the solutions — the values of $latex x$ that make them true — usually lie outside $latex \mathbb{Q}$.
尽管这些方程在其术语中只包含有理数，但解（使它们成立的 $latex x$ 的值）通常位于 $latex \mathbb{Q}$ 之外。
Our simple equation, for example, has the solutions $latex x = \sqrt{2}$ and $latex x =\kern0.5pt- \sqrt{2}$, which are irrational ; their digits begin $latex 1.41421\ldots\ $and keep going forever without repeating.
例如，我们的简单方程有解 $latex x = \sqrt{2}$ 和 $latex x =\kern0.5pt- \sqrt{2}$，它们是无理数；他们的数字以 $latex 1.41421\ldots\ $ 开头，并且永远持续下去，不会重复。
Mathematicians figured out that they can use these solutions to extend $latex \mathbb{Q}$.
数学家发现他们可以使用这些解决方案来扩展 $latex \mathbb{Q}$。
We can make a new field, $latex \mathbb{Q}$($latex \sqrt{2}$), by appending to $latex \mathbb{Q}$ the irrational numbers $latex \sqrt{2}$ and $latex -\sqrt{2}$, as well as all the irrationals you can get by arithmetically combining those solutions with other numbers in the field: Any number of the form $latex a + b\sqrt{2}$ (for any rationals $latex a$ and $latex b$), such as $latex 5 + 3\sqrt{2}\ or\ -\frac{\sqrt{2}}{7}$, is in the field.
我们可以通过将无理数 $latex \sqrt{2}$ 和 $latex -\sqrt{2}$ 附加到 $latex \mathbb{Q}$($latex \sqrt{2}$) 来创建一个新字段 $latex \mathbb{Q}$($latex \sqrt{2}$)，以及通过将这些解与该字段中的其他数字进行算术组合可以获得的所有无理数： $latex a + 形式的任何数字b\sqrt{2}$（对于任何有理数 $latex a$ 和 $latex b$），例如 $latex 5 + 3\sqrt{2}\ 或\ -\frac{\sqrt{2}}{7}$，位于该字段中。
If you picture the possible values of $latex a$ as a horizontal number line and $latex b$ as a vertical number line, the number field $latex \mathbb{Q}$($latex \sqrt{2}$) spans the whole plane.
如果将 $latex a$ 的可能值描绘为水平数轴，将 $latex b$ 视为垂直数轴，则数域 $latex \mathbb{Q}$($latex \sqrt{2}$) 跨越整个平面。
Crucially, $latex \mathbb{Q}$($latex \sqrt{2}$) has a “symmetry”: a transformation you can do to it that preserves all its elements.
至关重要的是， $latex \mathbb{Q}$($latex \sqrt{2}$) 具有“对称性”：您可以对其进行转换以保留其所有元素。
Namely, you can replace every $latex \sqrt{2}$ that appears in the field with the equation’s other solution, $latex -\sqrt{2}$, and vice versa.
也就是说，您可以将字段中出现的每个 $latex \sqrt{2}$ 替换为方程的其他解 $latex -\sqrt{2}$，反之亦然。
When you do this, the number field remains intact.
当您执行此操作时，数字字段保持不变。
Every $latex a + b\sqrt{2}$ becomes $latex a\kern0.5pt-\kern0.5ptb\sqrt{2}$, which was included in the field already, sitting on the opposite side of the horizontal axis.
每个 $latex a + b\sqrt{2}$ 都会变成 $latex a\kern0.5pt-\kern0.5ptb\sqrt{2}$，它已经包含在字段中，位于水平轴的另一侧。
So this symmetry transformation is like reflecting the field in a mirror.
所以这种对称变换就像在镜子中反射场一样。
In his short mathematical career, Évariste Galois identified key symmetries among the solutions to equations.
在他短暂的数学生涯中，埃瓦里斯特·伽罗瓦（Évariste Galois）发现了方程解中的关键对称性。
The collection of such symmetries is called the Galois group of the equation, after Évariste Galois, a mathematician who studied them in 1832.
这种对称性的集合被称为方程的伽罗瓦群，以 Évariste Galois（埃瓦里斯特·伽罗瓦）的名字命名，他是一位 1832 年研究这些对称性的数学家。
Galois’ big insight, which came at age 20, just weeks before he got himself killed in a duel, was that these symmetry groups reveal a great deal about equations and their solutions, even for equations that are too hard to solve.
伽罗瓦在 20 岁时，也就是他在决斗中身亡的几周前，获得了深刻的见解，即这些对称群揭示了有关方程及其解的大量信息，即使对于太难解的方程也是如此。
Let’s return to $latex x^{2}\kern0.5pt-\kern0.5pt2 = 0$.
