内容来源：https://www.quantamagazine.org/what-is-maths-mysterious-langlands-program-really-about-20260909/

内容总结：

数学界的神秘“朗兰兹纲领”究竟在讲什么？

被誉为“数学大统一理论”的朗兰兹纲领，是过去半个多世纪以来数学界最宏大也最令人困惑的研究方向之一。这个纲领源于加拿大数学家罗伯特·朗兰兹1967年一封著名的信件，其核心思想是揭示数论与调和分析（研究信号和波的数学分支）这两个看似毫无关联的数学领域之间存在着深刻的对应关系。然而，即便是许多专业数学家，对朗兰兹纲领的真正内涵也并不十分清楚。

从一道简单方程说起

要理解朗兰兹对应，我们需要从有理数域开始。有理数是可以表示为分数的数，它们构成一个封闭的数域——对有理数进行加减乘除（除零外），结果仍是有理数。但当我们引入多项式方程时，情况就变得复杂了。

以方程 x² - 2 = 0 为例，它的解是√2和-√2，这两个无理数不在有理数域中。数学家将这些解“附加”到有理数上，形成一个新的数域Q(√2)。这个新数域具有一种“对称性”：将√2替换为-√2，域的结构保持不变。

这种对称性的集合被称为“伽罗瓦群”，以纪念19岁就死于决斗的法国数学家埃瓦里斯特·伽罗瓦。对于x² - 2 = 0，它的伽罗瓦群是“阿贝尔的”（可交换的），意味着对称变换的顺序不影响结果。

通往朗兰兹的桥梁

再看一个稍复杂的方程 x³ - 2 = 0，它有三个解。这个方程生成的伽罗瓦群被称为S3，包含6种对称操作——可以想象为等边三角形的旋转和翻转。这个群是“非阿贝尔的”，即变换的顺序会改变结果。

S3的对称性可以用6个2×2矩阵表示，这就是所谓的“伽罗瓦表示”。朗兰兹在1967年的信中提出的惊人猜想是：每一个伽罗瓦表示都能生成一串类似条形码的数据，而这些数据恰好控制着另一个完全不同的数学对象——模形式的系数。

以函数 f(q) = q - q⁷ - q¹³ - q¹⁹ + q²⁵ + 2q³¹ - q³⁷ + ... 为例，其质数幂次项的系数（0、-1、2）恰好对应S3伽罗瓦群中各种对称矩阵的“迹”（矩阵对角线元素之和）。这种联系看似不可思议，却绝非巧合。

费马大定理的证明

朗兰兹纲领最著名的应用是1994年安德鲁·怀尔斯对费马大定理的证明。费马大定理声称，当n大于2时，方程 aⁿ + bⁿ = cⁿ 没有正整数解。这个猜想困扰了数学家357年。

怀尔斯的关键突破在于证明了一个朗兰兹型的对应关系——谷山-志村-韦伊猜想：每个有理数上的椭圆曲线都对应一个模形式。德国数学家格哈德·弗雷首先证明了如果费马大定理不成立，就能构造出一个特殊的椭圆曲线；而怀尔斯证明了这个曲线不可能对应任何模形式，从而排除了费马大定理反例存在的可能性，最终证明了该定理。

更广阔的图景

朗兰兹纲领的影响并不局限于数论。安德烈·韦伊在1940年给哲学家妹妹西蒙娜·韦伊的信中，描绘了“数学罗塞塔石碑”的愿景——数论、有限域上的曲线几何和光滑曲面几何三个领域之间存在着平行模式。如今，这三个领域各自都有自己的朗兰兹对应关系。

2024年，一组数学家发表了超过800页的系列论文，证明了几何朗兰兹对应的一个重要情形，引起了广泛关注。德国波恩大学的数学家杰西卡·芬岑表示：“我们可以在一个世界中使用另一个世界的结果来推导结论。”

深层含义的追寻

一些研究者认为，朗兰兹对应可能只是某个更深层数学结构的影子。伦敦帝国理工学院的数学家安娜·卡拉亚尼说：“肯定有某种预期，认为可能存在更大的东西来解释朗兰兹纲领中出现的各种关系。”

