两位研究人员正在从零开始重建数学
内容总结：
两位数学家试图从零开始重建数学根基
拓扑学常被通俗地描述为“橡皮泥几何”，认为甜甜圈和咖啡杯在拓扑学家眼中是一样的——因为它们都有一个洞。但这一描述忽略了关键问题：数学家究竟如何严格证明这种“拉伸”是合理的？实际上，他们并非靠直觉判断，而是通过一种能“遗忘”距离、同时保留底层结构的描述方式，允许物体弯曲和拉伸。
100多年前诞生的“拓扑空间”概念，曾是数学从日常数字和形状走向抽象思维的关键里程碑，成为现代数学的基石。然而，令人不安的是，拓扑空间在代数领域表现得极其糟糕——而代数恰恰是数学家非常喜欢做的事情。多年来，数学家们只能无奈接受这一局限。
但过去十年间，德国波恩大学的彼得·肖尔策和法国高等科学研究所的达斯廷·克劳森试图推翻拓扑空间，用名为“凝聚集”的新数学对象取而代之。凝聚集类似于一种无限精细的“尘埃”，既保留了拓扑空间的所有优点，又消除了其缺陷。斯坦福大学数学家拉维·瓦基尔评价说：“他们解决了一个我们都没意识到存在的问题，让一大片数学领域变得简单了许多。”
拓扑学的困境与突破
拓扑学的发展并非一帆风顺。1735年欧拉证明“七桥问题”时，拓扑学概念尚未诞生；莫比乌斯带等经典拓扑对象的研究也因缺乏合适语言而步履维艰。直到1912年，德国数学家费利克斯·豪斯多夫在《集合论基础》中首次系统定义了拓扑空间，才奠定了现代拓扑学的基础。
但问题随之而来：拓扑空间与代数的结合极其不畅。尽管“拓扑学是给代数的巨大礼物”，但爱丁堡大学的克拉克·巴威克指出，“它同时阻碍了进展，因为拓扑和代数就是不太合拍。”1945年范畴论诞生后，数学家们本希望借此连接拓扑与代数，但拓扑阿贝尔群缺乏范畴论所需的优良性质，就像“开着一辆总在抛锚的破车”。
凝聚集：数学的“新地基”
2013年，25岁的肖尔策与合作者在论文中无意定义了一种奇怪的集合，当时并未重视。后来克劳森出于不同原因独立发现了同样的结构，两人从2018年开始合作研究。2019年，肖尔策在波恩大学开设系列讲座，发布77页笔记，并给出了一个重要定理的全新优雅证明。
凝聚集的核心思想是用“完全不连通的尘埃”来构建连续对象。例如，康托尔集就是最简单的凝聚集——不断去掉线段中间三分之一后留下的点集，没有任何两个点相邻。肖尔策解释说：“这些完全不连通的碎片在代数上极其容易描述。”凝聚集构成的特殊范畴，最终让拓扑、代数和其他数学分支能够“以一种非常实用而精确的方式融合”。
一场悄然的数学革命
自2019年以来，肖尔策和克劳森不断用凝聚集构建新的数学结构，从“轻凝聚集”到“固体”“液体”乃至“气体”空间，发展出完整的“凝聚数学”。两人自认为并非拓扑学家，但他们的工作被一些数学家比作20世纪50-60年代格罗滕迪克对代数几何的重塑。巴威克说：“凝聚集的真正激动人心之处，在于它让你看到从未注意过的自然对象——就像一座未被攀登的山峰。”
肖尔策和克劳森的工作表明，选择正确的语言至关重要。正如肖尔策所言：“试图探究这些现象的本质，就是在寻找表达它们的语言。”而格罗滕迪克曾在回忆录中写道，数学家的角色不是发明，而是发现——最美丽的房子，是忠实反映事物隐藏结构与美的那一座。
中文翻译：
两位研究者正从零开始重建数学
克里斯蒂娜·阿米蒂奇/《量子杂志》
让我们从一个或许是最老套、最滥用的数学笑话开始：拓扑学家是那种分不清咖啡杯和甜甜圈的人。你看，这两者都有一个洞。
拓扑学通常被描述为一种“橡皮膜”几何学，在这种几何中，如果两个形状可以通过拉伸或压缩（而不撕裂）相互转化，就被视为相同。但这种概括遗漏了关键的一点：拓扑学家以及许多依赖其方法的其他数学家，如何严谨地解释所有这些拉伸行为？他们不会盯着甜甜圈和咖啡杯，眯起眼睛对自己说：“当然，我能凭直觉看出如何把一个挤成另一个，所以它们一定是相同的。”相反，他们以一种能够“忽略”距离的方式描述形状，同时灵活地尊重底层结构，允许它弯曲和拉伸。
当这些“拓扑空间”在100多年前被提出时，它们在逻辑和集合论的革命中扮演了重要角色，这标志着19世纪数学与现代数学的分界。拓扑空间的诞生是数学从人们日常生活中遇到的数字和形状，不可阻挡地向更抽象的思想洞穴迈进的关键里程碑。此后，拓扑空间成为数学巨大部分领域的基础。如果你把数学想象成一座摩天大楼，拓扑空间就是混凝土桩，深深打入最终支撑所有数学的常识基石中。
但令人不安的是，拓扑空间被证明极不适合现代数学的很大一部分：它们是进行代数的尴尬环境，而代数正是数学家们非常喜欢做的事情。
多年来，数学家们以为他们只能接受拓扑空间的局限性。如果你在一座摩天大楼的第87层工作，修复地下室的 foundations 是一个令人望而生畏的提议。
但在过去十年中，波恩大学的彼得·肖尔策和法国高等科学研究所的达斯廷·克劳森试图取代拓扑空间。他们定义了一类新的数学对象，称为凝聚集，它类似于一种无限精细的尘埃，保留了拓扑空间所有最佳性质，而没有其缺点。事实证明，尘埃是比拓扑空间那种卵石般、已被充分理解的土壤更好的基础材料。
“他们在解决我们不知道自己存在的问题，”斯坦福大学数学家、美国数学学会会长拉维·瓦基尔说，“因为我们已经有了我们认为是合理的解决方案。”结果是，“一大批数学领域变得简单多了。”
这是一个雄心勃勃的项目。肖尔策和克劳森引入的新定义和新概念既强大又复杂，难以学习。肖尔策本人也不确定它们会被广泛使用到何种程度。另一方面，他认为这只是更宏大计划的第一步——理解数字行为为何如此。
做数学有点像攀岩：就像你在陡峭岩壁上选择的路线可以通过技术动作的顺序展现出创造性和优雅一样，数学证明也是如此。两者都是在现有的地形上穿行。大多数研究——甚至一些顶尖研究——都以寻找通往已知高峰的新路线为形式。但在数学中，工具与地形之间存在一种奇特的关系，仿佛发明一种新型冰镐就会引发迄今为止未知的山脉出现。当这些新山脉在地平线上显现时，曾经看似陡峭难登的旧山开始变得像平缓的山丘一样。
开发这些新工具需要某种革命性的自信——尤其是当它要求搁置那些在学界使用了如此之久、以至于似乎已成为山脉一部分的工具时。
已知之点
即使没有良好的工作语言，也有可能发现强大的数学真理。
也就是说，拓扑学先于拓扑空间存在。早在1735年，莱昂哈德·欧拉就证明，不可能在不重复经过七座桥的情况下走遍柯尼斯堡（他居住的城市）。这是一个典型的拓扑学结果——城市中每块陆地的大小无关紧要，连接它们的桥的长度也不重要。只有它们相互连接的图案才重要。