让我们回到 $latex x^{2}\kern0.5pt-\kern0.5pt2 = 0$。
Its Galois group consists of two symmetries: the one that swaps $latex \sqrt{2}$ and $latex -\sqrt{2}$ throughout $latex \mathbb{Q}$($latex \sqrt{2}$), and an identity operation that does nothing (like multiplying everything by 1).
它的伽罗瓦群由两个对称性组成：一个在整个 $latex \mathbb{Q}$($latex \sqrt{2}$) 中交换 $latex \sqrt{2}$ 和 $latex -\sqrt{2}$ ，以及一个不执行任何操作的恒等运算（例如将所有内容乘以 1）。
This Galois group is what is known as “abelian”: You can execute its two symmetries in any order, and the number field $latex \mathbb{Q}$($latex \sqrt{2}$) will end up oriented the same way.
这个伽罗瓦群就是所谓的“阿贝尔群”：你可以以任何顺序执行它的两个对称性，并且数域 $latex \mathbb{Q}$($latex \sqrt{2}$) 最终将以相同的方式定向。
Next we’ll consider $latex x^{3}\kern0.5pt-\kern0.5pt2 = 0$, which, though it looks only a tick different than the last equation, is already complicated enough to illustrate what the Langlands program is about (but hopefully no more complicated than that).
接下来我们将考虑 $latex x^{3}\kern0.5pt-\kern0.5pt2 = 0$，虽然它看起来与最后一个方程只有一点点不同，但已经足够复杂来说明朗兰兹程序的含义（但希望不会比这更复杂）。
It has three solutions, let’s call them $latex {\ x}_{1}$, $latex x_{2}$ , and $latex x_{3}$.
它有三个解决方案，我们称它们为 $latex {\ x}_{1}$、$latex x_{2}$ 和 $latex x_{3}$。
These again create an extended number field, and this field has its own Galois group of symmetries — ways to rearrange the solutions — called S 3 .
这些再次创建了一个扩展的数域，并且该域有自己的伽罗瓦对称组 - 重新排列解的方法 - 称为 S 3 。
There are six symmetries: two ways to swap all three solutions, three ways to switch any two of them, and one way to do nothing.
有六种对称性：两种方法交换所有三种解决方案，三种方法交换其中任意两个，一种方法不执行任何操作。
It helps (and is mathematically accurate) to picture $latex x_{1}$, $latex x_{2}$, and $latex x_{3}$ as corners of an equilateral triangle.
它有助于（并且在数学上准确）将 $latex x_{1}$、$latex x_{2}$ 和 $latex x_{3}$ 描绘为等边三角形的角。
Swapping all three is equivalent to rotating the triangle by 120 degrees clockwise or counterclockwise.
交换所有三个相当于将三角形顺时针或逆时针旋转 120 度。
Switching two, however, is equivalent to flipping the triangle, thereby exchanging two corners.
然而，交换两个相当于翻转三角形，从而交换两个角。
Importantly, this combination of symmetries makes the Galois group S 3 “non-abelian,” meaning the order in which you transform the field matters.
重要的是，这种对称性组合使伽罗瓦群 S 3 成为“非阿贝尔群”，这意味着变换场的顺序很重要。
Rotating the triangle 120 degrees and then flipping it puts the three solutions in different positions than if you flip first and then rotate.
与先翻转然后旋转相比，将三角形旋转 120 度然后翻转它会将三个解置于不同的位置。
The S 3 symmetries can be expressed as a set of six little 2 × 2 matrices, or blocks of numbers; this is called a Galois representation.
S 3 对称性可以表示为一组六个 2 × 2 小矩阵，或数字块；这称为伽罗瓦表示。
The correspondence Langlands discussed in his 1967 letter is the fact that each Galois representation generates a barcode-like sequence of data that controls the form of a completely different mathematical object, one that differs so utterly in its origins and premise that the connection between them seems almost miraculous.
朗兰兹在 1967 年的信中讨论的对应关系是这样一个事实：每个伽罗瓦表示都会生成一个类似条形码的数据序列，控制着一个完全不同的数学对象的形式，这个数学对象的起源和前提截然不同，以至于它们之间的联系几乎是不可思议的。
It’s this unexpected link, and other, analogous wormholes, that mathematicians have been exploring ever since.
从那时起，数学家们一直在探索这种意想不到的联系以及其他类似的虫洞。
You have a taste by now of the strange cosmos mathematicians bop around in, thinking ultra-clearly about objects such as $latex \mathbb{Q}$ ($latex \sqrt{2})$ and extending the ideas in them to find new, more complex truths.
