内容来源：https://www.quantamagazine.org/what-is-the-positive-grassmannian-and-why-does-it-show-up-everywhere-20260625/

内容总结：

数学家发现神秘几何结构“正格拉斯曼流形”竟普遍存在于交通流、浅水波和量子物理中

近日，哈佛大学数学家劳伦·威廉姆斯（Lauren Williams）在其参与的播客节目中，深入揭秘了“正格拉斯曼流形”（Positive Grassmannian）这一看似抽象的数学概念，如何意外地成为连接交通流模型、浅水波方程乃至量子粒子散射等众多看似无关领域的“隐形纽带”。

威廉姆斯解释道，格拉斯曼流形从本质上说是一个“分类其他形状的形状”，可以将其想象成一个存放所有通过原点的直线、平面等几何对象的“图书馆”。当加上“正”的限制后，它变得更具结构，并展现出惊人的普适性。

从抽象理论到现实世界的惊人关联

威廉姆斯在博士后期发现，她用于计算正格拉斯曼流形不同维度碎片数量的多项式，竟然与一个模拟蛋白质合成过程中核糖体移动的模型概率完全吻合，而这个模型同样被用来模拟车流。这意味着，一个纯数学问题能直接回答“一段有n个空位的路段上有k辆车的概率”。

这种关联并非孤例。威廉姆斯指出，浅水波中的孤立子解（Soliton solutions）以及量子场论中的散射振幅计算，都离不开正格拉斯曼流形中的“Plücker坐标”。只要这些坐标保持非负，模型就能稳定描述现实世界；一旦出现负数或零，模型就会发散到无穷大，失去物理意义。她认为这种“排斥”特性是万物相连的内在原因：无论是相互排斥的粒子、交互的水波，还是碰撞的量子，其数学结构的内核都指向了同一个几何对象。

“歪问题”不会得到“美答案”

作为2024年麦克阿瑟“天才奖”得主，威廉姆斯在节目中分享了自己的数学美学观。她认为，数学家享有提出任何问题的自由，因此应当追求“美”的答案。一个简洁、优雅、能连接两个看似无关之物的定理，远胜于冗长、特例化的结论。她甚至直言：“如果你问了一个问题，答案却不美，那就说明你问错了问题。”在她看来，这种对美的追求并非主观臆断，而是指引数学家发现最深刻、最具广泛联系性理论的指南针。

AI能否取代数学家？新基准测试“First Proof”直面挑战

除了纯数学研究，威廉姆斯还主导了一项名为“First Proof”的项目，旨在客观衡量人工智能系统证明数学定理的能力。由于现有AI极易从互联网上“搜索”到已知证明，该项目特意从数学家未发表的研究中提取了10个中等难度的引理（约5页纸长度的证明）进行测试。结果显示，在单次“出题”且无交互的情况下，AI只能解决其中2个问题；而经过人类和团队的共同努力，最终有6个问题被攻克。

威廉姆斯强调，这项测试并非为了贬低AI，而是为了帮助数学家更好地适应被AI“颠覆”的研究生态。她期望未来10年内，AI能成为数学家可靠的“研究伙伴”，但在那之前，理解与提问的主动权仍掌握在人类手中。正如节目主持人在结尾时笑称：“当AI能作为嘉宾来上播客节目跟我们聊天时，我们才真正算被取代了。”

中文翻译：

什么是正格拉斯曼流形，它为何无处不在？
引言
交通流、浅水波和量子粒子散射的某些数学模型之间有什么联系？令人惊讶的答案隐藏在代数组合学的一个角落，名为正格拉斯曼流形。简而言之，正格拉斯曼流形是一种对其他形状进行分类的形状。值得注意的是，正格拉斯曼流形的各个部分可以重新组合成多种形式，揭示这些以及其他许多看似无关的数学系统中共享的结构。

我们之所以知道正格拉斯曼流形出现在许多现实场景中，很大程度上归功于哈佛大学劳伦·威廉姆斯的理论工作。在最新一期《为什么的乐趣》中，威廉姆斯与联合主持人史蒂文·斯特罗加茨谈论了她的工作，她如何意识到正格拉斯曼流形出人意料的普遍性，以及她如何将职业生涯投入到寻找乍看之下似乎无关的领域之间的联系上。随后，对话转向威廉姆斯正在从事的另一个项目，名为“第一证明”，该项目旨在客观衡量人工智能系统在研究级数学命题证明方面的能力，并由此探讨人工智能是否会（或不会）接管数学。

您可以在 Apple Podcasts、Spotify、TuneIn 或您最喜欢的播客应用中收听，也可以从 Quanta 流媒体播放。
注意：自本次对话录制以来，“第一证明”第二批次项目的结果已于2026年6月10日发布。

文字记录
[音乐播放]