在近200年的时间里，拓扑学研究断断续续地推进。19世纪中期，奥古斯特·费迪南德·莫比乌斯分析了以他命名的带子：一条扭转一次后两端接合的带子。这可以说是最有实际用途的最奇怪拓扑对象——例如，用于单面传送带，它在驱动机器时磨损均匀。大约在同一时期，莫比乌斯开始引入该领域的一些关键思想，例如如何通过观察在形状上绘制的环来分类具有不同数量孔洞的形状。
不久之后，伯恩哈德·黎曼、亨利·庞加莱等人取得了进一步进展。但他们因缺乏合适的语言而苦苦挣扎。正如澳大利亚数学家约翰·斯蒂尔韦尔在2009年写到庞加莱在拓扑学中的开创性工作时所说：“伴随着重大突破，也有混乱。”庞加莱有想法，但没有足够的词汇来恰当表达。
对外行人来说，似乎最接近拓扑学的数学分支应该是几何学。似乎拓扑学就是几何学，只是研究对象是可弯曲的而非刚性的。但解决庞加莱困惑的答案并非来自几何学，而是来自一个新兴的逻辑分支——集合论。
在20世纪初，研究人员试图将数学建立在更稳固的基础上。他们直到最近才意识到自己对数字的日常直觉是完全错误的；现在他们激烈辩论着应该把哪些公理（即显而易见的真理）作为构建理论的基础。他们最直接假设的表述方式中看似微小的差异，都会对什么可能或不可能被证明产生重大影响。
他们用集合论来梳理这些关于数学基础的争论。1912年，刚刚开始在波恩大学任教（数代之后肖尔策将在此工作）的费利克斯·豪斯多夫开始撰写第一部系统论述集合论的著作。当时，四十多岁的豪斯多夫已经是一位颇有成就的作家：他以笔名保罗·蒙格雷出版了一本诗集、两本试图调和尼采和康德的哲学著作，以及一部在40个城市上演的戏剧。作为数学家，他虽已成功但尚未成为巨星。
这种情况在1914年他的《集合论基础》一书出版后发生了改变。在书中，他首次描述了拓扑空间。拓扑空间简单说来就是一组项目，它们被分组为豪斯多夫所称的邻域——今天称为开集。开集为空间赋予了结构。
开集只需满足两个条件。第一，任何开集的组合也必须是开集。（如果布鲁克林构成一个邻域，皇后区构成一个邻域，那么布鲁克林和皇后区合起来必须算作一个更大的单一邻域。）第二，开集之间的任何有限交集也必须是开集。
拓扑空间可以是有限的或无限的。它们可以编码复杂结构，或完全没有结构。正如肖尔策所说，拓扑空间“无处不在。只要你有关于点彼此靠近的直觉概念，你就有拓扑学。”
考虑一个更熟悉的对象：数轴。当我们思考数字之间的相互关系时，我们实际上只考虑一个拓扑空间：所谓的标准拓扑，其中每个可能的区间（不包括其端点）构成一个邻域或开集。所有这些区间都是开集：
这种拓扑赋予了实数我们熟悉的结构。
但你可以另辟蹊径，拿掉你熟悉的数字，忘记你建立起来的所有日常直觉，然后在上面定义完全不同的拓扑空间。如果你把每个数字想象成图书馆里的一本书，这就像把所有的书从它们通常的书架上拿下来，用一种完全不同的方式重新排列。
例如，你可以把数轴揉成一个球，然后把这个球挤压成一个点，使每个数字任意接近其他每个数字，就像这样：
这类似于把所有书杂乱地堆成一堆。书之间的关系——比如历史小说都在同一书架上等等——就丢失了，因为所有的书现在都成了邻居。这被称为不可分拓扑，我们通过声明只有两个开集来创建它：空集和整个数轴。
另一个极端是“离散”拓扑，其中每个点构成自己的邻域：
在这种拓扑中，每个点都不与其他任何点接触，而不是每个点与其他每个点都有接触。这就像把图书馆里的每本书放在它自己的私人孤岛上。关系再次丢失，因为每本书都被孤立了。
在某种程度上，关于甜甜圈和咖啡杯的那个老笑话误解了拓扑思维的力量。关键不在于可以通过拉伸或压缩来改变距离。而在于，在距离根本不存在的情况下，仍然能够有意义地思考空间中的结构。
通过这种方式，拓扑空间使得探索数学图景中的新领域成为可能。例如，它们提供了一种新的、无距离的方式来理解连续性和连通性等概念，这是一种极其强大且反直觉的能力。这使得数学家能够将这些思想推广到更广泛的场景中——并在各种领域中证明重要的命题。例如，像代数基本定理这样的结果，其证明曾难倒像卡尔·弗里德里希·高斯这样的数学巨人，但一旦拓扑论证介入，就会得到非常简单的证明。
拓扑空间的引入也促使数学家提出他们原本可能不会想到的新问题。正如任何好的数学定义一样，拓扑空间既开启了新视野，又让穿越已知领域变得更加容易。
豪斯多夫的著作可以说是现代拓扑学的开端。正如后来被称为布尔巴基的数学集体所写，他精心挑选的定义赋予“他的理论以完全精确和充分的普遍性。他阐述这些公理后果的那一章 仍然是公理化理论的典范——抽象，却具有预见性。”
无尽海洋中的一滴水
现代数学中有许多子学科。每个子学科都有自己的词汇、语法和思想风格。但它们从未完全分离——它们以奇特的方式相互作用。部分原因在于，许多数学对象，如实数，同时作为多个学科的研究对象存在：它们具有代数结构、分析结构、组合结构和拓扑结构（以及其他结构！）。通常，重叠的领域最终会成为独特的研究领域。
1945年，两位美国数学家塞缪尔·艾伦伯格和桑德斯·麦克莱恩发表了一篇大胆的论文，创造了一个全新的学科，现在称为范畴论。范畴论一举在数学其他现有领域之间建立了一套高速公路。
艾伦伯格和麦克莱恩将“范畴”定义为对象以及对象之间的关系（称为态射）的集合。例如，一个范畴可以由集合以及将这些集合联系起来的函数组成，或者由向量空间以及将一个向量空间变换为另一个的线性映射组成。
但这对数学家理论真正的力量在于他们引入的下一个抽象层次：他们问道，如果你用一个他们称之为函子的东西，将整个范畴映射到另一个范畴上会怎样？函子以有序的方式将对象映射到对象，将态射映射到态射。换句话说，它不仅提供了一种将一组对象翻译成另一组对象的方法；它还保留了它们之间的关系——允许你将数学的不同领域相互连接。
除了其他目标外，艾伦伯格和麦克莱恩希望将拓扑学与数学的其他部分联系起来。拓扑学已知与数学最重要的领域之一——代数——不能很好地融合。尽管“拓扑学是给代数的巨大礼物，”爱丁堡大学的克拉克·巴威克说，但“它同时也阻碍了进步，因为拓扑学和代数不能很好地互相配合，这是由拓扑学构建的特定方式造成的。”
范畴论很快从陈述看似显而易见的事情，发展到基于其从业者通常亲切地称为“抽象废话”的东西，得出强大的数学结论。一些范畴具有特定的性质，使它们比其他范畴对数学家更有用。在这些范畴中，抽象废话变得强大有力——而在其他范畴中，它最终只是废话。
在拓扑学中，你可以定义一个由拓扑空间及其之间的连续映射组成的范畴。与此同时，在代数学中，一个重要的范畴是阿贝尔群范畴——这些群具有有用的对称性。如果你想对代数对象在拓扑意义上的行为方式，或者反过来，代数考量如何支配拓扑构造，进行一个宏大的综合，那么自然的做法是用同时具有拓扑和代数结构的对象来形成一个范畴，称为拓扑阿贝尔群。