现在你已经尝到了奇怪的宇宙数学家们的滋味，超清晰地思考诸如 $latex \mathbb{Q}$ ($latex \sqrt{2})$ 之类的对象，并扩展其中的想法以发现新的、更复杂的真理。
So far we’ve stayed on the side of the Langlands correspondence concerning numbers and algebra.
到目前为止，我们在数字和代数方面一直站在朗兰兹信件的一边。
Now we’ll pass through the wormhole to harmonic analysis, where mathematicians study how signals like lights and sounds can be broken up into component frequencies.
现在我们将穿过虫洞进行谐波分析，数学家们研究如何将光和声音等信号分解为分量频率。
Within the galaxy of harmonic analysis, we land on a special kind of mathematical object called a modular form — an object that, like sound and light waves, can be written as a sum of simple oscillating components.
在调和分析的星系中，我们发现了一种特殊的数学对象，称为模形式——这种对象像声波和光波一样，可以写成简单振荡分量的总和。
First studied in the 19th century, modular forms are “automorphic,” meaning they have a kind of symmetry where you can shift or transform the input (the value that you plug in to the function) by certain amounts and get the same output.
模块化形式首次在 19 世纪被研究，是“自同构”的，这意味着它们具有一种对称性，您可以将输入（插入函数的值）移动或变换一定量并获得相同的输出。
Mathematicians vividly illustrate these functions by assigning colors to different output values and then making a color map of the inputs that yield those outputs.
数学家通过为不同的输出值分配颜色，然后制作产生这些输出的输入的颜色图，生动地说明了这些函数。
Repetitions in the resulting patterns encode the modular form’s inherent symmetries.
由此产生的模式中的重复编码了模块化形式的固有对称性。
I mentioned that modular forms can be broken up into component parts.
我提到模块化形式可以分解为组成部分。
The modular form that concerns us here is defined by the following function, built from ever-larger powers of the input variable, $latex q$:
这里我们关心的模块化形式是由以下函数定义的，该函数由输入变量 $latex q$ 的更大幂构建而成：
$latex f(q)=q-q^{7}-q^{13}-q^{19}+q^{25}+2q^{31}-q^{37}+2q^{43}-q^{61}-q^{67}+\ldots$
$乳胶f(q)=q-q^{7}-q^{13}-q^{19}+q^{25}+2q^{31}-q^{37}+2q^{43}-q^{61}-q^{67}+\ldots$
The Langlands miracle here is that the coefficients in front of the prime-numbered powers of q in this function come from our little 2 × 2 matrices representing the Galois group S 3 , from our equation $latex x^{3}\kern0.5pt-\kern0.5pt2 = 0$.
这里的朗兰兹奇迹是，这个函数中 q 的素数幂前面的系数来自我们代表伽罗瓦群 S 3 的 2 × 2 小矩阵，来自我们的方程 $latex x^{3}\kern0.5pt-\kern0.5pt2 = 0$。
Prime numbers, those divisible only by 1 and themselves, are the atoms of arithmetic; all other whole numbers are built from them.
质数，即只能被 1 和它们本身整除的数，是算术的原子；所有其他整数都是由它们构建的。
Likewise, the coefficients of the terms with prime exponents shape a modular form.
同样，具有素数指数的项的系数形成模形式。
Let’s take a closer look at those coefficients:
让我们仔细看看这些系数：
$latex f(q)=\ldots+0q^{2}+0q^{3}+\ldots+0q^{5}+\ldots-1q^{7}+\ldots+0q^{11}$ $latex \qquad+\ldots-1q^{13}+\ldots+0q^{17}+\ldots-1q^{19}+\ldots+0q^{23}$ $latex \qquad+\ldots+2q^{31}+\ldots-1q^{37}+\ldots+2q^{43}+\ldots-1q^{61}$ $latex \qquad+\ldots-1q^{67}+\ldots$
$乳胶 f(q)=\ldots+0q^{2}+0q^{3}+\ldots+0q^{5}+\ldots-1q^{7}+\ldots+0q^{11}$ $乳胶\qquad+\ldots-1q^{13}+\ldots+0q^{17}+\ldots-1q^{19}+\ldots+0q^{23}$ $乳胶 \qquad+\ldots+2q^{31}+\ldots-1q^{37}+\ldots+2q^{43}+\ldots-1q^{61}$ $乳胶 \qquad+\ldots-1q^{67}+\ldots$
They’re all zero, $latex -1$, or 2.