史蒂夫·斯特罗加茨： 好了。
詹娜·莱文： 好的，我们现在开始了，开始了吗？
斯特罗加茨： 我是史蒂夫·斯特罗加茨，
莱文： 我是詹娜·莱文。
斯特罗加茨： 这里是《为什么的乐趣》。
莱文： 一档来自《量子杂志》的播客，我们在此探讨当今数学和科学中一些最大的未解问题。
斯特罗加茨： 我想从最近我与一位数学家——她的研究领域是我们所说的代数组合学——的讨论开始谈起。
莱文： 哦。
斯特罗加茨： 是吗？你为什么对这个说“哦”？
莱文： 哦，我不知道。听起来真的很难。我的意思是，代数，我们知道这两个词的意思。组合学，我猜是某种离散代数。
斯特罗加茨： 是的，差不多，对吧？它完全是关于离散对象或结构的。
莱文： 当我想到组合学时，我有时会想到组织事物的多种方式。这出现在热力学中，涉及到统计思维或熵的计数。例如，组织一个数字序列的可能方式有哪些？这是组合学的一个元素吗，还是说这太简单了？
斯特罗加茨： 不，你说得完全正确。这正是我们在思考的事情。对于不熟悉热力学或统计物理学的听众来说，我们玩纸牌游戏时就在做组合学。
莱文： 对，没错。
斯特罗加茨： 你知道，你可以问自己，一副52张牌，四种花色，拿到三条、一对或顺子的方式各有多少种？这类计数问题或概率问题，部分促成了组合学的诞生，但现在，正如你所说，当我们需要找到巧妙省时的方法来计数或按照某种结构化方式组织事物时，我们无时无刻不在使用它。
莱文： 是的，这是个很好的类比。这正是我对它的看法。如果你拿到五张牌，你有哪些可能的组合？你的第一张牌是K，那么接下来四张牌的可能组合是什么，然后是接下来的三张，接下来的两张，最后一张。
斯特罗加茨： 对。你说到了关键词，对吧？“组合学”源于许多组合可能性的概念。
莱文： 是的，很有趣。
斯特罗加茨： 但在我们深入探讨这期节目之前，有一件事我想听听你的看法，那就是关于美的问题。因为我知道在物理学中，你们偶尔会以美作为评判标准——哦，好吧，你开始对我坏笑了。
莱文： 是的！我感觉，我感觉物理学家是最后一批能板着脸将美作为非常严肃标准来讨论的人。你知道，如果某个东西看起来非常杂乱、丑陋、冗长、不简洁，你就会怀疑自己做错了什么，或者有更深层的东西值得探索，我们可能会摒弃它，但它似乎确实有效。
斯特罗加茨： 这就是有趣且有时不可思议的地方，对吧？
莱文： 是的。
斯特罗加茨： 首先，美，即使仅仅作为一种审美问题，也很难定义，但不知何故，我们期望宇宙会有如此好的品味，以至于它……
莱文： 是的。
斯特罗加茨： 你知道，我们可以把美作为选择理论的标准。
莱文： 是的。我不知道这是否仍然是一种强有力的趋势，但它已经取得了令人难以置信的成功，而且我们可以量化对称性，你知道，某些组织结构、简洁的表达式、将事物分组以便使其看起来更简化。我认为，在这种情况下，这些都是相当令人信服的对美的定义。
斯特罗加茨： 嗯。那么，这就是我们首先要讨论的。我们想听听劳伦·威廉姆斯对美的看法，不是在物理学中，而是在纯数学中。她是最近的麦克阿瑟天才奖获得者，哈佛大学的数学家，呃，你会在我们的对话中看到，她似乎专长于寻找看似不相关的领域或主题之间的联系。她总能找到共同的主线，通常不仅涉及组合学，还涉及组合学中一个名字更吓人的特定对象——正格拉斯曼流形。
莱文： 嗯。
斯特罗加茨： 所以，准备好深入那片水域吧。
莱文： 嗯，格拉斯曼很了不起，所以我已经很感兴趣了。
斯特罗加茨： 让我们开始吧。
[音乐播放]