但拓扑阿贝尔群缺乏范畴论学家所期望的特定性质。如果范畴论揭示了数学不同山脉之间一条迄今隐藏的高速公路网络，那么想要在这条高速公路上行驶的拓扑学家就只能开着一辆会不断抛锚、需要修理的破车。
这就是肖尔策和克劳森最终试图解决的问题。正如肖尔策告诉我的，“我认为拓扑学家实际上并不喜欢拓扑空间，因为它不是一个方便的范畴。”如果他们能定义新对象来取代拓扑空间——这些对象既能保留其力量，又能创建一种更好的范畴，最终让数学家能够将拓扑学与代数学及其他领域连接起来呢？
最友善的革命者
有时，新想法似乎即将涌现于世。
2013年，肖尔策25岁，已经作为一名深邃的数学思想家引起轰动，被誉为德国最年轻的正教授。
他与当时在新泽西州普林斯顿高等研究院工作的巴尔加夫·巴特合著了一篇论文，在文中他们提出了对特定类型范畴的新定义。顺便提一下，两人定义了一个有些深奥的数学集合，“一个点的pro-étale site上的层”。肖尔策当时并未多想。这些他们未命名的集合，“感觉像是故事中一个我并未完全理解的奇怪方面，”肖尔策告诉我。
当时，克劳森正在麻省理工学院完成博士学位。在哥本哈根大学做了五年博士后研究员后，他于2018年搬到波恩与肖尔策合作，肖尔策当时刚获得数学最高荣誉菲尔兹奖。克劳森出于不同原因独立得出了同一种集合，并说服肖尔策他们应该更仔细地研究它。肖尔策回忆道，到这个时候，克劳森已经瞥见了取代拓扑集的一个潜在机会。
大约在同一时间，巴威克和他当时的学生彼得·海恩独立提出了一个略有不同的定义，以回答他们感兴趣的范畴论中的一个特定问题。“我们想解决一个问题，”海恩说。“我们大致理解这个理论应该对很多东西有用，但我们真正想做的只有一件事：证明那种特定的结果，以推广我们之前所做的事。”
另一方面，谈到克劳森和肖尔策，他说，“我认为他们想要做的要多得多。”
的确如此。他们给他们的集合起了一个名字——“凝聚集”——然后开始工作。他们没有发表渐进式的进展。相反，2019年4月，肖尔策开始在波恩大学讲授“凝聚数学”；5月，他发布了一套77页的讲义，最终以对一条重要定理（称为凝聚对偶）的新颖优雅证明收尾。正如后来与肖尔策合作的约翰·科梅林回忆的那样，“凝聚对偶之前有一个极其迂回和技术的证明。”肖尔策和克劳森的证明干净而优雅。
这，科梅林说，“让人们有动力说：‘让我们在世界各地组织阅读小组和学习研讨会，来研读这些讲义，看看发生了什么。’”科梅林在弗莱堡大学组织了这样一个小组，但这很有挑战性。“我认为我们组里没有人理解全部细节，”他说。
尽管凝聚集最初是作为证明有用结果的工具而诞生的，肖尔策回忆道，它们很快就变得不仅仅是工具了。“对我个人而言，凝聚集改变了我思考数学的一些非常基本的东西，”他说。
“用凝聚视角取代拓扑视角，”他补充道，“渗透到我做的每一件事中。”
巴塞罗那大学的集合论学家杰弗里·伯格福克回忆起2019年第一次见到肖尔策和克劳森的情景，当时他和同为集合论学家的捷克科学院克里斯·兰比-汉森在线发布了一篇技术论文。集合论学家形成了一个特别小而紧密的社群，在某种程度上远离数学主流。“我们只期待从我们认识的人那里得到回复，”伯格福克说。但他们收到了克劳森的一封电子邮件，他注意到了这篇论文。邮件中说，他本人和肖尔策正在思考与凝聚数学相关的类似问题——这是一个伯格福克和兰比-汉森都没听说过的新兴学科。
伯格福克花了一会儿才意识到肖尔策和克劳森对凝聚数学的雄心有多大。“达斯廷和彼得非常乐于回复，非常酷和善于沟通，”他说。“他们并不需要像他们实际那样友善。”而且，尽管肖尔策和克劳森在他们专业领域之外工作，伯格福克指出，他们问到了那种你本以为只有集合论学家才会关心的事情。“他们让我们朝着我们本来就想要思考的方向去想，”他说。“这意味着我们必须学习凝聚数学。”
对于正在重塑20世纪数学一大块的两个人来说，克劳森和肖尔策相当低调。“在很大程度上，我所做的是用自己的话复述别人做过的事，”肖尔策在2021年一次采访中对数学家玛丽亚·亚克森说。“我对定理或证明不是那么感兴趣。”他说，他想做的是提出新的定义：“它们必须使陈述有趣的定理变得容易，并且必须使证明它们变得容易。”肖尔策不认为自己有创造力。他说，他只是“试图给已经存在的东西命名。”
至于克劳森——正如他大约在同一时间对亚克森在另一次采访中所说——他避免发表论文，因为他认为科学出版业存在根本缺陷。他甚至很大程度上避免非正式地撰写结果，将这事留给合作者。他只想专注于数学；像肖尔策一样，他不断寻找正确的名字，正确的语言。（事实上，他曾一度考虑从事文学翻译的职业。）
“我从未完全信服拓扑空间，”克劳森说。它们无法让他理解“那个存在的世界，那个我们试图触及但没有恰当语言来谈论的丰富世界。”
但这只进一步激励了他。“不理解让我非常快乐，”他说，“因为当我最终理解时，我更加快乐。”
建立在尘埃之上
这就是凝聚集的用武之地。凝聚集可以被视为一种从“完全不连通”空间（如肖尔策所述，就像用不同的面粉和糖颗粒做成蛋糕一样）构建连续对象（如实数）的配方。
以所谓的康托尔集为例。从包含0到1之间所有实数的线段开始，移除中间三分之一。然后从剩下的线段中移除中间三分之一。无限重复这个过程，你将得到一堆“尘埃”般的点。没有哪个点紧挨着另一个点。这个点的空间是完全不连通的。
康托尔集是最简单的凝聚集，也是制造其他凝聚集的基本构建块。肖尔策说，你可以通过以一种奇特方式将像康托尔集这样的点云撞在一起，来制造更复杂的凝聚集。
这种尘埃可能看起来陌生，但肖尔策指出我们一直在使用它。例如，当你用十进制展开表示数字时，你本质上就是把数字看作类似的一种尘埃。这就像取展开中的每个数字，并切出它在数轴上的部分，就像这样：
通过这种方式，产生一个给定的数字实际上涉及无限地“解剖”数轴。正如肖尔策所说，“十进制展开描述了一个完全不连通的空间，因为每增加一个新数字，你就越来越多次地切割你的数轴。”每个数字与其他所有数字完全不连通。
那么，如何用这样一个不连通的集合来得到一个连续的对象，比如我们习以为常的实数轴呢？你必须通过将诸如此类的数粘合回去，例如将0.49999999999999999…与0.5等同（0.50999999999999999…与0.51等同，依此类推）。
肖尔策和克劳森的凝聚集工作原理类似：它们是不连通的，但可以用来构建和研究连续对象，就像你在拓扑学中想要理解的那些。而且，如果从它们而不是拓扑空间开始，你会得到一个额外的好处：事实证明，肖尔策解释说，“这些完全不连通的片段在代数上极其容易描述。”
凝聚集形成了一种特殊类型的范畴，根据肖尔策在马克斯·普朗克数学研究所的合作者胡安·埃斯特班·罗德里格斯·卡马戈的说法，这终于使得能够“以一种非常实用和精确的方式”混合拓扑学、代数学和其他领域。
