How Alexander Grothendieck Revolutionized 20th-Century Mathematics

W hat Albert Einstein was to 20th-century physics, Alexander Grothendieck was to 20th-century mathematics.

亚历山大·格洛腾迪克 (Alexander Grothendieck) 之于 20 世纪数学，就如同阿尔伯特·爱因斯坦 (Albert Einstein) 之于 20 世纪物理学一样。

He is much less well known because math gets technical even more quickly than physics does.

他的知名度要低得多，因为数学比物理更快地变得技术化。

But as with Einstein, Grothendieck’s impact came not just from his own results, revolutionary though they were.

但与爱因斯坦一样，格洛腾迪克的影响不仅仅来自于他自己的成果，尽管这些成果具有革命性。

His work also reoriented his entire discipline in radical new directions.

他的工作也将他的整个学科重新定位为全新的方向。

Grothendieck was intense and ascetic from his early days.

格洛腾迪克从早年起就非常热情和禁欲。

Starting in the early 1950s, when he was in his 20s, he produced thousands of pages of formal and informal notes that changed the course of mathematics.

从 20 世纪 50 年代初开始，当他 20 多岁的时候，他写下了数千页的正式和非正式笔记，改变了数学的进程。

Then in 1970, he quit.

然后在 1970 年，他辞职了。

He left his post at a prestigious research institute just outside of Paris to teach at the provincial university in Montpellier where he studied as an undergraduate.

他辞去了巴黎郊外一家著名研究机构的职务，在蒙彼利埃的省立大学任教，并在那里攻读本科学位。

He mostly stopped talking to other mathematicians.

他基本上不再与其他数学家交谈。

In the early 1990s, he moved to a small village in the Pyrenees, where he lived as a hermit.

20世纪90年代初，他搬到比利牛斯山脉的一个小村庄，过着隐士生活。

Mathematicians are still grappling with the innovations he made half a century ago.

数学家们仍在努力研究他半个世纪前所做的创新。

His work pushed mathematics to a new level of abstraction by focusing on the relationships between objects rather than the objects themselves.

他的工作通过关注对象之间的关系而不是对象本身，将数学推向了一个新的抽象水平。

“If there is one thing in mathematics which fascinates me more than any other (and undoubtedly always has), it is neither ‘number’ nor ‘size,’ but invariably shape,” he wrote in his memoirs .

他在回忆录中写道：“如果说数学中有一件事比其他任何事情都更让我着迷的话（毫无疑问总是如此），它既不是‘数字’也不是‘大小’，而是总是形状。”

“And among the thousand and one faces under which shape chooses to reveal itself to us, that which has fascinated me more than any other and continues to do so is the structure hidden in mathematical things.”

“在形状选择向我们展示自己的一千零一张面孔中，最让我着迷并且继续如此的是隐藏在数学事物中的结构。”

His revolutionary mathematics centered around that search for hidden structure.

他的革命性数学集中于对隐藏结构的探索。

Grothendieck is most famous for his work in algebraic geometry.

格洛腾迪克最著名的是他在代数几何方面的工作。

The field first developed as the study of shapes defined by polynomial equations — equations that add together variables raised to fixed powers.

该领域最初是作为对多项式方程定义的形状的研究而发展起来的——多项式方程将变量的固定幂相加。

These can be as simple as a line ( x – y = 0) or a circle ( x 2 + y 2 – 1 = 0).

它们可以像直线 (x – y = 0) 或圆 (x 2 + y 2 – 1 = 0) 一样简单。

But as you consider more and more variables raised to higher powers and also look for solutions that satisfy sets of many equations instead of just one, things quickly get more complicated — and more abstract.

但当你考虑越来越多的变量的更高次幂，并寻找满足多个方程组而不是一个方程组的解决方案时，事情很快就会变得更加复杂，也更加抽象。

Grothendieck, seen here in 1954, was fascinated by hidden geometric structure.

格洛腾迪克于 1954 年在此见到，他对隐藏的几何结构着迷。

“If there is one thing in mathematics which fascinates me more than any other (and undoubtedly always has), it is neither ‘number’ nor ‘size,’ but invariably shape,” he wrote.