2000年代，理论物理学家安东·卡普斯京和爱德华·威滕发现，几何朗兰兹对应实际上是某些量子理论中“电-磁对偶”的结果。在没有电荷和电流的情况下，电场和磁场可以互换而不被发现。这种对偶性暗示，朗兰兹对应的两边可能只是同一个未知对象的两种视角。

加州大学伯克利分校的爱德华·弗伦克尔用一个生动的比喻来解释：一个咖啡杯在地面上的投影是圆盘，在墙上的投影是矩形。乍看之下，这些投影中点的对应关系令人惊讶，但一旦找到了光源本身，就一切了然了。“人们对这些联系感到兴奋，恰恰是因为它们指向了数学中我们尚未发现的更深层结构，”弗伦克尔说，“我们希望通过收集越来越多的信息，最终找到真正的解释、真正的原因。”

中文翻译：

数学中神秘的朗兰兹纲领究竟是怎么回事？
引言
数学宇宙广袤无边、异彩纷呈：这边，是关于数字与方程的真理性认识；那边，是形状与空间的逻辑；再远处，是变化与概率的研究。数百年来，这些领域各自有机生长，每个领域都围绕着自己的研究对象、方法和问题展开。正因如此，当数学不同领域之间发现直接联系时，才会如此令人震惊——就像连接遥远星系的虫洞一样。

这些虫洞被称为朗兰兹对应，而数百位数学家数十年来致力于扩展和利用这些对应关系的事业，就是朗兰兹纲领。由于它暗示着数学真理的宇宙具有某种深层的统一性，朗兰兹纲领被称作“数学的大统一理论”。

然而，即便是大多数数学家，也不太清楚该如何理解它。我曾询问几位与朗兰兹纲领相关的专家，问他们认为最近一届国际数学家大会的普通参会者是否能对朗兰兹纲领有像样的理解，还是完全不知所云。“我觉得后者更有可能，”研究几何朗兰兹对应的得克萨斯大学奥斯汀分校数学家戴维·本-兹维说，他的回答代表了普遍的反应，“当然，每个人都听说过它。”

即使在专家之间，对朗兰兹纲领的描述也各不相同。有人说它关乎“意想不到的对称性”；另一个人将其定义为“两片数学领域之间的桥梁”；还有人说是“我们理解非阿贝尔版本傅里叶理论的最佳愿景”——这句话我稍后会展开解释，因为它可能是目前所能给出的最深层的解释。有人提到了盲人摸象的寓言。这个纲领有太多面向、太多角落、太多推论，以至于很难加以阐释。而且数学家天性回避阐释，因为他们所说的任何事情都尚未被证明。另一个挑战在于，虽然朗兰兹纲领气势恢宏、极具统一性，但数学对应本身却极其具体且深奥难懂。

以下是我作为一个多年报道数学和物理的人，尝试将朗兰兹纲领从其起源梳理到目前这种空前数学工程形态的尝试。我还将探讨这一系列联系意味着什么。这并不容易，因为数学家尚未发掘出朗兰兹纲领最深层的意义——这条隐秘的信息似乎不仅与数学宇宙有关，也与物理宇宙相关。

罗伯特·朗兰兹，一位现年八十多岁的加拿大数学家，在新泽西州普林斯顿高等研究院占用着阿尔伯特·爱因斯坦曾经的办公室。1967年，他在一封写给同事的信中启动了以他名字命名的纲领。他在信中捕捉到了两个相距甚远的数学领域之间的联系：数论与调和分析（对信号与波的研究）。朗兰兹由此推测，在对象、对称性和性质之间，存在着一族对应关系。

要理解朗兰兹最初的虫洞，我们可以从一列名为$latex \mathbb{Q}$的数字说起。具体来说，$latex \mathbb{Q}$是所有有理数的集合，也就是可以表示为分数的数，比如1，或$latex \frac{5}{16}$，或$latex -5.3277$。这些数构成了一种特殊的集合，称为数域，因为对有理数进行加、减、乘、除（除以零除外）运算后，得到的结果仍然在这个集合中。