肖尔策和克劳森首先使用他们的凝聚集重新证明了以前依赖拓扑空间的旧结果——比如代数基本定理。这些新证明为数学家提供了一种新颖的理解，“如同被适当润滑、涂油并变得柔韧，”瓦基尔说。“你越能将其融入直觉，就理解得越好。”
然后，两人决定更进一步。
一部凝聚史
自2019年以来，肖尔策和克劳森不断用他们的凝聚集构建新型结构——并发布新的讲义。“波恩的 ideas 发展得比世界其他地方吸收它们的速度快得多，”科梅林说。有“轻”凝聚集，还有固态、液态，再到气态空间——一整套凝聚数学。
克劳森和肖尔策都不认为自己是一名拓扑学家。如果两人不那么友善，或者他们的 ideas 不那么有效，那么他们试图重建一个他们不常涉足的领域的基础，可能会招致一些不满。“我不想强加任何东西，”当被问及他认为凝聚数学会产生什么影响时，肖尔策说。他和克劳森听起来像是在享受乐趣，试图提出他们自己觉得有用的 idea。
但一些数学家将他们的工作比作20世纪50年代和60年代发生的一场类似的数学革命——当时亚历山大·格罗滕迪克根据范畴论重新构想了代数几何领域，极大地扩展了其范围和力量。格罗滕迪克对现代数学的影响是深远的。现在，根据阿尔伯塔大学博士后研究员、克劳森的前研究生达古尔·阿斯盖尔松的说法，“我认为在这个意义上把彼得与格罗滕迪克相提并论是公平的。他正在以某种方式重塑一切。”
“对我而言，凝聚相关的东西真正令人兴奋之处在于定义新的研究对象的可能性，”巴威克说。它“向你展示存在这样一个你从未注意过的自然对象类，就像一座未被攀登过的山。我们只是在啃噬这片广阔领域的角落。”
在一组讲义中，克劳森和肖尔策引用了著名数学家大卫·芒福德的一句名言。芒福德的代数几何领域，他说，“似乎已经获得了深奥、排他且非常抽象的名声，其拥护者秘密策划接管数学的其他所有部分。”克劳森和肖尔策接着指出，他们的计划是利用同样深奥、排他且抽象的凝聚数学，继续这一努力未竟的事业。他们并不完全是在开玩笑。
无论如何，他们不那么秘密的“接管数学”仍在继续。在最近几年里，克劳森和肖尔策定义了其他新颖的数学对象，例如“解析叠”和“gestalten”。一些数学家认为这些比凝聚集更为重要。
肖尔策怀疑凝聚数学甚至可能在他们和克劳森的核心兴趣——数论之外的领域证明是有用的。肖尔策指出，量子场论——现代物理学的一个核心方面，长期以来一直与其基础作斗争——使用了非常复杂的代数和范畴论。“与此同时，”他补充道，“量子场论本质上是非常解析和拓扑的。混合这两个世界并非易事，但凝聚数学提供了一个可能的框架来做到这一点。”
肖尔策和克劳森的全部工作表明，选择正确的语言有多么重要——重新构架概念如何使穿越已知领域变得更容易，并探索新的数学视界。“试图触及这些现象的本质，就是试图找到一种语言来表达它们，”肖尔策说。
在格罗滕迪克晚年出版的一本回忆录中，他将数学家描述为建造者，尽管他主张他们当然不是在发明任何东西，只是在发现已经存在、等待被发现的结构。他写道：“最美丽的房子，即建造者的爱意体现得最明显的房子，不是比其他房子更大或更高的那一座。相反，一栋房子如果忠实地反映了事物中隐藏的结构和美丽，它就是美丽的。”
英文来源：
Two Researchers Are Rebuilding Mathematics From the Ground Up
Kristina Armitage/Quanta Magazine
Let’s start with what’s probably the most tired, overused joke in math: A topologist is someone who can’t tell a coffee cup from a doughnut. Both, you see, have a hole in them.
Topology is usually described as a sort of “rubber sheet” geometry in which two shapes are considered the same if one can be stretched or compressed into the other without tearing it. But this summary leaves out something essential: How do topologists, and the many other mathematicians who rely on their methods, rigorously account for all this stretching? They don’t look at a doughnut and a coffee cup, squint, and say to themselves, “Sure, I can intuitively see how to squeeze one into the other, so they must be the same.” Rather, they describe a shape in a way that can “forget” about distance while respecting the underlying structure in a flexible way, allowing it to bend and stretch.
When these “topological spaces” were developed over 100 years ago, they played a major part in the revolutions in logic and set theory that marked the boundary between 19th-century and modern mathematics. Their birth was a crucial waypoint on math’s inexorable march from the numbers and shapes that people encounter in everyday life into ever more abstract caverns of thought. Topological spaces have since become the foundation for huge chunks of mathematics. If you think of math as a skyscraper, topological spaces are concrete pilings, driven deep into the bedrock of common sense that all of math ultimately rests on.