斯特罗加茨： 欢迎来到《为什么的乐趣》。
威廉姆斯： 非常感谢。很高兴来到这里。
斯特罗加茨： 劳伦，我得说，除了你做过的所有美丽数学之外，让我如此兴奋地想和你谈话的原因是，我觉得你的工作触及了数学中一个带有完整哲学维度的不可思议的方面。如果你不介意的话，让我们从一些个人问题开始。我有两个女儿，我和我妻子。如果我没理解错的话，你是四姐妹之一？
威廉姆斯： 是的。我是四个女孩中的老大。
斯特罗加茨： 那么，你的姐妹们也有数学天赋吗？
威廉姆斯： 是的。我所有的姐妹也都喜欢数学。我们之间的年龄差分别是两岁、两岁和六岁，所以我小时候，家里总是很热闹。我记得教我妹妹吉纳维芙做带进位的数字加法，那时她大概四五岁，我教她怎么做，然后给了她一些作业。她回来时咧嘴大笑，她用了不同的方法。她从左到右而不是从右到左计算，她觉得这太滑稽了。所以我不想替她们发言，但我的记忆是，她们觉得数学很有趣。而对于和我年龄最接近的妹妹埃莉诺，当我在四年级或五年级，她比我小两岁时，我们决定与其学习另一门语言，不如学习密码，于是我们开始了解凯撒密码，然后自己编密码。我们花了很多时间做这个，非常有趣。那是一种秘密语言。
斯特罗加茨： 我还在网上看到一些你拉小提琴的照片，这让我想知道你对美好事物的兴趣，无论是在音乐还是数学中……我是说，你通常对美感兴趣吗？
威廉姆斯： 非常感兴趣。我四岁开始拉小提琴，这是我很热衷的一件事。我也写诗和故事。但可以肯定的是，我们在数学中看到的美与我们在音乐或诗歌中看到的美有很大关系，关乎优雅、对称、和谐。两者之间有很强的相关性。我不知道是早期学习音乐帮助我欣赏数学，还是让我更倾向于数学，亦或是对数学的欣赏与对音乐的欣赏是相辅相成的。我真的不知道。
斯特罗加茨： 现在，我可以稍微快进一下吗？你能为我们解释一下组合学的概念是什么吗？你会在这个领域研究什么样的问题？
威廉姆斯： 粗略地说，组合学是研究有限或离散结构的学科。刚开始学习组合学时，人们往往会学习很多计数技巧，或者研究不同类型的对象，并用通常的非负整数（零、一、二等等）来计数它们。实际上，大概三四十年前，组合学还被看不起，被认为是数学的一个低等领域。纯粹主义者中曾有一种观点，认为组合学不过是一袋子技巧。但事实上，我的导师理查德·斯坦利是真正让组合学成为一个更受尊敬领域的先驱之一。他写了那些美丽的书。他的工作揭示了这一切背后存在着更多的结构和更深层次的理论。特别是，他在代数组合学方面有很多出色的工作，即使用代数技巧来研究组合问题。
斯特罗加茨： 我想回到你刚才说的一件事，关于数学的某些部分被认为是更外围或更边缘的，比如它们会流行或过时。我只是想知道你能否展开谈谈，因为这涉及到数学的社会学。
威廉姆斯： 这是一种社会学现象。我认为部分原因与一个领域的年龄有关。例如，数论是研究数字及其算术性质的学科，比如素数。它是一个非常非常古老的学科，享有很高的声誉。你知道，费马大定理是数论中最辉煌的成就之一。而不知何故，组合学到最近才真正获得它应有的良好声誉。我的意思是，事实上，理查德·斯坦利在20世纪60年代是哈佛大学的博士生，当时哈佛大学没有教授研究组合学。他想师从在麻省理工学院工作的组合学家吉安-卡洛·罗塔，而哈佛的一些教员告诉他，“别再搞这种米老鼠数学了。”幸运的是，他坚持了下来，并且改变了这个领域。然后，你知道，最近十年里，数学家许埈珥获得了菲尔兹奖，很大程度上是因为他在组合学和代数几何交叉领域的工作，这当然可能是它终于得到认可的迹象。
斯特罗加茨： 你确实说到了点子上。数学本身内在联系的理念是一个学科地位的一部分。如果一个学科被认为过于边缘化，没有与数学的其他领域联系起来，那往往会降低它的声誉。
威廉姆斯： 我同意。
斯特罗加茨： 对吧？而核心的东西，无论那意味着什么，地位就更高，比如你提到的代数几何。
威廉姆斯： 对。
斯特罗加茨： 所以，我一开始问了你关于美的问题，数学的审美维度。我想把它与你的一段引文联系起来。你说：“如果你问一个问题，而答案不美，那就意味着你问错了问题。”我不知道你是怎么想的。我想多听听你的看法。
威廉姆斯： 是的。从我研究生阶段一开始，我就去参加研讨会，上课。我试图弄清楚的一件事是，不同的人在数学中发现什么有趣，或者发现什么美？我觉得什么有趣？我觉得什么美？我部分从我的导师那里学到了这种审美。你知道，工作应该是美的。呃。作为一个纯数学家，我们不受特定问题或应用的约束，那么我们对自己应该有什么标准呢？嗯，我们应该寻找美丽的东西。作为纯数学家，我们拥有完全的自由。如果我们拥有完全的自由去问任何我们想问的问题，那么，我想做的就是找到有美丽答案的问题。而且我认为，那些问题和答案往往最终会变得最重要，或者与数学内外的其他领域产生最多的联系。
斯特罗加茨： 在公众心目中，数学被认为是一个非黑即白的学科。答案要么对，要么错。当你开始谈论美时，听起来非常主观。有句老话说，美在观察者的眼中。考虑到定义美是如此困难，数学家如此痴迷于美似乎有点令人惊讶。我想首先知道，不美的数学是什么样子的？
威廉姆斯： 一段美丽的数学是简短、干净、优雅的，可能是一个连接两个你意想不到会联系起来的数学对象的陈述。这样的东西，我比那些连陈述都要用好几个段落甚至好几页纸的定理更有可能觉得美。另一方面，你可以提出一个真实的定理或事实，但它看起来非常随意或具体。也许你问：“好吧，如果我在平面上随机取73条直线，然后看……我不知道，看某些通过这四个交点的曲线，会发生什么？”这就不太可能像“存在无穷多个素数”这样的陈述那样被认为美。一个简短、简单、干净、感觉更具普遍性的陈述，会比另一种陈述被认为更美。
斯特罗加茨： 让我看看我是否跟上了你的思路。你谈到的审美倾向于手段的经济性或极简主义。你喜欢普遍性。你不想要很多非常具体的东西。越通用，加分。越简洁，加分。但我听到你对某些类型的数字有些轻视，而偏爱其他类型的数字。
威廉姆斯： 嗯，毕竟我是个组合学家。
斯特罗加茨： 好吧。
威廉姆斯： 但这更多地是关于那些答案是对无穷多种情况都成立的定理的问题，而不是答案是一个特定的实数的问题。
斯特罗加茨： 那么，如果我们为数学家写一本自助手册，假设你正在研究某样东西，根据这些以及可能还有其他一些标准，它得出的结果不美，你会怎么做？有没有一些方法可以促使你走上更美的轨道？
威廉姆斯： 我想我会尝试调整问题。如果看起来我快要得到答案了，但它就是不优雅、不有趣，那么我会尝试调整问题。我的意思是，在极端情况下，我可能会放弃那个努力，但如果我觉得附近应该有什么有趣的东西，那么我会尝试调整问题。
斯特罗加茨： 这是个很好的策略，因为在很多领域，你基本上只能接受那个问题。但在数学中，你提到了可以自由思考任何东西，所以你是说你可以利用这种自由来改变问题。也许你那丑陋的答案表明你的问题不太对劲。好了，我想现在我明白你这话的意思了。
威廉姆斯： 是的。我想音乐或艺术界的人能理解这种说法。也许他们在创作音乐时试图运用某个主题，但和声不对。我相信艺术中会有很多对框架或方法的调整，从而产生更好的最终产品。