但当你引入多项式方程时，情况就变了，比如$latex x^{2}-2=0$。虽然这些方程的系数全部是有理数，但解——使方程成立的$latex x$的值——通常不在$latex \mathbb{Q}$之内。例如，我们这个简单方程的解是$latex x = \sqrt{2}$和$latex x =-\sqrt{2}$，它们都是无理数；它们的小数从$latex 1.41421\ldots$开始，永无止境且不循环。

数学家发现，他们可以利用这些解来扩展$latex \mathbb{Q}$。我们可以构造一个新的数域$latex \mathbb{Q}$($latex \sqrt{2}$)，方法是将无理数$latex \sqrt{2}$和$latex -\sqrt{2}$加入$latex \mathbb{Q}$，再加入所有通过算术运算将这些解与域中其他数组合所能得到的无理数：任何形如$latex a + b\sqrt{2}$的数（其中$latex a$和$latex b$为任意有理数），如$latex 5 + 3\sqrt{2}$或$latex -\frac{\sqrt{2}}{7}$，都在这个域中。如果你将$latex a$的可能取值想象为水平数轴，将$latex b$想象为垂直数轴，那么数域$latex \mathbb{Q}$($latex \sqrt{2}$)就覆盖了整个平面。

关键在于，$latex \mathbb{Q}$($latex \sqrt{2}$)具有一种“对称性”：一种可以作用于它并保持其所有元素不变的变换。具体来说，你可以将域中出现的每个$latex \sqrt{2}$替换为方程的另一个解$latex -\sqrt{2}$，反之亦然。当你这样做时，数域保持不变。每个$latex a + b\sqrt{2}$变成$latex a-b\sqrt{2}$，而这个数本来就在域中，位于水平轴的另一侧。所以这个对称变换就像是让数域照了一次镜子。

这类对称性的集合被称为方程的伽罗瓦群，得名于数学家埃瓦里斯特·伽罗瓦，他于1832年研究了这些对称性。伽罗瓦的重大洞见——产生于他20岁时，也就是他在决斗中丧生前几周——是这些对称群能揭示关于方程及其解的大量信息，即使是那些难以求解的方程也不例外。

让我们回到$latex x^{2}-2 = 0$。它的伽罗瓦群包含两个对称：一个是将$latex \mathbb{Q}$($latex \sqrt{2}$)中所有的$latex \sqrt{2}$与$latex -\sqrt{2}$互换，另一个是恒等操作（什么都不做，就像将所有数乘以1）。这个伽罗瓦群是所谓的“阿贝尔”群：你可以以任意顺序执行它的两个对称，数域$latex \mathbb{Q}$($latex \sqrt{2}$)最终的取向是一样的。

接下来我们考虑$latex x^{3}-2 = 0$，虽然它看起来只比上一个方程多了一点点变化，但已经足够复杂，可以用来阐释朗兰兹纲领的内容（但希望不会再复杂到哪里去）。

它有三个解，我们称之为$latex x{1}$、$latex x{2}$和$latex x{3}$。它们同样生成了一个扩展数域，这个数域有它自己的伽罗瓦对称群——即重新排列解的方式——称为S3。共有六个对称：两种轮换全部三个解的方式，三种交换其中任意两个解的方式，还有一种什么都不做的方式。将$latex x{1}$、$latex x{2}$和$latex x{3}$想象为等边三角形的三个顶点是有帮助的（而且在数学上是准确的）。轮换全部三个解相当于将三角形顺时针或逆时针旋转120度。交换其中两个解则相当于翻转三角形，从而交换两个顶点。

重要的是，这种对称组合使得伽罗瓦群S3是“非阿贝尔”的，这意味着你变换数域的顺序是有影响的。先旋转三角形120度再翻转，与先翻转再旋转，三个解所在的位置是不同的。

S3的对称可以用一组六个小2×2矩阵（即数字方块）来表示；这被称为伽罗瓦表示。朗兰兹在1967年那封信中讨论的对应关系就是：每个伽罗瓦表示都会生成一串类似条形码的数据序列，而这一序列控制着另一个完全不同的数学对象的形式——这个对象的起源和前提如此迥异，以至于它们之间的联系看起来近乎奇迹。正是这种出人意料的联系，以及其他类似的虫洞，成为数学家们此后一直在探索的对象。