But disconcertingly, topological spaces turn out to be extremely poorly suited for a big chunk of modern math: They are an awkward setting in which to do algebra, which is something mathematicians quite like doing.
For years, mathematicians figured they just had to live with the limitations of topological spaces. If you’re working on the 87th story of a skyscraper, fixing the foundations in the subbasement is a scary proposition.
But over the past decade, Peter Scholze of the University of Bonn and Dustin Clausen of the Institute of Advanced Scientific Studies in France have sought to replace topological spaces. They have defined a new category of mathematical objects called condensed sets, which resemble a sort of infinitely fine dust and retain all the nicest properties of topological spaces without the drawbacks. Dust, it turns out, is a better foundational material than the pebbly, well-understood soil of topological spaces.
“They are solving a problem we didn’t know we had,” said Ravi Vakil, a mathematician at Stanford University and president of the American Mathematical Society, “because we already had what we thought were reasonable solutions.” As a result, “a whole slate of mathematics has become much simpler.”
It’s an ambitious project. The new definitions and concepts that Scholze and Clausen have introduced are powerful but also complicated and hard to learn. Scholze, for his part, is not sure how widely used they will become. On the other hand, he sees them as just the first step in a far bigger program to understand why numbers behave the way they do.
Doing math can be a little like rock climbing: Just as the route you take up a sheer face can incorporate creativity and even elegance in the way it sequences technical maneuvers, so too can a proof. Both traverse existing terrain. Most research — even some of the best research — takes the form of finding new routes to known peaks. But in mathematics, there is a weird relationship between the equipment and the landscape, as though developing a new type of ice axe causes hitherto unknown mountain ranges to emerge. As those new ranges appear on the horizon, older mountains that had seemed forbiddingly steep begin to resemble gentle hills.
Christophe Peus/IHES
Developing these new tools takes a certain revolutionary confidence — especially when it requires setting aside implements that have been used in the community for so long that they seem to be part of the mountains themselves.
Point of Know Return
It’s possible to discover powerful mathematical truths without having a good language to work in.