“如果数学中有一件事比其他任何事情都更让我着迷（毫无疑问总是如此），它既不是‘数字’也不是‘大小’，而是总是形状，”他写道。

Paul R.

保罗·R。

Halmos photograph collection, e_ph_08592_pub, The Dolph Briscoe Center for American History, The University of Texas at Austin

Halmos 照片集，e_ph_08592_pub，多尔夫·布里斯科美国历史中心，德克萨斯大学奥斯汀分校

The discipline took flight in the late 19th century, when mathematicians started asking questions about what happens if instead of plugging ordinary numbers into your equations, you plug in numbers from other, more abstract sets.

该学科在 19 世纪末兴起，当时数学家们开始提出这样的问题：如果不是将普通数字代入方程，而是代入其他更抽象的集合中的数字，会发生什么情况。

Before Grothendieck, algebraic geometry was an interesting and vibrant subdiscipline within mathematics.

在格洛腾迪克之前，代数几何是数学中一个有趣且充满活力的分支学科。

But it was also somewhat in crisis, as the mathematician David Mumford later wrote .

但正如数学家大卫·芒福德后来写道的那样，它也陷入了某种危机。

“Every researcher used his own definitions and terminology, in which the ‘foundations’ of the subject had been described in at least half a dozen different mathematical ‘languages.’”

“每个研究人员都使用自己的定义和术语，其中该主题的‘基础’至少用六种不同的数学‘语言’进行了描述。”

Then “Grothendieck came along and turned a confused world of researchers upside down, overwhelming them with [a] new terminology …

然后，“格洛腾迪克的出现彻底颠覆了研究人员的困惑世界，用[一个]新术语压倒了他们......

as well as with a huge production of new and very exciting results.”

以及大量令人兴奋的新成果。”

Grothendieck came along and turned a confused world of researchers upside down.

格洛腾迪克的出现彻底颠覆了研究人员的困惑世界。

Grothendieck is most famous for introducing mathematical constructions that helped him and others prove longstanding conjectures, and that eventually became central objects of study in their own right.

格洛腾迪克最著名的是引入了数学结构，这些数学结构帮助他和其他人证明了长期存在的猜想，并最终成为了研究的中心对象。

His work also put algebraic geometry in the center of a web of many other areas of math — among them topology, number theory, representation theory, and logic.

他的工作还将代数几何置于许多其他数学领域网络的中心——其中包括拓扑、数论、表示论和逻辑。

“Grothendieck never worked directly in number theory,” said Brian Conrad of Stanford University, “but the ideas he introduced into algebraic geometry totally transformed how number theory is done.”

斯坦福大学的布赖恩·康拉德说：“格洛腾迪克从未直接从事数论研究，但他引入代数几何的思想完全改变了数论的研究方式。”

His first major result in algebraic geometry was his 1957 generalization of the Riemann-Roch theorem, a proof from a century earlier that dictates how the shape of a surface limits which functions can be defined on it.

他在代数几何方面的第一个重大成果是 1957 年对黎曼-罗赫定理的推广，该定理是一个世纪前的证明，它规定了表面的形状如何限制可以在其上定义的函数。

As Leila Schneps of the French National Center for Scientific Research wrote, Grothendieck’s proof “propelled him to instant stardom in the world of mathematics.”

正如法国国家科学研究中心的莱拉·施奈普斯（Leila Schneps）所写，格洛腾迪克的证明“使他立即成为数学界的明星”。

Thanks to his techniques, “a whole new wealth of operations becomes available,” Conrad said.

康拉德说，由于他的技术，“全新的丰富的操作变得可用”。

“It opens up a whole new way to think about why the theorem is true.”

“它开辟了一种全新的方式来思考为什么这个定理是正确的。”

Then, just as quickly, Grothendieck moved on to the next thing.

然后，格洛腾迪克同样迅速地转向下一件事。

At the 1958 International Congress of Mathematicians, he announced his intention to remake all of algebraic geometry.

在 1958 年国际数学家大会上，他宣布打算重塑整个代数几何。

He was going to do it with something called a scheme.

他打算用一种叫做计划的东西来做到这一点。

A decade earlier, the mathematician André Weil had conjectured a link between solutions to polynomial equations defined in two very different mathematical settings.