至此，你已经略微品尝到了数学家们游走其中的奇异宇宙——他们对$latex \mathbb{Q}$($latex \sqrt{2}$)这样的对象进行极其清晰的思考，并将其中的思想加以延伸以发现新的、更复杂的真理。到目前为止，我们一直停留在朗兰兹对应中涉及数字和代数的那一侧。现在我们穿过虫洞，来到调和分析一侧，数学家在这里研究光、声等信号如何分解为各构成频率。

在调和分析的星系中，我们落到一种特殊的数学对象上——模形式。这种对象和声波、光波一样，可以写成简单振荡分量的和。模形式在19世纪首次被研究，它们是“自守”的，意味着它们具有一种对称性：你可以将输入（即代入函数的值）按一定量进行平移或变换，得到的输出不变。数学家们通过为不同的输出值赋予不同颜色，然后绘制产生这些输出的输入的彩色图谱来生动地展示这些函数。结果图案中的重复模式编码了模形式内在的对称性。

我提到模形式可以分解为分量。这里与我们相关的模形式由以下函数定义，该函数由输入变量$latex q$的越来越大的幂构成：
$latex f(q)=q-q^{7}-q^{13}-q^{19}+q^{25}+2q^{31}-q^{37}+2q^{43}-q^{61}-q^{67}+\ldots$

这里的朗兰兹奇迹在于：此函数中$latex q$的素数幂项前面的系数，恰好来自我们表示伽罗瓦群S3的那些小2×2矩阵，而那个S3来自我们的方程$latex x^{3}-2 = 0$。素数——只能被1和自身整除的数——是算术的原子；所有其他整数都由它们构成。同样，素数指数的项前面的系数塑造了模形式的形态。让我们仔细看看这些系数：
$latex f(q)=\ldots+0q^{2}+0q^{3}+\ldots+0q^{5}+\ldots-1q^{7}+\ldots+0q^{11}$
$latex \qquad+\ldots-1q^{13}+\ldots+0q^{17}+\ldots-1q^{19}+\ldots+0q^{23}$
$latex \qquad+\ldots+2q^{31}+\ldots-1q^{37}+\ldots+2q^{43}+\ldots-1q^{61}$
$latex \qquad+\ldots-1q^{67}+\ldots$

它们全部是0、$latex -1$或2。（其他系数是由这些系数根据指数的质因子做算术组合得到的。）同样，我们S3伽罗瓦表示中的每个矩阵都有一个称为迹的数，即左上和右下元素之和。而反射矩阵、120度旋转矩阵和恒等矩阵的迹分别恰为0、–1和2。

我们的伽罗瓦表示中哪个迹附着在带素数指数的每一项上，取决于方程$latex x^{3}-2 = 0$在该特定素数所定义的数域中的行为。每种情况下，方程都在所谓模算术中求解。模算术就像时钟一样运作；它不使用所有数字，而是计数到某个数——时钟的情况是12——然后重置为零。以“模12”计数（例如）意味着13和25这样的数在数值上等同于1，而12的所有倍数都为零。

例如，$latex q^{31}$前面的迹取决于当模数为31时$latex x^{3}-2 = 0$的行为。在这种情况下，方程有三个简单解：4、7或20。（例如，将4代入$latex x^{3}-2$，得到62，62是31的倍数，因此在这个有限算术中等同于零。）

回顾一下，伽罗瓦对称是重新排列方程解的方式。要计算这些对称在模算术中的效果，你需要将每个解提升到模数的幂次。在这种情况下，我们的解提升到31次幂后完全不变；例如，$latex 4^{31} = 4$（模31）。因此，伽罗瓦群对这些解的作用是恒等操作，这就是为什么$latex q^{31}$前面的系数取的是恒等矩阵的迹，即2。

相比之下，$latex x^{3}-2 = 0$模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。现在，将每个解提升到13次幂会将其变换为另一个解。例如，$latex (\sqrt[3]{2})^{13} = \sqrt[3]{2}[(\sqrt[3]{2})^3]^4 = \sqrt[3]{2}(2)^4 = 16\sqrt[3]{2} = 3\sqrt[3]{2}$（模13）。这就是我们伽罗瓦群S3中的120度旋转元素，其迹为–1。相应地，模形式中$latex q^{13}$的系数为–1。