它们都是零、$latex -1$ 或 2。
(The other coefficients are arithmetic combinations of those, based on the prime factors of the power.) Likewise, each of the matrices in our Galois representation of S 3 has a number called a trace, which is the sum of its top left and bottom right elements.
（其他系数是这些系数的算术组合，基于幂的素因数。）同样，S 3 的伽罗瓦表示中的每个矩阵都有一个称为迹的数字，它是其左上角和右下角元素的总和。
And the traces of the reflection, 120-degree rotation, and identity matrices are zero, –1, and 2, respectively.
反射、120 度旋转和单位矩阵的迹分别为零、–1 和 2。
This visual representation reflects the modular form cited above, which is surprisingly linked to the equation $latex x^{3}\:\!-\:\!2 = 0$.
这种视觉表示反映了上面引用的模块化形式，令人惊讶的是，它与方程 $latex x^{3}\:\!-\:\!2 = 0$ 相关。
Which trace from our Galois representation clings to each of these terms with prime exponents depends on how our equation, $latex x^{3}\:\!-\:\!2 = 0$, behaves in a number field defined by that particular prime.
我们的伽罗瓦表示中的哪一条轨迹与这些具有素数指数的项相关，取决于我们的方程 $latex x^{3}\:\!-\:\!2 = 0$ 在由该特定素数定义的数域中的表现。
In each case, the equation is solved in what is known as modular arithmetic.
在每种情况下，方程都通过所谓的模运算来求解。
Modular arithmetic works like a clock; instead of using all numbers, it counts up to a certain number — 12 in the case of a clock — and then resets to zero.
模算术的工作原理就像一个时钟；它不是使用所有数字，而是计数到某个数字（对于时钟来说是 12），然后重置为零。
Counting “modulo 12” (for example) means that numbers such as 13 and 25 are numerically equivalent to 1, and all multiples of 12 would be zero.
计算“模 12”（例如）意味着 13 和 25 等数字在数值上等于 1，并且所有 12 的倍数都为零。
The trace that goes in front of $latex q^{31}$, for example, depends on how $latex x^{3}\kern0.5pt-\kern0.5pt2 = 0$ behaves when the modulus is 31.
例如，$latex q^{31}$ 前面的跟踪取决于模数为 31 时 $latex x^{3}\kern0.5pt-\kern0.5pt2 = 0$ 的行为。
In that case, the equation has three simple solutions: 4, 7, or 20.
在这种情况下，方程有三个简单的解：4、7 或 20。
(Plugging 4 into $latex x^{3}\kern0.5pt-\kern0.5pt2$, for example, gives 62, which is a multiple of 31, and therefore the same as zero in this limited arithmetic.)
（例如，将 4 代入 $latex x^{3}\kern0.5pt-\kern0.5pt2$ 得到 62，它是 31 的倍数，因此在这个有限算术中与零相同。）
Recall that the Galois symmetries are ways of rearranging an equation’s solutions.
回想一下，伽罗瓦对称性是重新排列方程解的方法。
To compute the effect of these symmetries in modular arithmetic, you raise each solution to the power of the modulus.
要计算模算术中这些对称性的影响，需要对每个解求模的幂。
In this case, our solutions, raised to the power of 31, don’t change at all; for example, $latex 4^{31} = 4$ (modulo 31).
在这种情况下，我们的解决方案，提高到 31 次方，根本不会改变；例如，$latex 4^{31} = 4$（模 31）。
Therefore, the Galois group acts on these solutions as the identity operation, and that’s why $latex q^{31}$ picks up the trace of the identity matrix, 2, as its coefficient.
因此，伽罗瓦群将这些解视为恒等运算，这就是为什么 $latex q^{31}$ 选择恒等矩阵 2 的迹作为其系数。
Video : Rutgers University mathematician Alex Kontorovich takes us on a journey through the continents of mathematics to learn about the awe-inspiring symmetries at the heart of the Langlands program.
视频：罗格斯大学数学家亚历克斯·康托罗维奇带领我们踏上数学大陆之旅，了解朗兰兹纲领核心的令人惊叹的对称性。
For comparison, the solutions of $latex x^{3}\kern0.5pt-\kern0.5pt2 = 0$ modulo 13 are $latex \sqrt[3]{2}$, $latex 3\sqrt[3]{2}$, and $latex 9\sqrt[3]{2}$, which in this case all lie outside the number field defined by this prime.