十年前，数学家安德烈·韦尔（André Weil）推测在两种截然不同的数学环境中定义的多项式方程的解之间存在联系。

The first was finite fields, number systems that operate according to a cyclical form of arithmetic.

第一个是有限域，根据循环算术形式运行的数字系统。

The second was complex numbers, which take our familiar, everyday numbers and add the square root of -1, called i .

第二个是复数，它采用我们熟悉的日常数字并加上 -1 的平方根，称为 i 。

Weil made four conjectures that related polynomials from one setting to those from the other.

韦尔提出了四个猜想，将一种设置的多项式与另一种设置的多项式联系起来。

These conjectures, Conrad said, “sound like communication between parallel universes.”

康拉德说，这些猜想“听起来像是平行宇宙之间的通信”。

André Weil posed four conjectures that not only became foundational pillars of algebraic geometry, but also linked the field to other major areas of study, including number theory.

安德烈·韦尔提出的四个猜想不仅成为代数几何的基础支柱，而且还将该领域与包括数论在内的其他主要研究领域联系起来。

As part of the effort to prove these conjectures, Grothendieck proposed his notion of a scheme.

作为证明这些猜想的努力的一部分，格洛腾迪克提出了他的方案概念。

The attempted proofs were “a primary motivation for the theory of schemes,” said Daniel Litt of the University of Toronto, but “what it really bought you was a whole lot more.”

多伦多大学的丹尼尔·利特（Daniel Litt）表示，尝试的证明是“图式理论的主要动机”，但“它真正带给你的远不止于此。”

Before Weil, mathematicians only really talked about equations like x 2 + y 2 – 1 = 0 by specifying the particular number system they wanted to work in.

在韦尔之前，数学家只是通过指定他们想要使用的特定数字系统来真正讨论像 x 2 + y 2 – 1 = 0 这样的方程。

The solutions to such equations would look quite different if x and y could only be integers, for example, versus if they could be any real number, or any complex number.

例如，如果 x 和 y 只能是整数，与它们可以是任何实数或任何复数，则此类方程的解看起来会非常不同。

Grothendieck found the right way to define abstract notions of space, new ways of thinking about spaces.

格洛腾迪克找到了定义空间抽象概念的正确方法，以及思考空间的新方法。

After Grothendieck came up with an explanation for why Weil’s conjectures are true, mathematicians came to believe that equations had meaningful structure independent of whether x and y were complex numbers, or elements of a finite field, or bananas.

在格洛腾迪克对韦尔猜想为何正确提出解释后，数学家们开始相信方程具有有意义的结构，与 x 和 y 是复数、有限域元素还是香蕉无关。

At first, this belief seems to make as little sense as saying that a sentence has meaning regardless of which language you choose its words from.

乍一看，这种信念似乎毫无意义，就像说一个句子无论你选择哪种语言的单词都有意义一样。

But Grothendieck defined mathematical structures that made it possible to make such statements rigorous and even intuitive to those who mastered his new language.

但格洛腾迪克定义的数学结构使得这些陈述对于那些掌握他的新语言的人来说变得严格甚至直观。

As Conrad explained, “Grothendieck found the right way to define abstract notions of space, new ways of thinking about spaces.” He recognized that “the way you probe the geometry of a space is not by looking at the points, but by studying other things.”

正如康拉德解释的那样，“格洛腾迪克找到了定义空间抽象概念的正确方法，以及思考空间的新方法。”他认识到“探索空间几何形状的方式不是通过观察点，而是通过研究其他事物。”

That’s where Grothendieck’s schemes came into play.

这就是格洛腾迪克的计划发挥作用的地方。

It takes some effort to construct even a simple scheme.

即使构建一个简单的方案也需要付出一些努力。

But if you read on, it’s possible to understand what schemes are and develop an intuition for why they’re useful.

但如果您继续阅读，就有可能了解什么是方案，并对它们为何有用产生直觉。

Schemes are geometric spaces that are built out of abstract algebraic ingredients.

方案是由抽象代数成分构建的几何空间。

Start with an abstract generalization of the integers called a ring.

从称为环的整数的抽象概括开始。

A ring is a set of elements that can be added, subtracted, and multiplied together, but that can’t always be divided.