定义我们模形式的其余素数指数项的系数，由$latex x^{3}-2 = 0$模该素数时解的特殊对称性决定：如果所有解都在该数域中则保持不变，如果两个不在则交换它们，或者轮换全部三个解，分别提供系数2、0和–1。不知何故，来自遥远数论星系的伽罗瓦对称性绘制出了这个模形式的漩涡状图案。这不是一个简单的联系，但也不可能是巧合。

我在这里描述的对应关系远远超出了$latex x^{3}-2 = 0$和$latex f(q)$。朗兰兹推测，所有合适的伽罗瓦表示与自守形式（模形式的一种推广）之间都存在对应关系。虽然这听起来深奥且孤立，但事实并非如此。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联，因此在数学的遥远领域之间发现虫洞——涉及数字、方程、群、函数和对称性等如此基本的对象——产生了广泛的影响。穿越虫洞常常能借助一侧来证明或洞悉另一侧的问题。

例如，1994年，这个虫洞使得数学家安德鲁·怀尔斯证明了费马大定理——数学中最著名、持续时间最长的开放问题之一。你大概记得毕达哥拉斯定理将直角三角形的三边联系起来：$latex a^{2} + b^{2} = c^{2}$。费马猜测，对于任何大于2的整数$latex n$，都不存在三个非零整数$latex a$、$latex b$和$latex c$满足方程$latex a^{n} + b^{n} = c^{n}$。357年来无人能证明。

最终奏效的方法是证明一个朗兰兹式的对应关系。首先，德国数学家格哈德·弗雷表明，如果费马大定理为假，那么你可以用那些数写出一个公式$latex y^{2} = x(x-a^{n})(x + b^{n})$来定义一条所谓的椭圆曲线。其他研究人员随后表明，这样的曲线不可能对应于一个具有模性的无穷和，就像我们上面提到的$latex f(q)$那样。

这意味着弗雷曲线违背了名为谷山-志村-韦伊猜想的朗兰兹式对应——这是朗兰兹1967年思想的先驱，结果证明是他所设想的更一般现象的一个特例。该猜想说，有理数上的每条椭圆曲线都对应一个模形式。怀尔斯在一个足够大的类中证明了这个猜想，从而排除了弗雷反例的存在，进而证明了费马大定理。

同样的虫洞在不同数学领域中也出现了类似的版本。

罗伯特·朗兰兹写那封著名信件的对象——法国数学家安德烈·韦伊，他自己在更早的1940年，也就是27年前，曾给他的妹妹——哲学家西蒙娜·韦伊写过一封著名的信。在那封信中，他描述了一个三栏数学罗塞塔石碑的愿景。一栏是数论，另一栏涉及有限域上的曲线（一种在模算术世界中发生的奇异几何），第三栏涉及更为熟悉的平滑曲面几何。韦伊观察到在这三个领域中都在出现的模式。

事实上，韦伊的每一栏如今都被认为拥有各自的朗兰兹对应家族：虫洞将一种数学对象连接到另一种截然不同、但不知何故编码着相同信息的对象。在数论这一栏中，伽罗瓦表示连接到遥远调和分析星系中的自守形式。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守形式相匹配。对于被称为黎曼曲面的光滑弯曲二维空间，描述曲面周围对称性和运动的几何对象与调和分析中称为层的更奇异的对象相匹配。

数十年的工作和该领域的许多重大奖项都投入于在各种情境中证明那些被推测存在的朗兰兹对应。例如，2024年，一组数学家以一系列论文——总计超过800页——证明了几何朗兰兹对应的一个重要情形，登上了新闻头条。证明这些对应关系也加速了相关研究领域的进展。“我们可以用另一个世界的结果来推导出这个世界的结论，”波恩大学的数学家杰西卡·芬岑说。