斯特罗加茨： 嗯，那我现在想深入探讨一下与格拉斯曼流形和正格拉斯曼流形相关的一系列概念，它们在很长一段时间里都是纯数学家的游乐场。那么，这到底是什么东西？对于那些从未听说过格拉斯曼流形的人，请给我们开个头。我们应该如何在最简单的情况下理解它？
威廉姆斯： 好的。格拉斯曼流形是一个几何对象，有点像存放更简单对象的图书馆。例如，二维空间中一平面的格拉斯曼流形就是xy平面上所有过原点的直线的集合。所以我们把它看作一个单一的数学对象，它记录了所有这些直线。你可以想象所有这些过原点的直线在二维空间中。而这个格拉斯曼流形的正部分，就是那些穿过正象限的直线的集合。
斯特罗加茨： 让我想想这些直线。如果我看一个罗盘，有一条直线指向南北。
威廉姆斯： 是的。
斯特罗加茨： 我认为那是一条线。所以南北是同一条线的一部分。
威廉姆斯： 是的，没错。
斯特罗加茨： 东西也是同一条线的一部分。
威廉姆斯： 对。然后我们可以通过倾斜这两条直线中的任意一条，得到无穷多条直线。
斯特罗加茨： 是的。所以我在想有无穷多条不同的直线，你告诉我可以用一个形状来描述它们整体？
威廉姆斯： 是的。如果你愿意，考虑二维空间中所有过原点的直线的集合，每一条都穿过你罗盘的北半部分。所以实际上，你可以通过它们与圆的交点来记录每条直线。然后，如果你只考虑顶部的正半圆，那么那个正半圆上的每个点都将唯一地指定一条过原点的直线。
斯特罗加茨： 在罗盘上，那上面会有小标记，标着北、东北，我们甚至可以标北北北东北。
威廉姆斯： 是的。
斯特罗加茨： 我的意思是，那样做会很繁琐。但你是说在这个北半圆上有无穷多个点，所有这些点的集合构成了你所说的半圆。但是我是否要考虑……赤道或者说东点和西点的情况呢？
威廉姆斯： 是的，没错。我们应该只包含其中一个，因为只有一条线同时穿过最西点和最东点。是的。所以我们只保留其中一个，把另一个点扔掉。
斯特罗加茨： 就所有这些直线的集合而言，我这样想对不对：它就像一个线段，从东点（但不包括东点）开始，一直绕到顶部再到达西点（但我包括西点）？
威廉姆斯： 没错。
斯特罗加茨： 所以它就像一个有一个端点但没有另一个端点的线段？
威廉姆斯： 没错，是的。这就是二维空间中一平面的格拉斯曼流形。我们可以这样来理解它。
斯特罗加茨： 你可以把它想象成一个半开区间。
威廉姆斯： 是的。
斯特罗加茨： 嗯。好吧。那是一种有点奇怪的对象。我的意思是，它感觉有点不对称，一个半开区间，我迫切地想闭合另一个端点，但我不应该。如果我要的是格拉斯曼流形，就不能。
威廉姆斯： 嗯，我们某种程度上识别了这两个点，所以几乎可以认为我们把它们粘在了一起。
斯特罗加茨： 好的。然后对于正格拉斯曼流形，你只想要正象限。
威廉姆斯： 对。首先，三维空间中直线的格拉斯曼流形，在这个稍高维度的设定下，看起来或多或少像北半球。然后，如果我们限制到格拉斯曼流形的正部分，我们只看到那个半球与正象限的交集。所以它实际上看起来像一个三角形，一种弯曲的三角形。
斯特罗加茨： 好的。这个词 "orthant"（象限）不太熟悉，它是象限的三维版本。
威廉姆斯： 没错。
斯特罗加茨： 对。有八个。所以你才说 "orthant"。
威廉姆斯： 是的。那个正象限就是 X、Y、Z 坐标都为正或非负的区域。
斯特罗加茨： 好的。你说它看起来有点像弯曲的三角形。那么到现在，如果听众们还跟得上，为什么会有人考虑这个对象？这看起来并不是一个显而易见值得考虑的东西。
威廉姆斯： 是的，是的。这并非显而易见的事情，但早在20世纪，数学家们就在研究一类叫做全正矩阵的矩阵，它们有很好的性质。它们与一些不同的系统（如振荡）有联系。然后在20世纪90年代末、21世纪初，Lusztig 和 Postnikov 意识到有一种方法可以将全正矩阵的概念推广到格拉斯曼流形内的一个对象上。所以，尝试研究全正性，不再仅仅针对矩阵，而是针对像格拉斯曼流形这样的几何对象，这本身就是一个纯粹有趣的数学想法。所以，有各种各样的矩阵集合描述了现实世界中的运动和对称性。
斯特罗加茨： 它们出现在量子理论中。现在经常用于人工智能，但即使在数学内部，正如你所说，它们可以像机器一样，对其他数学对象起作用。
威廉姆斯： 是的，在数学中，一个常见的主题是我们不仅研究数学对象，也研究对象之间的关系。矩阵可以为我们提供一种创建或分析不同数学对象之间关系的方法。
斯特罗加茨： 现在，你的一项著名成果是关于我们谈到的正格拉斯曼流形，你用一种组合学的方式研究了非常普遍的情况。那么，请跟我们谈谈你当时所做工作的要点。
威廉姆斯： 好的，当然可以。早在我还是研究生或博士后的时候，我想我和其他一些数学家讨论过组合学作为一个领域到底是什么。在那次晚宴上，我们讨论的一点是，我们可以把组合学不一定看作一个领域，而是看作一种态度。你知道，我们可以带着组合学的态度生活，并对不同的问题采取组合学的方法。正格拉斯曼流形可以被划分成不同维数的片块。我喜欢用的一个类比是立方体，比如三维立方体。如果一个组合学家看它，他可能会说：“嗯，它有六个二维面，”这些不同侧面上的正方形，“它有12条一维边，还有八个零维块，”这八个顶点。所以你可以把这些数字——六、十二和八——与立方体联系起来。这就是组合学家可能会做的。现在，有无限多个正格拉斯曼流形，它们可以有任意高的维数，但我在研究生时解决的第一个问题是提出一个显式公式，用于计算每种维数的片块有多少个。所以，我写了一个多项式，对于任何 k 和 n，它能告诉你那个正格拉斯曼流形中每种维数的片块有多少个。
斯特罗加茨： 好吧，现在让我们从这个相当抽象的矩阵、格拉斯曼流形和正格拉斯曼流形领域，稍微转向一个更平凡的世界：交通、海洋波浪以及细胞内制造的蛋白质，因为事实证明所有这些事情都可以被视为一个故事的一部分。
威廉姆斯： 是的。有三个不同的领域，我都有亲身经历，正格拉斯曼流形与它们联系了起来。在我做博士后期间，我了解到另一位数学家西尔维·科特尔写了一篇论文，论文中说，我那些按维数计算正格拉斯曼流形片块数的多项式，也在计算一个模型中出现的概率，这个模型最初被引入是为了研究蛋白质合成中的翻译过程，同时也被用作交通流的模型。当我看到这篇论文时，我惊呆了，没想到我的多项式竟然与所谓的“现实世界”有关。她的结果非常漂亮。基本上，她说的是，我的多项式给出了在一个有 n 个位点的晶格中，或者在一个能容纳 n 辆车的道路上，恰好有 k 辆车的概率。这就是我的多项式计算的内容。太好了，所以我们知道了在一个能容纳 n 辆车的道路上恰好有 k 辆车的概率。那么，如果我们想知道汽车出现在位置一、四、五、八的概率呢？你知道，如果你想知道任何给定汽车配置出现的所有概率呢？这正是要问的自然问题。这实际上开启了一段长达数十年的、与西尔维的合作。我的意思是，到现在我们已经一起写了很多篇论文了。
[音乐播放]