为了进行比较，$latex x^{3}\kern0.5pt-\kern0.5pt2 = 0$ modulo 13 的解是 $latex \sqrt[3]{2}$、$latex 3\sqrt[3]{2}$ 和 $latex 9\sqrt[3]{2}$，在这种情况下，它们都位于该素数定义的数域之外。
Now, raising each solution to the 13th power transforms it into one of the others.
现在，将每个解的 13 次方转化为其他解之一。
For example, $latex (\sqrt[3]{2})^{13} = \sqrt[3]{2}[(\sqrt[3]{2})^3]^4$ $latex = \sqrt[3]{2}(2)^4$ $latex = 16\sqrt[3]{2}$ $latex = 3\sqrt[3]{2}$ (modulo 13).
例如，$latex (\sqrt[3]{2})^{13} = \sqrt[3]{2}[(\sqrt[3]{2})^3]^4$ $latex = \sqrt[3]{2}(2)^4$ $latex = 16\sqrt[3]{2}$ $latex = 3\sqrt[3]{2}$（模 13）。
That’s the 120-degree-rotation element of our Galois group S 3 , which has a trace of –1.
这是伽罗瓦群 S 3 的 120 度旋转元素，其迹为 –1。
Accordingly, the coefficient of $latex q^{13}$ in the modular form is –1.
因此，模形式的 $latex q^{13}$ 的系数为 –1。
The other coefficients of prime-exponent terms that define our modular form are determined by the symmetry properties of the solutions of $latex x^{3}\kern0.5pt-\kern0.5pt2 = 0$ modulo that prime; it leaves them unchanged if they are all in the number field, or switches the two that are not, or cycles all three solutions, providing the coefficients 2, zero, and –1, respectively.
定义我们的模形式的素数指数项的其他系数由 $latex x^{3}\kern0.5pt-\kern0.5pt2 = 0$ 模该素数的解的对称性决定；如果它们都在数字字段中，则保持它们不变，或者交换不在数字字段中的两个，或者循环所有三个解决方案，分别提供系数 2、零和 –1。
Somehow, Galois symmetries from the faraway galaxy of number theory paint the swirly pattern of this modular form.
不知何故，来自遥远的数论星系的伽罗瓦对称性描绘了这种模形式的漩涡图案。
It’s not a simple connection, but it can’t be a coincidence.
这不是简单的联系，但也不可能是巧合。
The correspondence I described here goes far, far beyond $latex x^{3}\kern0.5pt-\kern0.5pt2 = 0$ and $latex f(q)$.
我在这里描述的对应关系远远超出了 $latex x^{3}\kern0.5pt-\kern0.5pt2 = 0$ 和 $latex f(q)$。
Langlands conjectured correspondences between all suitable Galois representations and automorphic forms (a generalization of a modular form).
朗兰兹猜想了所有合适的伽罗瓦表示和自同构形式（模形式的推广）之间的对应关系。
Although this sounds esoteric and isolated, it isn’t.
尽管这听起来深奥且孤立，但事实并非如此。
Every mathematical object and idea logically relates to many others, so the discovery of a wormhole between distant parts of math, involving such fundamental objects as numbers, equations, groups, functions, and symmetries, has had widespread ramifications.
每个数学对象和思想在逻辑上都与许多其他对象和思想相关，因此在数学的遥远部分之间发现虫洞，涉及数字、方程、群、函数和对称性等基本对象，产生了广泛的影响。
Traversing the wormhole has often yielded proofs or insights about one side with the aid of the other.
穿越虫洞通常会在另一方的帮助下产生关于另一方的证据或见解。
In 1994, for example, the wormhole enabled the mathematician Andrew Wiles to prove Fermat’s Last Theorem , one of the most famous and long-standing open problems in math.
例如，1994 年，虫洞使数学家安德鲁·怀尔斯 (Andrew Wiles) 证明了费马大定理，这是数学中最著名和长期存在的开放问题之一。
You’ll recall that the Pythagorean theorem relates the three sides of a right triangle: $latex a^{2} +\kern0.5ptb^{2} = c^{2}$.
您会记得毕达哥拉斯定理与直角三角形的三边相关：$latex a^{2} +\kern0.5ptb^{2} = c^{2}$。
Fermat guessed that for any whole number $latex n$ larger than 2, there are no three nonzero integers $latex a$, $latex b$, and $latex c$ that satisfy the equation $latex a^{n} + b^{n} = c^{n}$.
费马猜测，对于任何大于 2 的整数 $latex n$，不存在满足方程 $latex a^{n} + b^{n} = c^{n}$ 的三个非零整数 $latex a$、$latex b$ 和 $latex c$。
No one could prove it for 357 years.
357年来无人能证明这一点。
What finally worked was proving a Langlands-like correspondence.