当数学家说这些神秘的对应没有明显的理由时，我们必须相信他们。“这是最理想的世界，但很难确切指出它意味着什么，”芬岑说。

本-兹维将朗兰兹纲领视为傅里叶分析的非阿贝尔版本，而傅里叶分析是数学和物理中使用最广泛的工具之一。正如约瑟夫·傅里叶在1807年发现的，任何信号——比如声波或光束——都可以分解为构成它的纯音或纯色。在数学上，这些纯频率是正弦波，当按规则间隔平移时它们不会改变。想象将正弦波左右滑动。如果移动量合适，它看起来就完全没有移动过。正弦波的对称是阿贝尔的；前后平移波的顺序无关紧要。而在这些波中，决定每个纯频率强度的系数都是简单的数字。自守形式就像是更复杂的正弦波，它们可以具有非阿贝尔对称性。在这种情况下，系数与矩阵和伽罗瓦表示相关联——这就是最初的朗兰兹对应。

因此，或许朗兰兹纲领加深了一个更广泛的谜团：为什么傅里叶分解是可能的，为什么它让我们能够分析信号、压缩图像、重建医学扫描结果以及做无数其他事情。一个复杂的对象可以分解为一系列纯分量，这在某种程度上是现代科学的一种组织原则。

我采访的一些研究者怀疑，朗兰兹对应两侧的对象可能是某种其他仍然隐藏的数学对象或结构的投影或侧面。“确实存在某种期待，可能有一个更大的东西来解释人们在朗兰兹纲领中看到的那类关系，”伦敦帝国理工学院的数学家安娜·卡拉亚尼说。

支持这种观点的证据——在我看来也是关于朗兰兹对应深层起源最重要的线索之一——来自对物理宇宙而非数学宇宙的思考。在2000年代，理论物理学家安东·卡普斯廷和爱德华·威滕意识到，几何朗兰兹对应是某些量子理论所展现的“对偶性”（一种系统具有两种不同物理描述的情形）的一个推论。最简单的例子就是所谓的电磁对偶：当周围没有电荷或电流时，电场和磁场是可以互换的；你可以交换它们而无法察觉任何区别。（带电粒子如电子的存在打破了两者之间的镜像对称，因为不存在等效的磁荷。）

卡普斯廷和威滕研究了一个时空模型（宇宙的四维结构），在这个模型中，两个空间维度形成一个黎曼曲面，也就是几何朗兰兹对应中涉及的那种二维曲面。物理学家发现，在这个时空区域中交换电场和磁场，恰好产生了在几何朗兰兹对应两侧之间切换的效果。

这很难解读，但根据本-兹维的说法，它表明几何朗兰兹的两侧在我们尚未理解的某种方式下在概念上是相近的。电磁对偶反映了这样一个事实：两个场实际上是一个底层量子“电磁场”的不同方面。“它们不是像伽罗瓦群和自守形式那样听起来完全无关的两个超级奇异世界，”他说，而是一对相互交织的量子场。类似地，各种朗兰兹对应最终可能被理解为单一对象或系统的两种对偶视角。

爱德华·弗伦克尔，加州大学伯克利分校数学家，从事几何朗兰兹纲领研究数十年，也持这种怀疑。“依我之见，真正的原因是表面之下还有尚未被发现的东西，”他说。

在我们的视频通话中，他举起一个亮粉色的咖啡杯，我们观察了它的影子。“在桌上的投影是一个圆盘，”他说，“在墙上的投影是一个矩形。然后你开始做标记说，‘哦，这里有一个点连着那里的一个点。’对你来说，一个投影或影子中的点与另一个中的点对应似乎令人惊讶。但如果你找到了这一切的真正来源，如果你开始看到那个来源而不只是投影，那将为你提供对这种对偶或对应远为更有说服力的解释。”

“有意或无意地，”弗伦克尔补充说，“人们对这些联系感到兴奋，正是因为它们指向数学中一些我们尚未发现的更深层结构。最终我们希望找到这些结构，通过获取越来越多的信息——希望这些信息将引导我们找到真正的解释、真正的原因。”