斯特罗加茨： 詹娜，这里有很多内容需要梳理。你听明白了吗？
莱文： 嗯，我理解了一部分，对吧？这个想法是把某个数学对象切成片块，找出某种数学规则来计算特定类型的片块出现的数量。她提到了维数。听了所有这些，这非常吸引人，但请从宏观角度告诉我什么是格拉斯曼流形。
斯特罗加茨： 好的，有道理。对，这不是我们每天都会遇到的概念。那它是什么呢？有一个技术性的定义方式，但在我给你定义之前，我能先告诉你“它能为我们做什么”吗？它有什么用处？所以，它是一个非常好的元概念。它是一个告诉我们关于其他形状的形状。
莱文： 嗯。
斯特罗加茨： 它是一个可以作为其他类型结构或形状的图书馆或目录的形状。
莱文： 所以它是一个形状还是一族形状？
斯特罗加茨： 它是一族形状。有不同的格拉斯曼流形。我的意思是，技术性定义是这样的，可能对你有用，但我不想在上面花太多时间，因为我不认为这是最有帮助的理解方式。从技术上讲，它涉及到考虑所有不同的方式，即 k 维线性空间（比如二维空间是一个平面，一维空间是一条直线，三维空间就是我们习惯的普通三维空间）在 n 维空间中穿过原点的所有可能性。
莱文： 好的。
斯特罗加茨： 所以有两个参数，k 和 n。现在，最简单的情况是考虑穿过原点的直线。那就是在平面中穿过原点的一维空间。
莱文： 所以 n 是2，k 是1。
斯特罗加茨： 对。那就是 1-2 格拉斯曼流形或类似的东西。
莱文： 哦，我明白了。好的。
斯特罗加茨： 对吧？有无限多个直线，一个完整的连续统。但是如果你想用我们的语言来参数化它们，如果你想对它们进行分类，你可以通过说“它们的罗盘方向是什么？”来做。比如有一条直线指向南北，或者有一条直线指向北东北和南西南之类的，对吧？所以，如果我列出所有可能的直线，它可以是整个上半圆。所以那是一个形状。
莱文： 所以人们研究不同的格拉斯曼流形，有些是高维空间，有些是低维空间。
斯特罗加茨： 没错。现在，劳伦专攻其中的一部分，叫做正格拉斯曼流形。在我们那个平面中穿过原点的直线的例子中，这意味着只考虑那些斜率为正的直线。事实证明，这具有额外的结构，使其在众多应用中更有用。在这一点上，这似乎是纯几何学家会考虑的东西。这是一个对其他形状进行分类的形状。令人毛骨悚然的是，这东西在现实世界环境中到处出现。比如她提到交通流。我希望你脑海中不是汽车在高速公路上飞驰的画面，因为她指的不是那种交通。想象一下COVID肆虐的时候，我们不得不在超市收银台排队，你必须在前面的人后面保持六英尺的距离，对吧？所以假设有大约10个你可以站的位置。那就好比我们的 n。然后人们开始到达排队，同时队伍前面的人可以离开。游戏规则是，无论你在哪个位置，你都有一定概率向前移动一个位置，除非那里有人站着。有一个约束。如果你让整个过程长时间运行，人们随机到达和离开，并在可能时随机向前移动一个位置，你就可以对所有可能的、这10个位置被比如说4个人占据的方式进行分类。那就是 k。事实证明，4,10 格拉斯曼流形告诉我一些关于在这个队列中看到特定人数可能性的信息。
莱文： 现在，我很好奇。我可以想象在COVID期间，如你所说，必须解决这个问题，对吧？这是一个必须解决的问题，因为我们有疫苗分发中心，这种事正在发生或类似的情况。一个人是如何注意到他们为解决这个实际问题而生成的多项式，恰好与一个非常抽象的数学家为某个正格拉斯……找到的多项式相同？我的意思是，他们是怎么注意到这种相关性的？
斯特罗加茨： 这可能是劳伦·威廉姆斯和她的合作者西尔维·科特尔的独特天才之处。而且这不只是关于排队。如果你想到核糖体在 mRNA 分子上移动，进行蛋白质合成，它也会出现在那个场景中。你明白我的意思了吧？这是一组非常有趣、多样的应用，都神秘地归于正格拉斯曼流形这个标题之下。但在休息之后，劳伦·威廉姆斯将向我们解释为什么这种现象会发生，为什么它如此普遍，以及人工智能可能会或可能不会如何接管数学。
[音乐播放]