最终起作用的是证明了类似朗兰兹的对应关系。
First the German mathematician Gerhard Frey showed that if Fermat’s Last Theorem is false, then you could use those numbers to write a formula $latex y^{2} = x(x\kern0.5pt-\kern0.5pta^{n})(x + b^{n})\ $defining a so-called elliptic curve.
首先，德国数学家 Gerhard Frey 表明，如果费马大定理为假，那么您可以使用这些数字编写公式 $latex y^{2} = x(x\kern0.5pt-\kern0.5pta^{n})(x + b^{n})\ $定义所谓的椭圆曲线。
Other researchers then showed that such a curve could not possibly correspond to an infinite sum that’s modular, like our $latex f(q)$ above.
其他研究人员随后表明，这样的曲线不可能对应于模块化的无限和，就像上面的 $latex f(q)$ 一样。
In 1940, André Weil saw patterns connecting different areas of mathematics, patterns that would be realized with the Langlands program.
1940 年，安德烈·韦尔 (André Weil) 看到了连接不同数学领域的模式，这些模式将通过朗兰兹计划得以实现。
Konrad Jacobs/Archives of the Mathematisches Forschungsinstitut Oberwolfach
康拉德·雅各布斯/奥伯沃尔法赫数学研究所档案馆
This meant Frey’s curve ran afoul of a Langlands-type correspondence called the Taniyama-Shimura-Weil conjecture, a precursor to Langlands’ 1967 idea that turned out to be a special case of the more general phenomenon he envisioned.
这意味着弗雷的曲线与称为谷山-志村-韦尔猜想的朗兰兹型对应发生了冲突，该猜想是朗兰兹 1967 年想法的先驱，结果证明这是他所设想的更普遍现象的一个特例。
The conjecture says that every elliptic curve over the rational numbers corresponds to a modular form.
该猜想表明有理数上的每条椭圆曲线都对应于一个模形式。
Wiles proved this conjecture true in a large enough class of cases to rule out the existence of Frey’s counterexample, thereby proving Fermat’s Last Theorem.
怀尔斯在足够多的案例中证明了这个猜想的正确性，从而排除了弗雷反例的存在，从而证明了费马大定理。
Analogues of the same wormhole have shown up in other areas of math.
相同虫洞的类似物也出现在数学的其他领域。
The person to whom Robert Langlands wrote his famous letter, the French mathematician André Weil, had sent a famous letter of his own in 1940, 27 years earlier, to his sister, the philosopher Simone Weil.
罗伯特·朗兰兹（Robert Langlands）写这封著名信的对象是法国数学家安德烈·韦尔（André Weil），他在 27 年前的 1940 年也给他的妹妹、哲学家西蒙娜·韦尔（Simone Weil）写了一封著名的信。
In it, he had described a vision of a mathematical Rosetta stone with three columns.
在其中，他描述了一个具有三根柱子的数学罗塞塔石碑的愿景。
One column was number theory, another concerned curves over finite fields (a strange kind of geometry that takes place in the world of modular arithmetic), and the third involved the more familiar geometry of smooth surfaces.
其中一栏是数论，另一栏涉及有限域上的曲线（模算术世界中发生的一种奇怪的几何），第三栏涉及更熟悉的光滑表面的几何。
Weil observed patterns that were appearing in all three areas.
韦尔观察到所有三个区域都出现了模式。
Indeed, each of Weil’s columns is now known to feature its own family of Langlands correspondences: wormholes linking one kind of mathematical object to another very different kind that somehow encodes the same information.
事实上，现在我们知道韦尔的每一篇专栏都以自己的朗兰兹对应系列为特色：虫洞将一种数学对象与另一种非常不同的数学对象联系起来，而另一种数学对象以某种方式编码相同的信息。
In the number theory column, Galois representations connect to automorphic forms in the distant galaxy of harmonic analysis.
在数论专栏中，伽罗瓦表示与调和分析的遥远星系中的自守形式相关联。
For curves over finite fields, representations of their own Galois groups are likewise matched with automorphic forms.
对于有限域上的曲线，它们自己的伽罗瓦群的表示同样与自同构形式匹配。
And for smoothly curving 2D spaces called Riemann surfaces, geometric objects describing symmetries and motions around the surface match more exotic objects of harmonic analysis called sheaves .
对于称为黎曼曲面的平滑弯曲的二维空间，描述表面周围对称性和运动的几何对象与称为滑轮的调和分析中更奇特的对象相匹配。
Decades of work and many of the field’s major prizes have gone toward proving conjectured Langlands correspondences in various settings.