环是一组可以相加、相减和相乘的元素，但不能总是被分割。

(In the ring of integers, for instance, you can’t divide 2 by 3, because 2 / 3 isn’t an integer.)

（例如，在整数环中，您不能将 2 除以 3，因为 2 / 3 不是整数。）

Grothendieck fathered five children.

格洛腾迪克是五个孩子的父亲。

Here he holds his fourth, Mathieu, who was born in 1965.

他在这里怀着他的第四个孩子马蒂厄 (Mathieu)，他出生于 1965 年。

Now look at a subset of your ring that is “closed,” meaning that if you add or subtract two elements of the subset, the result is also in the subset.

现在看看环的一个“闭合”子集，这意味着如果您添加或减去子集中的两个元素，结果也在子集中。

For example, take all multiples of 5.

例如，取 5 的所有倍数。

This subset is not only closed, it has another property: You can multiply any number in the ring by an element in the subset, and the result is inevitably also in the subset.

这个子集不仅是封闭的，它还有另一个性质：你可以将环中的任意数字与子集中的元素相乘，结果也必然在子集中。

That makes the subset what mathematicians call an ideal.

这使得该子集成为数学家所说的理想子集。

Moreover, if you multiply any two numbers from the ring and end up in this subset (3 × 5 = 15), then one of the numbers you multiplied (5) must have been in this subset, too, even though the other number (3) isn’t.

此外，如果将环中的任意两个数字相乘并最终进入该子集 (3 × 5 = 15)，则相乘的其中一个数字 (5) 也一定位于该子集中，即使另一个数字 (3) 不在该子集中。

This second property makes the subset a prime ideal.

第二个性质使该子集成为素理想。

(To see why, look at the multiples of 6.

（要了解原因，请查看 6 的倍数。

These form an ideal, but not a prime ideal, because 2 × 3 is in the ideal, but neither 2 nor 3 is.)

它们形成一个理想，但不是素理想，因为 2 × 3 属于理想，但 2 和 3 都不是。）

In the case of the integers, the prime ideals are sets of multiples corresponding to each of the prime numbers, along with zero.

在整数的情况下，素数理想是与每个素数相对应的倍数集合以及零。

It’s possible to study the set of all the prime ideals of a ring as a single geometric space.

可以将环的所有素理想的集合作为单个几何空间来研究。

First, represent each prime ideal as a point.

首先，将每个素理想表示为一个点。

Then define a “topology” on those points that puts them into neighborhoods, depending on their shared elements.

然后在这些点上定义一个“拓扑”，根据它们的共享元素将它们放入邻域中。

(Strangely, the zero ideal ends up being “close” to every single prime, illustrating a previously unknown structure hidden behind the integers.)

（奇怪的是，零理想最终“接近”每个素数，说明了隐藏在整数后面的先前未知的结构。）

Grothendieck’s innovation was to add a layer on top of this space — a recently discovered mathematical superstructure called a sheaf, which carries additional algebraic information.

格洛腾迪克的创新是在这个空间的顶部添加一个层——一个最近发现的称为层的数学上层结构，它携带额外的代数信息。

At each point in your space, for instance, this sheaf attaches another set, called a stalk.

例如，在您空间的每个点上，这捆捆都附有另一组，称为茎。

Let’s return to one of the prime ideals of the integers: the point in our space representing the subset of all multiples of 5.

让我们回到整数的素理想之一：空间中代表 5 的所有倍数子集的点。

The stalk attached to this point would contain all the fractions whose denominators are not divisible by 5.

附着在该点上的茎将包含所有分母不能被 5 整除的分数。

(The stalk attached to 0 contains all possible fractions.) In this simple example, it’s hard to see what the stalks accomplish, but in more elaborate schemes, computing the contents of stalks and the ways they interact with each other turned out to be a mathematically powerful machine.

（附加到 0 的茎包含所有可能的分数。）在这个简单的例子中，很难看出茎完成了什么，但在更复杂的方案中，计算茎的内容以及它们相互作用的方式被证明是一个数学上强大的机器。

This entire object — the space of prime ideals, with the sheaf (and all its stalks) built on top of it — is called an affine scheme.

这整个物体——素理想的空间，其上建有捆（及其所有茎）——被称为仿射方案。

In general, schemes are constructed by gluing affine schemes together in a precise mathematical way.