英文来源：

What Is Math’s Mysterious Langlands Program Really About?
Introduction
The 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. Its domains have grown organically over centuries, each centered around its own objects, methods, and questions. That’s why it’s so surprising when direct connections between different areas of math are discovered, like wormholes linking distant galaxies.
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. Because it suggests an underlying unity to the universe of mathematical truths, the Langlands program has been called a “grand unified theory of mathematics.”
Yet even most mathematicians don’t quite know what to make of it. 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. “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. “Everyone will have heard of it, certainly.”
Even among experts, descriptions of the Langlands program sound nothing alike. 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. References were made to the parable of the blind men and the elephant. The program has so many aspects and corners and consequences that it can be hard to interpret. Mathematicians shy away from interpretation by nature, anyway, since anything they say will be unproven. Another challenge is that though the Langlands program is sweeping and unifying, the mathematical correspondences themselves are excruciatingly specific and esoteric.
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. I’ll also explore what this set of connections means. 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.
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. 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). From there, Langlands conjectured the existence of a family of correspondences between objects, symmetries, and properties.
To understand that original Langlands wormhole, we can start with a collection of numbers called $latex \mathbb{Q}$. 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 $latex – 5.3277$. 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.
But things change when you introduce a polynomial equation, such as $latex x^{2}\kern0.5pt -2=0$. 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 $latex \mathbb{Q}$. 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.
Mathematicians figured out that they can use these solutions to extend $latex \mathbb{Q}$. 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. 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.
Crucially, $latex \mathbb{Q}$($latex \sqrt{2}$) has a “symmetry”: a transformation you can do to it that preserves all its elements. 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. When you do this, the number field remains intact. 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. So this symmetry transformation is like reflecting the field in a mirror.
The collection of such symmetries is called the Galois group of the equation, after Évariste Galois, a mathematician who studied them in 1832. 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.
Let’s return to $latex x^{2}\kern0.5pt-\kern0.5pt2 = 0$. 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). 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.
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).
It has three solutions, let’s call them $latex {\ x}{1}$, $latex x{2}$ , and $latex x{3}$. These again create an extended number field, and this field has its own Galois group of symmetries — ways to rearrange the solutions — called S3. 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. It helps (and is mathematically accurate) to picture $latex x{1}$, $latex x{2}$, and $latex x{3}$ as corners of an equilateral triangle. Swapping all three is equivalent to rotating the triangle by 120 degrees clockwise or counterclockwise. Switching two, however, is equivalent to flipping the triangle, thereby exchanging two corners.
Importantly, this combination of symmetries makes the Galois group S3 “non-abelian,” meaning the order in which you transform the field matters. Rotating the triangle 120 degrees and then flipping it puts the three solutions in different positions than if you flip first and then rotate.
The S3 symmetries can be expressed as a set of six little 2 × 2 matrices, or blocks of numbers; this is called a Galois representation. 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. It’s this unexpected link, and other, analogous wormholes, that mathematicians have been exploring ever since.
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. So far we’ve stayed on the side of the Langlands correspondence concerning numbers and algebra. 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.
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. 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. 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. Repetitions in the resulting patterns encode the modular form’s inherent symmetries.
I mentioned that modular forms can be broken up into component parts. The modular form that concerns us here is defined by the following function, built from ever-larger powers of the input variable, $latex q$:
$latex f(q)=q-q^{7}-q^{13}-q^{19}+q^{25}+2q^{31}-q^{37}+2q^{43}-q^{61}-q^{67}+\ldots$
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 S3, from our equation $latex x^{3}\kern0.5pt-\kern0.5pt2 = 0$. Prime numbers, those divisible only by 1 and themselves, are the atoms of arithmetic; all other whole numbers are built from them. Likewise, the coefficients of the terms with prime exponents shape a modular form. Let’s take a closer look at those coefficients:
$latex f(q)=\ldots+0q^{2}+0q^{3}+\ldots+0q^{5}+\ldots-1q^{7}+\ldots+0q^{11}$
$latex \qquad+\ldots-1q^{13}+\ldots+0q^{17}+\ldots-1q^{19}+\ldots+0q^{23}$
$latex \qquad+\ldots+2q^{31}+\ldots-1q^{37}+\ldots+2q^{43}+\ldots-1q^{61}$
$latex \qquad+\ldots-1q^{67}+\ldots$
They’re all zero, $latex -1$, or 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 S3 has a number called a trace, which is the sum of its top left and bottom right elements. And the traces of the reflection, 120-degree rotation, and identity matrices are zero, –1, and 2, respectively.