斯特罗加茨： 欢迎回到《为什么的乐趣》。我们正在与哈佛大学的数学家劳伦·威廉姆斯谈论代数组合学和正格拉斯曼流形。
斯特罗加茨： 我很想问为什么这些联系会发生在更平凡的世界里。我知道没人知道确切原因。
威廉姆斯： 嗯，你知道，对我来说非常有趣的是，你知道，在交通流模型中，它是一个粒子相互排斥的模型。在浅水波中，这些波是聚集在一起并相互作用的。在散射振幅中，是关于粒子被抛在一起并相互作用。不知何故，总是关于粒子或波被抛在一起，然后它们以某种方式排斥。你知道，用于格拉斯曼流形的坐标是普吕克坐标。如果你有一个 k×n 矩阵，你把它看作是一个列向量的列表。嗯，如果两个向量靠得如此之近以至于它们实际上重叠在一起，那么你的普吕克坐标就消失了。所以，格拉斯曼流形上的普吕克坐标的本质里就内置了这种排斥性质。所以我一直想知道是否有可能将这三个不同的场景联系起来，无论是粒子相互排斥、波还是散射振幅。但不知何故，我认为这一切都可以归结为格拉斯曼流形上的普吕克坐标。
斯特罗加茨： 很好。这是一个非常非常好的答案。我的意思是，感觉这比你最初说的更精炼。因为我见过一些图表，比如当你看到费曼图中的粒子时，它们彼此靠近然后弹开。或者如果你看水波的图表，你只看，我不知道，波峰之类的，有画图的方法，看起来你一遍又一遍地画着同样的图。
威廉姆斯： 对，对，对。
斯特罗加茨： 是的。所以，我的意思是，这很奇怪，但也许又不那么令人惊讶，如果在非常深的层面上，我们一遍又一遍地画着同样的图，而大自然在解释它，或者数学在不同环境下以不同的方式解释它，但基本上是同样的机制。但你的普吕克坐标和……那个零点是否意味着那是排斥的类比？它们不能相互穿过，就是因为那个零？
威廉姆斯： 嗯，它们可以，但那样会发生符号变化。
斯特罗加茨： 哦。
威廉姆斯： 然后不知何故，就像浅水波的情况一样，我在想为什么正性会出现在画面中。你知道，为了分析这些解，分析这些水波，你使用 KP 方程的孤子解，这涉及到一个 tau 函数，你需要取某个函数的对数，而这个函数是以某种方式用普吕克坐标构建的。只要普吕克坐标都是非负的，你取对数的东西就总是正数。但如果你失去了这种正性，如果你现在讨论的是格拉斯曼流形上的所有点，而不仅仅是正部分，那么你可能在某些时候会对零或非常接近零的东西取对数。但这样一来，你的浅水波模型就会趋向正无穷或负无穷，这显然不代表现实世界。所以，这里有一点，如果你想要保持在现实世界中，你就必须远离这个零。这意味着要限制在正格拉斯曼流形上。所以，是的。
斯特罗加茨： 但是，如果我们稍微不那么严谨一点，但我想可能更易懂一点，是不是说，在我们的思想或自然界中，存在着一组可能的模式。有时这些模式，你知道，如果它们足够基础，就会出现在我们思想和观察的许多部分中。所以就像有一族特定的模式在你周围盘旋，而这个正格拉斯曼流形的故事正在编码它们，它们在数学和现实世界中有着不同的表现形式，但这基本上是一遍又一遍的相同模式。
威廉姆斯： 是的，也许是这样。也许是这样。我的意思是，格拉斯曼流形是如此普遍，而正性则不知为何捕捉到了现实世界的某些属性。
斯特罗加茨： 这个故事还没有结束，因为后来你自然（我特意用了这个词）地卷入了量子物理学中发生的事情，特别是与一个非常美丽的量子场论有关的事情：N=4 超对称杨-米尔斯理论。
威廉姆斯： 是的，是的。
斯特罗加茨： 如果我没说错的话。但是，尼马·阿尔卡尼-哈米德和其他合作者正在研究这个奇妙的模型，不知何故你与他们建立了联系。你愿意为我们搭建那座桥梁吗？
威廉姆斯： 好的。他们开始意识到，正格拉斯曼流形的结构在某种程度上帮助理解散射振幅。散射振幅基本上是概率，告诉你如果你将一堆具有给定动量的粒子抛在一起，然后有更多粒子出来，你可能预期会发生什么。嗯，我想那种经典的散射振幅方法是使用一些复杂的图表，称为费曼图。但是物理学家尼马·阿尔卡尼-哈米德和合作者们意识到有更紧凑的方法来理解这些散射振幅。这涉及到了正格拉斯曼流形的许多机制。呃，是的。这反过来又导致了一个美丽的几何对象，他们称之为“振幅多面体”，其体积可以计算散射振幅。
斯特罗加茨： 在我们开始深入探讨振幅多面体之前（如果我说得没错的话），我在做一些背景阅读时有一个关于你提到的费曼图的问题。费曼图是一种奇妙的技术，用于计算物理学家所需的那类信息，以尝试与他们实验中所见相匹配，或预测未来实验的结果。但它可能非常繁琐。可能有成千上万张图，有时甚至更多，他们需要在计算机上计算。在物理学家所做的振幅多面体工作中，似乎出现了一件疯狂的事情：成千上万的计算有时可以简化为一个计算。也就是说，当你提到计算体积时，这类似于求一个形状的体积。这似乎是一个奇迹。成千上万甚至上百万的东西怎么能被一个东西取代？这让我想起我教微积分时讲的相消。我们教一些东西，你可能也不时地需要教微积分。我们称之为“伸缩级数”，其中有一系列项，内部有很多项被加上然后减去，再加再减，然后全部抵消。我感觉从我读到的内容来看，在你的图景中——因为我们谈到将“正”这个形容词应用于格拉斯曼流形——当你把这种额外的正性加入其中时，不知何故会产生这种……它不像伸缩级数中的那种抵消，但感觉有那种味道。大量内部抵消，将一个庞大杂乱的东西简化为简单得多的东西。我的理解对吗？我的意思是，即使不是在细节上，在精神上也是。