Which is to say that topology predates topological spaces. As far back as 1735, Leonhard Euler proved that it was impossible to traverse the city of Königsberg (where he lived) by crossing each of its seven bridges only once. This is a recognizably topological result — the size of each of the city’s landmasses doesn’t matter, nor does the length of the bridges between them. Only the pattern of how they connect to each other does.
For nearly 200 years, research in topology proceeded in fits and starts. In the mid-19th century, August Ferdinand Möbius analyzed the strip that bears his name: a ribbon twisted on itself one time before its ends are joined. It is arguably the strangest topological object to have practical utility in the real world — for instance, in one-sided conveyor belts that wear evenly as they drive machines. Around the same time, Möbius began introducing some of the field’s key ideas, such as how to classify shapes with varying numbers of holes by looking at how loops can be drawn on them.
Shortly afterward, Bernhard Riemann, Henri Poincaré, and others made further advances. But they struggled for lack of the right language. As the Australian mathematician John Stillwell wrote in 2009 of Poincaré’s groundbreaking work in topology: “Along with great breakthroughs, there is also confusion.” Poincaré had ideas that he lacked the vocabulary to properly express.
To an outsider, it seems as if the branch of math closest to topology ought to be geometry. It seems as if topology is geometry, just with flexible objects instead of rigid ones. But the resolution to Poincaré’s confusion would come not from geometry, but from a nascent branch of logic called set theory.
At the turn of the 20th century, researchers were trying to wrestle mathematics onto firmer footing. They had only recently realized that their everyday intuition about numbers was completely wrong; now they were fervently debating which axioms, or obvious truths, they should build their theories on. Seemingly small differences in how they stated their most straightforward assumptions had major consequences for what would be possible or impossible to prove.
They used set theory to hash out these debates about the foundations of mathematics. In 1912, Felix Hausdorff, who had recently started teaching at the University of Bonn — where Scholze would end up generations later — set out to write the first comprehensive treatment of set theory. At the time, Hausdorff, then in his mid-40s, was already an accomplished writer: Under the pseudonym Paul Mongré, he had published a collection of poetry, two books of philosophy that tried to reconcile Nietzsche and Kant, and a play that was produced in 40 cities. As a mathematician, he was successful but not yet a superstar.
That changed after the 1914 publication of his book Fundamentals of Set Theory. In it, he gave the first description of topological spaces. A topological space is simply a collection of items that are grouped together into what Hausdorff called neighborhoods — today known as open sets. The open sets give structure to the space.
Open sets must satisfy just two conditions. First, any combination of open sets must also be an open set. (If Brooklyn forms a neighborhood and Queens forms a neighborhood, Brooklyn and Queens together must count as a single, bigger neighborhood.) And second, any finite overlap between open sets must also be an open set.
Mark Belan/Quanta Magazine
Topological spaces can be finite or infinite. They can encode intricate structure, or no structure at all. As Scholze put it, topological spaces are “just everywhere. If you have this intuitive idea of points being close to each other, you have topology.”
Consider a more familiar object: the number line. When we think about the way numbers relate to one another, we’re thinking about just one topological space: the so-called standard topology, in which every possible interval (not including its end points) forms a neighborhood, or open set. All these intervals are open sets:
This topology gives the real numbers the structure we’re used to.
But you can instead take the numbers you are used to dealing with, forget all the everyday intuition you’ve built up about them, and define entirely different topological spaces on them. If you think of each number as a book in a library, this would be like taking all the books off their usual shelves and organizing them in a completely different way.
For instance, you might crumple the number line into a ball and squeeze that ball down to a point, making every number arbitrarily close to every other number, like so:
This is akin to throwing all your books into a disorderly heap. The relationships between the books — historical fiction all being on the same shelf, and so on — are lost, because all the books are now neighbors. This is called the indiscrete topology, and we create it by declaring that there are only two open sets: the empty set and the entire number line.
At the other extreme is the “discrete” topology, in which every point forms its own neighborhood:
In this topology, instead of every point being in contact with every other point, no point touches any other one. It’s like putting every book in your library on its own private island. The relationships are again lost, because each book is isolated.
In a way, the old joke about the doughnut and coffee cup misunderstands the power of topological thinking. It’s not so much that it is possible to change distances by stretching or compressing things. It’s that it’s possible to meaningfully think about structure in spaces where distance simply does not exist.
In this way, topological spaces made it possible to explore novel parts of the mathematical landscape. For instance, they provided a new, distance-free way of understanding concepts like continuity and connectedness, which is a profoundly powerful and counterintuitive capability to have. This has allowed mathematicians to generalize those ideas to a much broader range of settings — and to prove important statements in a wide variety of fields. For example, results like the fundamental theorem of algebra, whose proof tripped up even mathematical giants like Carl Friedrich Gauss, end up with very simple proofs once topological arguments come into play.
The introduction of topological spaces also led mathematicians to ask new questions they might not otherwise have thought to ask. As is true of any good mathematical definition, topological spaces both opened new vistas and made it much easier to traverse known ones.
Hausdorff’s book was arguably the beginning of modern topology. As the mathematical collective known as Bourbaki would later write, his well-chosen definitions imbued “his theory with both the full precision and full generality desired. The chapter in which he develops the consequences of these axioms remains a model of axiomatic theory — abstract, yet anticipatory.”
Just a Drop of Water in an Endless Sea
There are many subdisciplines within modern mathematics. Each has its own vocabulary, grammar, and intellectual flavor. But they are never wholly separate — they interact in strange ways. In part, this is because many mathematical objects, like the real numbers, exist as objects of study in multiple disciplines: They have an algebraic structure, an analytic structure, a combinatorial structure, and a topological structure (among others!). Often, areas of overlap end up becoming distinct areas of study.
In 1945, two American mathematicians, Samuel Eilenberg and Saunders MacLane, published an audacious paper that created an entirely new discipline, now known as category theory. In one fell swoop, category theory created a set of expressways between other, existing areas of math.
Eilenberg and MacLane defined “categories” as collections of objects and the relationships (called morphisms) between them. For instance, a category might consist of sets and functions that relate these sets to each other, or of vector spaces and the linear maps that transform one vector space into another.