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. In each case, the equation is solved in what is known as modular arithmetic. 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. 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.
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. In that case, the equation has three simple solutions: 4, 7, or 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.)
Recall that the Galois symmetries are ways of rearranging an equation’s solutions. To compute the effect of these symmetries in modular arithmetic, you raise each solution to the power of the modulus. In this case, our solutions, raised to the power of 31, don’t change at all; for example, $latex 4^{31} = 4$ (modulo 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.
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. Now, raising each solution to the 13th power transforms it into one of the others. 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). That’s the 120-degree-rotation element of our Galois group S3, which has a trace of –1. Accordingly, the coefficient of $latex q^{13}$ in the modular form is –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. Somehow, Galois symmetries from the faraway galaxy of number theory paint the swirly pattern of this modular form. It’s not a simple connection, but it can’t be a coincidence.
The correspondence I described here goes far, far beyond $latex x^{3}\kern0.5pt-\kern0.5pt2 = 0$ and $latex f(q)$. Langlands conjectured correspondences between all suitable Galois representations and automorphic forms (a generalization of a modular form). Although this sounds esoteric and isolated, it isn’t. 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. Traversing the wormhole has often yielded proofs or insights about one side with the aid of the other.
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. You’ll recall that the Pythagorean theorem relates the three sides of a right triangle: $latex a^{2} +\kern0.5ptb^{2} = c^{2}$. 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 $latex a^{n} + b^{n} = c^{n}$. No one could prove it for 357 years.
What finally worked was proving a Langlands-like correspondence. 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. 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.
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. The conjecture says that every elliptic curve over the rational numbers corresponds to a modular form. 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.
Analogues of the same wormhole have shown up in other areas of math.
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. In it, he had described a vision of a mathematical Rosetta stone with three columns. 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. Weil observed patterns that were appearing in all three areas.
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. In the number theory column, Galois representations connect to automorphic forms in the distant galaxy of harmonic analysis. For curves over finite fields, representations of their own Galois groups are likewise matched with automorphic forms. 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.
Decades of work and many of the field’s major prizes have gone toward proving conjectured Langlands correspondences in various settings. 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. Proving these correspondences also hastened progress in the connected research areas. “We can deduce things in one world using results in the other world,” said Jessica Fintzen, a mathematician at the University of Bonn.
We have to trust our mathematicians when they say there’s no apparent reason for the mysterious correspondences. “It’s the best possible world, but it’s very difficult to actually pinpoint what it means,” Fintzen said.
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. 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. Mathematically, these pure frequencies are sine waves, which, when shifted by regular intervals, don’t change. Picture sliding a sine wave left or right. If you shift it the right amount, it will appear not to have moved at all. The symmetries of sine waves are abelian; the order in which you shift the wave back and forth doesn’t matter. And in those waves, the coefficients that set the strength of each pure frequency are all simple numbers. Automorphic forms are like more sophisticated sine waves, and they can have non-abelian symmetries. In that case, the coefficients are tied to matrices and Galois representations — the original Langlands correspondence.
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. That a complicated object can be broken down into a spectrum of pure components is something of an organizing principle of modern science.
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. “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.
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. 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. 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. (The presence of electrically charged particles such as electrons breaks the mirror between them, because no equivalent magnetic charges exist.)
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. 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.
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. 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. Similarly, the various Langlands correspondences may eventually be understood as dual views of a single object or system.
Edward Frenkel, a mathematician at the University of California, Berkeley who has worked on the geometric Langlands program for decades, suspects so. “The real reason in my view is that there are things below the surface that have not yet been discovered,” he said.
On our video call, he held up a hot pink coffee cup, and we considered its shadows. “The projection onto the table will be a disk,” he said. “A projection on the wall will be a rectangle. 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. 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.
“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. Eventually we hope to find them, by finding more and more information which hopefully will lead us to the true explanation, the true reason.”