威廉姆斯： 是的，从费曼图表达式到我们所知道的最紧凑的表达式，会发生许多抵消。这个领域的一个重大进展是 BCFW（布里托、卡查佐、冯和威滕）递归关系，他们写下了这个美丽且紧凑得多的递归关系来计算散射振幅。然后几年后，一位名叫霍奇斯的物理学家注意到，在一些特殊情况下，如果你取这个递归关系并将你的振幅表示为若干项的和，看起来这些项的和是通过将一个几何对象切成片块然后加总这些片块的体积来计算它的体积。所以，这是霍奇斯在少数非常特殊情况下的观察，然后他提出了一个问题：“这在一般情况下成立吗？”我们能否将所有这散射振幅都写成计算某个几何对象的体积，通过将它切成片块然后求和？于是尼马·阿尔卡尼-哈米德和雅罗斯拉夫·特林卡发明/发现了振幅多面体，作为这个问题的答案。所以他们定义了这个对象，它与正格拉斯曼流形密切相关，并且他们在2013年的论文中提出，这就是霍奇斯问题的答案。这个对象的体积确实在计算所讨论的散射振幅。
斯特罗加茨： 那么，也许我们应该结束我们的讨论了，只是简单谈谈你最近在做的一个名为“第一证明”的项目。你能告诉我们这个项目是关于什么的，以及你试图用它达到什么目的吗？
威廉姆斯： 好的。“第一证明”是我们去年秋天启动的一个项目，其动机和想法是试图提出一个客观的衡量标准，来评估人工智能系统在证明数学命题方面有多好。媒体有很多噪音，要么炒作人工智能的能力，要么贬低它，我们认为数学家自己应该弄清楚如何在我们的研究中最好地利用人工智能，特别是弄清楚人工智能在证明命题方面有多好。但这测试起来非常棘手，因为 LLM（大型语言模型）、AI 模型非常擅长搜索文献。所以，如果你让你最喜欢的 AI 模型对一个数学命题给出证明，如果那个命题和证明在互联网某处，它会找到它。所以我们想知道它在提出尚未存在的新证明方面有多好。所以我们决定需要做的是，从我们自己的研究中提取数学命题、引理，这些引理或命题我们已经证明过了，但解决方案还未发布到互联网上任何地方，然后将这类命题作为问题提出，作为对 AI 系统的挑战。于是，我们11个人组成了一个小组，从我们的工作中整理出这类问题，并于2月6日将它们作为挑战发布在互联网上的一篇论文中。
斯特罗加茨： 那是2026年2月6日，供未来收听此节目的听众参考。
威廉姆斯： 没错。是的。然后当时我们想做的就是明确表示我们已经自己解决了这些问题。我们对解决方案进行了加密，将加密后的解决方案放到网上，然后我们说我们会在一个星期后公布解密的密钥，公开解决方案。在那段时间里，我们非常欣慰地看到，无论是来自数学界，比如专业数学家或数学爱好者，还是来自大型人工智能公司，都表现出了难以置信的兴趣，他们纷纷投身挑战，看看他们能做些什么。
斯特罗加茨： 是的。因为这些不是奥林匹克竞赛题或高中数学竞赛题之类的。这些是真正的研究级问题，但都是小型的。
威廉姆斯： 是的。
斯特罗加茨： 正如你所说，它们是引理，不是整篇论文。
威廉姆斯： 对，对，对。所以这是一个新型的挑战，因为正如你所说，以前大多数基准测试包含的都是有数值答案的问题，而不是由证明构成的答案。所以对于我们所有的问题，我们都确保我们有大约五页或更短的证明。
斯特罗加茨： 那么 AI 做得怎么样？有可能评估吗？
威廉姆斯： 是的，我们在提出这10个问题的时候，自己做了私人评估。实际上，决定测试协议也是一件棘手的事情，因为你可以给 AI 模型一次机会来回答问题。你知道，你可以直接把问题给它，看看它做得怎么样。或者你可以与模型进行长时间的对话，试图引导它给出更好的答案。但在我们事先的私人测试中，我们只是给了每个 AI 模型一次机会来回答问题。我们没有进行任何来回的交流。当时我们发现，这些模型能够解决我们10个问题中的两个。
斯特罗加茨： 哦，好的。那不错了。这些都是难题。
威廉姆斯： 是的，是的，是的。不，不错。在那周期间，各种个人以及公司的人都在研究这些问题并提出解决方案。如果你把提交答案的所有不同个人和团体的最佳努力汇总起来，我们确实得到了可能正确的另外四个问题的解决方案，所以总共是10个中的六个。但是我们试图避免对个人或团体的表现做出任何正式的声明，因为我们没有制定任何基本规则。因为不同的人、不同的团体和不同的公司会有不同的程序，以及不同数量的反馈，所以很难比较这些模型的表现。
斯特罗加茨： 那么，就在最近，就在我们这次对话的前几天，你发布了你称之为……你管它叫什么？
威廉姆斯： 第二批。
斯特罗加茨： 第二批。
威廉姆斯： 是的。“第一证明”是一个烘焙双关语。关于在烘烤之前让面团发酵。所以我们在二月份发布了我们的第一批问题。就在几天前，2026年3月14日，π Day，我们发布了一个公告，我们将在今年晚些时候的春天发布第二批问题。它们同样将是来自数学家研究的数学不同领域的类似小问题。但这一次，我们打算让我们的问题成为一个更正式的基准测试。并且我们确实打算在最后对解决方案进行评分。
斯特罗加茨： 好的。这将很有趣，值得关注。关于我们真正讨论的两个主题——格拉斯曼流形及其相关对象，或这个人AI工作——你有什么最希望在比如说十年后看到的发现吗？
威廉姆斯： 就格