But the real power of the pair’s theory rested on the next level of abstraction they introduced: What if, they asked, you mapped one whole category onto another, with something they called a functor? A functor takes objects to objects and morphisms to morphisms, in an orderly way. In other words, it doesn’t just give you a way to translate from one group of objects to another; it also preserves the relationships between them — allowing you to connect different areas of math to each other.
Among other aims, Eilenberg and MacLane wanted to connect topology with the rest of math. Topology was already known not to gel particularly well with one of math’s most important areas: algebra. Although “topology has been this enormous gift to algebra,” said Clark Barwick of the University of Edinburgh, it has also “obstructed progress because topology and algebra don’t play all that nicely with each other, because of the particular way topology was built up.”
Category theory quickly goes from stating things that sound obvious to drawing powerful mathematical conclusions on the basis of what its practitioners typically call, with affection, “abstract nonsense.” Some categories have particular properties that make them more useful to mathematicians than others. In those categories, abstract nonsense becomes powerful — in other categories, it ends up just being nonsense.
In topology, you can define a category consisting of topological spaces and the continuous maps between them. In algebra, meanwhile, an important category is that of abelian groups — groups that have useful symmetries. If you want to produce a grand synthesis of the ways in which algebraic things behave topologically, or, conversely, how algebraic considerations govern topological constructions, the natural thing to do is to form a category out of objects that have both topological and algebraic structure, called topological abelian groups.
But topological abelian groups lack the particular properties that category theorists desire. If category theory unveiled a hitherto hidden network of highways between different mountain ranges of math, topologists who wanted to travel on that highway were stuck driving a janky car that kept breaking down and needing repairs.
That’s the problem that Scholze and Clausen wound up trying to solve. As Scholze told me, “I think topologists don’t actually like topological spaces, because it’s not a convenient category.” What if they could define new objects to replace topological spaces — ones that would retain their power, but also create a better kind of category that would finally allow mathematicians to connect topology to algebra and other fields?
The Nicest Revolutionaries
Sometimes new ideas seem ready to burst into the world.
In 2013, Scholze was 25 years old and already making waves as a deep mathematical thinker, hailed as the youngest full professor in Germany.
He and Bhargav Bhatt, then a researcher at the Institute for Advanced Study in Princeton, New Jersey, coauthored a paper in which they came up with a new definition for a particular type of category. In passing, the pair defined a somewhat abstruse mathematical set, a “sheaf on the pro-étale site of a point.” Scholze didn’t think much of it at the time. Those sets, which they didn’t name, “felt like a weird aspect of the story that I didn’t fully comprehend,” Scholze told me.
Simons Foundation
At the time, Clausen was completing his doctorate at the Massachusetts Institute of Technology. After spending five years as a postdoctoral fellow at the University of Copenhagen, he moved to Bonn in 2018 to work with Scholze, who had just been awarded a Fields Medal, math’s highest honor. Clausen had independently come up with the same kind of set for different reasons, and he persuaded Scholze that they should study it more closely. By this time, Scholze recalled, Clausen had glimpsed a potential opportunity to replace topological sets.
Around the same time, Barwick and his then-student, Peter Haine, independently came up with a slightly different definition in order to answer a particular question in category theory that interested them. “We wanted to solve one problem,” Haine said. “We kind of understood that this theory should be useful for quite a lot of stuff, but there is one thing we really wanted to do: prove this kind of specific result which generalized what we had previously done.”
On the other hand, he said of Clausen and Scholze, “I think they had a lot more that they wanted to do.”
Indeed they did. They gave their sets a name — “condensed sets” — and got to work. They didn’t publish their incremental progress. Instead, in April 2019, Scholze began giving lectures on “condensed mathematics” at the University of Bonn; in May, he posted a 77-page set of notes that culminated in a new, elegant proof of an important theorem called coherent duality. As Johan Commelin, who would go on to collaborate with Scholze, remembers, “coherent duality had an extremely roundabout and technical proof before.” Scholze and Clausen’s proof was clean and elegant.
That, Commelin said, “gave people the motivation to say, ‘Let’s organize reading groups and study seminars all over the world to go through these lecture notes and find out what’s happening.’” Commelin organized one such group at the University of Freiburg, but it was challenging. “I don’t think anybody in our group understood all the details,” he said.
Although condensed sets had started as a tool for proving useful results, Scholze recalled, they quickly became something more. “For me personally, condensed sets changed something very basic about how I think about mathematics,” he said.
“Replacing the topological perspective by the condensed one,” he added, “creeps into everything I’m doing.”
Jeffrey Bergfalk, a set theorist at the University of Barcelona, remembers first meeting Scholze and Clausen in 2019 after he and his fellow set theorist Chris Lambie-Hanson of the Czech Academy of Sciences posted a technical paper online. Set theorists form a particularly small and tight-knit community, somewhat removed from the mathematical mainstream. “We were only expecting to hear from people that we knew,” Bergfalk said. But they got an email from Clausen, who had noticed the paper. He and Scholze, the email said, were thinking about similar things in the context of condensed mathematics — a nascent subject that neither Bergfalk nor Lambie-Hanson had heard of.
It took a moment for Bergfalk to realize the scale of Scholze and Clausen’s ambitions for condensed math. “Dustin and Peter were just wonderfully responsive, really cool and communicative,” he said. “They don’t have to be as nice as they are.” And even though Scholze and Clausen were working well outside their areas of expertise, Bergfalk noted, they asked about the sorts of things you wouldn’t expect anybody but a set theorist to care about. “They were asking us to think in the direction we would want to think anyway,” he said. “Which meant we had to learn condensed math.”
For two people who are reinventing a big chunk of 20th-century mathematics, Clausen and Scholze are unassuming. “To a large extent, what I am doing is rephrasing what others have done in my own words,” Scholze told the mathematician Maria Yakerson in a 2021 interview. “I’m not that much interested in theorems or proofs.” What he wanted to do, he said, was to come up with new definitions: “They must make it easy to state interesting theorems, and they must make it easy to prove them.” Scholze doesn’t see himself as creative. He is, he said, just “trying to give names to what is there.”