数十年的工作和该领域的许多重大奖项都致力于证明各种环境下的朗兰兹对应猜想。
In 2024, for example, a group of mathematicians made headlines with a set of papers — more than 800 pages in total — that proved a major case of the geometric Langlands correspondence .
例如，2024 年，一群数学家发表了一组论文（总共 800 多页），成为头条新闻，证明了几何朗兰兹对应的一个重大案例。
Proving these correspondences also hastened progress in the connected research areas.
证明这些对应关系也加速了相关研究领域的进展。
“We can deduce things in one world using results in the other world,” said Jessica Fintzen , a mathematician at the University of Bonn.
波恩大学数学家杰西卡·芬岑 (Jessica Fintzen) 表示：“我们可以利用另一个世界的结果来推断一个世界的事物。”
We have to trust our mathematicians when they say there’s no apparent reason for the mysterious correspondences.
当我们的数学家说这些神秘的对应关系没有明显的原因时，我们必须相信他们。
“It’s the best possible world, but it’s very difficult to actually pinpoint what it means,” Fintzen said.
“这是最好的世界，但很难真正确定它的含义，”芬岑说。
Ben-Zvi thinks of the Langlands program as a non-abelian version of Fourier analysis, which is one of the most ubiquitous tools in math and physics .
本-兹维认为朗兰兹纲领是傅里叶分析的非阿贝尔版本，傅里叶分析是数学和物理学中最普遍的工具之一。
As Joseph Fourier figured out in 1807, any signal — a sound wave, say, or a beam of light — can be decomposed into the pure tones or colors that make it up.
正如约瑟夫·傅立叶 (Joseph Fourier) 在 1807 年指出的那样，任何信号（例如声波或光束）都可以分解为构成它的纯音或颜色。
Mathematically, these pure frequencies are sine waves, which, when shifted by regular intervals, don’t change.
从数学上讲，这些纯频率是正弦波，当按规则间隔移动时，它们不会改变。
Picture sliding a sine wave left or right.
想象一下向左或向右滑动正弦波。
If you shift it the right amount, it will appear not to have moved at all.
如果你移动正确的量，它看起来根本没有移动。
The symmetries of sine waves are abelian; the order in which you shift the wave back and forth doesn’t matter.
正弦波的对称性是阿贝尔的；来回移动波的顺序并不重要。
And in those waves, the coefficients that set the strength of each pure frequency are all simple numbers.
在这些波中，设定每个纯频率强度的系数都是简单的数字。
Automorphic forms are like more sophisticated sine waves, and they can have non-abelian symmetries.
自守形式就像更复杂的正弦波，它们可以具有非阿贝尔对称性。
In that case, the coefficients are tied to matrices and Galois representations — the original Langlands correspondence.
在这种情况下，系数与矩阵和伽罗瓦表示法联系在一起——原始的朗兰兹对应关系。
So perhaps the Langlands program sharpens a broader mystery of why Fourier decomposition is possible, why it lets us analyze signals, compress images, reconstruct medical scans, and do a million other things.
因此，也许朗兰兹计划加深了一个更广泛的谜团：为什么傅立叶分解是可能的，为什么它可以让我们分析信号、压缩图像、重建医学扫描以及做一百万个其他事情。
That a complicated object can be broken down into a spectrum of pure components is something of an organizing principle of modern science.
一个复杂的物体可以分解成一系列纯净的成分，这是现代科学的一个组织原则。
Some of the researchers I spoke to suspect that objects on both sides of the Langlands correspondence might be shadows or facets of some other, still-hidden mathematical object or structure.
我采访过的一些研究人员怀疑朗兰兹对应两侧的物体可能是其他一些仍然隐藏的数学物体或结构的阴影或侧面。
“There is certainly some expectation that there could be something bigger that explains the kinds of relationships one sees in the Langlands program,” said Ana Caraiani , a mathematician at Imperial College London.
伦敦帝国理工学院的数学家安娜·卡拉阿尼 (Ana Caraiani) 表示：“当然，人们期望有更大的东西可以解释人们在朗兰兹计划中看到的各种关系。”
Supporting evidence for that view, and what seems to me to be one of the most important clues about the deep origins of the Langlands correspondence, came from thinking about the physical universe rather than the mathematical one.
支持这一观点的证据，在我看来，是关于朗兰兹对应的深层起源的最重要线索之一，来自对物理宇宙而不是数学宇宙的思考。
In the 2000s, the theoretical physicists Anton Kapustin and Edward Witten realized that the geometric Langlands correspondence is a consequence of a “duality” (a situation in which one system has two different physical descriptions ) that’s exhibited by certain quantum theories.