一般来说，方案是通过以精确的数学方式将仿射方案粘合在一起来构造的。

So what does all that have to do with an equation like x 2 + y 2 – 1 = 0?

那么这一切与 x 2 + y 2 – 1 = 0 这样的方程有什么关系呢？

Well, instead of starting with the ring of integers, you can study a particular ring associated with that polynomial.

那么，您可以研究与该多项式相关的特定环，而不是从整数环开始。

You can then build the scheme for that ring.

然后您可以为该环构建方案。

But crucially, the variables x and y can be whatever you want them to be: integers, real numbers, complex numbers, elements of a finite field.

但至关重要的是，变量 x 和 y 可以是您想要的任何内容：整数、实数、复数、有限域的元素。

By studying the scheme’s properties, you can gain insight about the structure of the equation, independent of any particular number system.

通过研究该方案的属性，您可以深入了解方程的结构，而与任何特定的数字系统无关。

It is, impossible though it may sound, a way to study the sentence apart from the language its words are written in.

尽管听起来不可能，但这是一种脱离句子所用语言来研究句子的方法。

Grothendieck lived the final decades of his life as a recluse in the French countryside.

格洛腾迪克在法国乡村隐居，度过了他生命的最后几十年。

This 2013 photo was taken a year before his death.

这张 2013 年的照片拍摄于他去世前一年。

Broadly speaking, this is why Grothendieck and others could use schemes — and a series of ideas building on them — to re-prove one of the four Weil conjectures and prove two more.

从广义上讲，这就是为什么格洛腾迪克和其他人可以使用方案以及基于它们的一系列想法来重新证明四个韦尔猜想之一并证明另外两个。

(Grothendieck’s student Pierre Deligne would later use other structures that Grothendieck developed to prove the fourth, which is a version of the famous Riemann hypothesis in the setting of finite fields.) Grothendieck continued to come up with even more abstract and powerful concepts, including topoi, stacks, motives, and étale cohomology.

（格洛腾迪克的学生皮埃尔·德利涅后来使用格洛腾迪克开发的其他结构来证明第四个结构，这是有限域背景下著名的黎曼假设的一个版本。）格洛腾迪克继续提出更抽象和更强大的概念，包括拓扑、堆栈、动机和étale上同调。

All play a major role in algebraic geometry and other areas of math today.

所有这些在当今代数几何和其他数学领域都发挥着重要作用。

Schemes gave mathematicians a novel, systematic way to study the relationships between objects in algebraic geometry.

方案为数学家提供了一种新颖、系统的方法来研究代数几何中对象之间的关系。

And because schemes allow you to study rings, which appear all over math, as geometric spaces, they can be used to import geometric techniques into algebra, number theory, and beyond.

由于方案允许您研究作为几何空间出现在数学中的环，因此它们可用于将几何技术导入代数、数论等领域。

Grothendieck died in 2014 after years of solitude, estranged from the mathematical community he had helped create.

格洛腾迪克在经历了多年的孤独之后于 2014 年去世，他与他帮助创建的数学界疏远了。

Nonetheless, mathematicians remember him with reverent affection.

尽管如此，数学家们仍然怀着崇敬的感情纪念他。

As the Harvard mathematician Barry Mazur wrote, “During the early ’60s, his conversations had a secure calmness.

正如哈佛大学数学家巴里·马祖尔 (Barry Mazur) 所写：“在 20 世纪 60 年代初，他的谈话充满了安全的平静。

He would offer mathematical ideas with a smile that always had an expanse of generosity in it …

他会带着微笑提出数学想法，脸上总是带着慷慨的笑容……

a sense that ‘nothing could be easier in the world’ than to view things as he did.”

一种‘世界上没有什么比像他那样看待事物更容易的感觉了’。”

His ideas were complicated, but “most of the arguments are very straightforward once you set things up,” Litt said.

他的想法很复杂，但“一旦你把事情弄清楚，大多数论点都非常简单，”利特说。

“You just keep going and going and going.

“你就继续走啊走啊。

He found us the highway.”

他给我们找到了高速公路。”

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获取发送到您的电子邮件收件箱的最重要新闻的摘要

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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。

Moderators are staffed during regular business hours (New York time) and can only accept comments written in English.

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。