Clausen, for his part — as he told Yakerson in a separate interview around the same time — avoids publishing papers, because he believes that the scientific publishing industry is fundamentally flawed. He also largely avoids even informally writing up results, leaving that to collaborators. He just wants to focus on the math; like Scholze, he’s constantly looking for the right names, the right language. (At one point, in fact, he considered pursuing a career in literary translation.)
“I was never completely convinced by topological spaces,” Clausen said. They couldn’t give him an understanding of “this world that’s there, this rich world that we’re trying to get at but we don’t have the proper language to talk about.”
But that only motivated him further. “I’m extremely happy not understanding,” he said, “because I’m even happier when I finally do understand.”
Building on a Foundation of Dust
That’s where condensed sets come in. Condensed sets can be seen as a sort of recipe for building continuous objects, such as the real numbers, out of “totally disconnected” spaces — like making a cake out of disparate grains of flour and sugar, in Scholze’s telling.
Take the so-called Cantor set. Start with the line segment containing all the real numbers between 0 and 1, and remove the middle third. Then remove the middle third from the remaining line segments. Repeat this process infinitely many times, and you’ll end up with a “dust” of points. No point is right next to any other. The space of points is totally disconnected.
The Cantor set is the simplest kind of condensed set, as well as a building block for making other condensed sets. You can make more complicated condensed sets, Scholze said, by smashing together clouds of points like the Cantor set in a weird way.
Such dust might seem foreign, but Scholze points out that we use it all the time. When you represent numbers as decimal expansions, for instance, you’re essentially thinking of the numbers as a similar kind of dust. It’s like taking each number in the expansion and cutting out its section of the number line, like so:
In this way, producing a given number actually involves infinitely “dissecting” the number line. As Scholze put it, the “decimal expansion describes a totally disconnected space because with every new digit, you are chopping up your line more and more.” Every number is totally disconnected from every other one.
How, then, can you use such a disconnected set to get a continuous object like the real number line we’re so used to? You have to glue all the disconnected segments back together by equating, say, 0.49999999999999999… with 0.5 (and 0.50999999999999999… with 0.51, and so on).
Scholze and Clausen’s condensed sets work similarly: They’re disconnected, but they can be used to build and study continuous objects, like those you want to understand in topology. And if you start with them instead of topological spaces, you get an additional benefit: It turns out, Scholze explained, that “these totally disconnected pieces are extremely simple to describe algebraically.”
Condensed sets form a special type of category that, according to Juan Esteban Rodríguez Camargo, a collaborator of Scholze’s at the Max Plank Institute for Mathematics, finally makes it possible to mix topology, algebra, and other fields “in a very practical and precise way.”
Scholze and Clausen started by using their condensed sets to re-prove old results that previously relied on topological spaces — like the fundamental theorem of algebra. These new proofs have given mathematicians a novel understanding “suitably oiled and greased up and made supple,” Vakil said. “The more you can fit into your intuition, the better you understand.”
Then the pair decided to push even further.
A Condensed History
Since 2019, Scholze and Clausen have kept building new types of structures out of their condensed sets — and sharing new sets of lecture notes. “The ideas were evolving in Bonn way faster than the rest of the world could consume them,” Commelin said. There were “light” condensed sets, and also solid, then liquid, then gaseous spaces — an entire condensed mathematics.
Neither Clausen nor Scholze thinks of himself as a topologist. If the two were less friendly, or their ideas less effective, there might be some resentment over their attempt to rebuild the foundations of a field they don’t tend to work in. “I wouldn’t want to impose anything,” Scholze said when asked what he thinks the impact of condensed math will be. He and Clausen sound as though they are having fun, trying to come up with ideas that they themselves find useful.
But some mathematicians are likening their work to a similar mathematical revolution that took place in the 1950s and ’60s — when Alexander Grothendieck reimagined the field of algebraic geometry in accordance with category theory, vastly extending its reach and power. Grothendieck’s impact on modern mathematics was profound. And now, according to Dagur Asgeirsson, a postdoctoral researcher at the University of Alberta and Clausen’s former graduate student, “I think it’s fair to compare Peter to Grothendieck in this sense. He is reinventing everything somehow.”
“The real excitement about condensed stuff for me is the possibility of defining new objects of study,” Barwick said. It’s “showing you there is this natural class of objects that you just never looked at before, like an unclimbed mountain. We are just chewing at the corners of this vast territory.”
In one set of lecture notes, Clausen and Scholze quoted a well-known saying by the prominent mathematician David Mumford. Mumford’s field of algebraic geometry, he said, “seems to have acquired the reputation of being esoteric, exclusive, and very abstract, with adherents who are secretly plotting to take over all the rest of mathematics.” Clausen and Scholze went on to note that their plan was to use condensed math — also esoteric, exclusive, and abstract — to continue where this effort had left off. They were not entirely joking.
Regardless, their not-so-secret “takeover of mathematics” is continuing. In the last few years, Clausen and Scholze have defined other novel mathematical objects, such as “analytic stacks” and “gestalten.” Some mathematicians consider these to be even more significant than condensed sets.
Scholze suspects that condensed mathematics might prove useful even in areas distant from his and Clausen’s core interest in number theory. Quantum field theory — a central aspect of modern physics, which has long struggled with its foundations — makes use of very sophisticated algebra and category theory, Scholze noted. “At the same time,” he added, “quantum field theories are by their nature very analytic and topological. Mixing these two worlds is a nontrivial matter, but condensed mathematics gives a possible framework to do so.”
Scholze and Clausen’s body of work shows just how much choosing the right language matters — how reframing concepts makes it possible to traverse known terrain more easily and to explore new mathematical vistas. “Trying to get to the bottom of these phenomena is trying to find a language in which to express them,” Scholze said.
In a memoir Grothendieck published late in life, he described mathematicians as builders, even though he argued that they are most certainly not inventing anything, only finding structures that are already there, waiting to be discovered. He wrote: “The most beautiful house, that in which the love of the builder is most evident, is not that which is larger or higher than the others. Rather, a house is beautiful if it faithfully reflects the structure and beauty hidden in things.”