2000 年代，理论物理学家安东·卡普斯汀 (Anton Kapustin) 和爱德华·维滕 (Edward Witten) 意识到，几何朗兰兹对应是某些量子理论所表现出的“对偶性”（一种系统有两种不同物理描述的情况）的结果。
The simplest instance is the so-called electric-magnetic duality: When there are no charges or currents around, electric and magnetic fields are interchangeable; you could swap them and there would be no way to tell.
最简单的例子就是所谓的电磁二象性：当周围没有电荷或电流时，电场和磁场是可以互换的；当周围没有电荷或电流时，电场和磁场是可以互换的；当周围没有电荷或电流时，电场和磁场是可以互换的。你可以交换它们，但没有办法分辨。
(The presence of electrically charged particles such as electrons breaks the mirror between them, because no equivalent magnetic charges exist.)
（带电粒子（例如电子）的存在打破了它们之间的镜子，因为不存在等效的磁荷。）
Kapustin and Witten studied a model of space-time (the four-dimensional fabric of the universe) in which two spatial dimensions form a Riemann surface, the kind of 2D surface involved in the geometric Langlands correspondence.
卡普斯汀和威滕研究了时空模型（宇宙的四维结构），其中两个空间维度形成黎曼曲面，这是几何朗兰兹对应中涉及的一种二维曲面。
The physicists found that switching the electric and magnetic fields in this patch of space-time had the effect of switching between sides of the geometric Langlands correspondence.
物理学家发现，在这片时空中切换电场和磁场会产生几何朗兰兹对应关系两侧之间切换的效果。
That’s hard to parse, but according to Ben-Zvi, it showed that the two sides of geometric Langlands are conceptually close to each other in some way we don’t yet understand.
这很难解析，但根据 Ben-Zvi 的说法，它表明几何朗兰兹的两侧在概念上以某种我们尚不理解的方式彼此接近。
Electric-magnetic duality reflects the fact that both fields are really aspects of a single, underlying quantum “electromagnetic field.” “It’s not two super-exotic worlds like Galois groups and automorphic forms, which sound totally unrelated,” he said, but rather a pair of intertwined quantum fields.
电磁二象性反映了这样一个事实：这两个场实际上都是单个基础量子“电磁场”的各个方面。 “这不是像伽罗瓦群和自守形式这样听起来完全无关的两个超级奇异的世界，”他说，而是一对相互交织的量子场。
Similarly, the various Langlands correspondences may eventually be understood as dual views of a single object or system.
类似地，各种朗兰兹对应关系最终可以被理解为单个对象或系统的双重视图。
Edward Frenkel , a mathematician at the University of California, Berkeley who has worked on the geometric Langlands program for decades, suspects so.
加利福尼亚大学伯克利分校的数学家爱德华·弗伦克尔（Edward Frenkel）对此表示怀疑，他几十年来一直致力于几何朗兰兹纲领。
“The real reason in my view is that there are things below the surface that have not yet been discovered,” he said.
“在我看来，真正的原因是表面之下还有一些东西尚未被发现，”他说。
On our video call, he held up a hot pink coffee cup, and we considered its shadows.
在我们的视频通话中，他举起了一个亮粉色的咖啡杯，我们看着它的影子。
“The projection onto the table will be a disk,” he said.
“桌子上的投影将是一个圆盘，”他说。
“A projection on the wall will be a rectangle.
“墙上的投影将是一个矩形。
And then you start marking and say, ‘Oh, there is a point here that connects to a point here.’ To you it appears surprising that points in one projection or shadow correspond to points in the other.
然后你开始标记并说，“哦，这里有一个点连接到这里的一个点。”对你来说，一个投影或阴影中的点与另一个投影或阴影中的点相对应似乎令人惊讶。
But if you find the real source of this, if you start seeing the source and not just the projections, that would give you a much more convincing explanation of this duality or correspondence.
但如果你找到了它的真正来源，如果你开始看到来源而不仅仅是预测，这将为你提供对这种二元性或对应性的更令人信服的解释。
“Consciously or unconsciously,” Frenkel added, “people are excited about these connections precisely because they point to some deeper structures in mathematics that we have not found.
“有意识或无意识地，”弗伦克尔补充道，“人们对这些联系感到兴奋，正是因为它们指出了一些我们尚未发现的更深层次的数学结构。
Eventually we hope to find them, by finding more and more information which hopefully will lead us to the true explanation, the true reason.”
最终我们希望通过找到越来越多的信息来找到它们，这些信息有望引导我们找到真正的解释、真正的原因。”
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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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