英文来源：

What Is the Positive Grassmannian and Why Does It Show Up Everywhere?
Introduction
What links certain mathematical models of traffic flow, shallow-water waves, and quantum particle scattering? The surprising answer lies in a corner of the algebraic combinatorics world that goes by the name of positive Grassmannian. In simple terms, the positive Grassmannian is a shape that classifies other shapes. Remarkably, pieces of the positive Grassmannian can be reassembled in forms that reveal shared structures in these and many other seemingly unrelated mathematical systems.
That we know the positive Grassmannian crops up in many real-world settings is largely down to the theoretical work of Lauren Williams at Harvard University. In this latest episode of The Joy of Why, Williams talks to co-host Steven Strogatz about her work, how she realized the surprising pervasiveness of the positive Grassmannian, and how she has made a career of finding connections among fields that don’t at first sight seem connected. The conversation then switches to another project Williams is working on, called First Proof, which is trying to measure objectively how good AI systems are at coming up with proofs of research-level mathematical statements, and which leads to an exploration of whether AI may or may not take over mathematics.
Listen on Apple Podcasts, Spotify, TuneIn or your favorite podcasting app, or you can stream it from Quanta.
Note: Since this conversation was recorded, results from the First Proof Second Batch project were released on June 10, 2026.
Transcript
[Music plays]
STEVE STROGATZ: All right.
JANNA LEVIN: Okay, now we’re starting, starting?
STROGATZ: I’m Steve Strogatz,
LEVIN: And I’m Janna Levin.
STROGATZ: And this is The Joy of Why.
LEVIN: A podcast from Quanta Magazine where we explore some of the biggest unanswered questions in math and science today.
STROGATZ: I can start us off with a discussion I had recently with a mathematician who works in the area of math that we call algebraic combinatorics.
LEVIN: Oof.
STROGATZ: Yeah? What makes you say “oof” to that?
LEVIN: Oh, I don’t know. It sounds really hard. I mean, algebraic, we know what those two words mean. Combinatorics, I assume some kind of discrete algebra.
STROGATZ: Yeah, something like that, right? It’s all about discrete objects or structures.
LEVIN: When I think of combinatorics, I sometimes think of the many ways in which you can organize something. Like this comes up in thermodynamics where statistical thinking or in counting of entropy. What are the possible ways of organizing a sequence of numbers, for instance? And is that an element in combinatorics, or is that too simplistic?
STROGATZ: No, that’s exactly on the money. That’s the kind of thing that we’re thinking about. And for our listeners who may not be in the world of thermodynamics or statistical physics, we do combinatorics any time we play card games.
LEVIN: Right, exactly.
STROGATZ: You know, so you can ask yourself, with a deck of 52 cards and the four suits, how many ways are there to get three of a kind versus two of a kind or a straight? Those kinds of counting questions or probability questions, that’s part of what gave rise to combinatorics, but now it’s used all the time when we have to find clever time-saving ways to count things or to organize things, as you said, in some very structured manner.
LEVIN: Yeah, that’s a great analogy. That’s exactly how I think of it. If you’re handed five cards, what are your possible combinatorics? Your first card is a king, then what are the possible combinations that follow with the next four cards, and then the next three, and then the next two, and then the one?
STROGATZ: Right. and you’ve put your finger on the key word there, right? Combinatorics coming from the idea of many combinations of possibilities.
LEVIN: Yeah, fascinating.
STROGATZ: But one of the things that came up that I really would like to have your take on before we dive into the episode is the question of beauty. Because I know that in physics, you all use beauty occasionally as a cri- oh, okay, so you’re smirking at me here.
LEVIN: I am! Well, I feel like, I feel like physicists are the last ones with a straight face that really talk about beauty as a very serious criteria. You know, something looks really messy, ugly, long, not compact, you suspect you’ve done something wrong or there’s something deeper worth pursuing, and that, we might dismiss it, but it seems to work.
STROGATZ: That’s the interesting and sometimes uncanny thing, right?
LEVIN: Yeah.
STROGATZ: That, that first of all, beauty, even just as an aesthetic issue, is hard to define, but then somehow we expect that the universe is going to have such good taste that it’s gonna…
LEVIN: Yeah.
STROGATZ: You know, that we can use beauty as a selection criterion for our theories.
LEVIN: Yeah. And I don’t know if that is still as forceful a kind of trend, but it has been incredibly successful, and we can quantify symmetries, you know, certain organizational structures, compact expressions, grouping things together so that it seems to streamline. These are pretty compelling definitions of beauty, I think, in this context.
STROGATZ: Uh-huh. Well, so that’s where we’re gonna go first. We wanna hear Lauren Williams’s take on the issue of beauty, not in physics, but in pure math. She is a recent MacArthur Genius Prize winner, a mathematician at Harvard University, and, uh, you’ll see in, in our conversation that she seems to specialize in finding connections between fields or topics that don’t seem like they’re related.
She finds a common thread, often not just with combinatorics, but with a very specific object in combinatorics that goes by the even more intimidating name of the positive Grassmannian.
LEVIN: Mmmmm.
STROGATZ: So get ready to go plunge into those waters.
LEVIN: Well, Grassmann was amazing, so I’m already interested.
STROGATZ: Let’s dive in.
[Music plays]
STROGATZ: Welcome to The Joy of Why.
WILLIAMS: Thank you so much. It’s great to be here.
STROGATZ: The thing that’s got me really fired up to talk to you, Lauren, I have to say, besides all the beautiful math that you’ve done, I feel like your work touches on this uncanny aspect of math that has a whole philosophical dimension to it.
If you don’t mind, let’s start with some personal stuff. I have two daughters, my wife and I, and if I understand right, you are one of four daughters?
WILLIAMS: That’s right. I’m the eldest of four girls.
STROGATZ: And so, are your sisters also mathematically inclined?
WILLIAMS: Yes. All of my sisters liked math as well. The age gaps between us are two, two, and six, so when I was growing up, the house was pretty full.
I remember teaching my youngest sister, Genevieve, addition of numbers with carries when she was probably four or five, and I showed her how to do it, and then I gave her, you know, some homework. She came back and she had a big grin on her face, and she had done it a different way. She was doing things left to right instead of right to left, and she thought that was hysterical.
So I don’t want to put words in their mouths, but my recollection is that it seemed fun for them. And with my sister Eleanor, who’s closest to me in age, when I was in fourth or fifth grade and she was two years younger, we decided that instead of learning another language, we could learn codes, and so we started learning about Caesar cipher and then making up our own codes. We spent a lot of time doing that, and it was a lot of fun. It was a secret language.
STROGATZ: I also saw some pictures of you, looking on the internet, playing the violin, and it makes me wonder if your interest in beautiful things, whether in music or in math… I mean, are you generally interested in beauty?
WILLIAMS: Very much so. I started playing the violin when I was four, it was quite a big passion of mine. I was also writing poetry and stories as well. But, for sure, it feels like the beauty that we see in mathematics has a lot to do with the beauty that we see in music or in poetry in terms of elegance or the symmetry, the harmony. There’s a strong correlation there.
I don’t know if learning music early on helped me to appreciate math or made me more inclined to math, or if having a mathematical appreciation also went hand in hand with having an appreciation of music. I really don’t know.
STROGATZ: Now, if I can fast forward a little bit can you unpack for us what is the idea of combinatorics? What are the kinds of questions you would study in that field?
WILLIAMS: Roughly speaking, combinatorics is the study of finite or discrete structures. When one starts out learning combinatorics, one tends to learn a lot of techniques for counting or studying different kinds of objects and counting them with the usual sort of non-negative integers, zero, one, two, and so on.
Actually, probably 30, 40 years ago, combinatorics was looked down on as a field of math. There was a viewpoint among purists that combinatorics was sort of a bag of tricks. But in fact, my advisor, Richard Stanley, was one of the people who really made combinatorics a much more respectable field. He wrote these beautiful books. His work sort of pointed out that there was a lot more structure and deeper theory behind it all. And in particular, he has a lot of wonderful work in algebraic combinatorics, which is using techniques from algebra to study combinatorial problems.
STROGATZ: So I, I wanna go back to something you said a minute ago, the idea that some parts of math are considered more peripheral or more marginal, like they can come in and out of fashion, and I, I’m just wondering if you could expand because it gets into the sociology of math.
WILLIAMS: It is a sociological phenomenon. I think partly it has to do with the age of a field. So for example, number theory is the study of numbers and their arithmetic properties, things like primes. It’s a very, very old subject and had a very prestigious reputation. You know, Fermat’s last theorem is one of the crowning achievements in number theory, whereas somehow combinatorics didn’t really come into its presence, good reputation, until fairly recently.
I mean, in fact, Richard Stanley was a PhD student at Harvard in the 1960s, and at the time, Harvard had no professor doing combinatorics. He wanted to work with Gian-Carlo Rota, who was a combinatorist working at MIT, and he had faculty members at Harvard telling him, you know, “Stop doing this Mickey Mouse mathematics.”
Fortunately, he persisted, and he, and he changed the field. And then, you know, more recently, in the last 10 years, the mathematician June Huh won a Fields Medal largely for his work at the interface of combinatorics and algebraic geometry, and that, of course, it’s maybe a sign that this has finally been recognized.
STROGATZ: You’re really putting your finger on it there. This idea of the interconnectedness of math itself is part of a status of a subject. If a subject is seen as too far in the outskirts, not connecting to the other domains in math, that tends to lower its reputation.
WILLIAMS: I agree.
STROGATZ: Right? Whereas the things that are in the core, whatever that means, have higher status like algebraic geometry you mentioned.
WILLIAMS: Right.
STROGATZ: So I began by asking you questions about beauty, the aesthetic dimension of math, and so I would like to try to connect that to a quote that I read of yours. It says, “If you ask a question and the answer is not beautiful, that means you asked the wrong question.” I don’t know. I’d like to hear more.
WILLIAMS: Yes. So starting from the beginning of my time as a graduate student, I was going to seminars, I was taking classes, and one of the things I was trying to figure out was, what do different people find interesting in mathematics, or what do they find beautiful in mathematics? What do I find interesting? What do I find beautiful? And I picked up this aesthetic in part from my advisor. You know, the work should be beautiful. Uh. And as a pure mathematician, we’re not beholden to particular questions or applications, and so what are the standards to which we should hold ourselves? Well, we should be looking for something beautiful.
And we have complete freedom as pure mathematicians, and if we have complete freedom to ask whatever question we want, well, what I want to do is find questions with beautiful answers. And I think those are the questions and answers that tend to wind up becoming the most important or having the most connections to other fields, whether it’s within math or outside of math.
STROGATZ: In the popular mind, math is known as being a very black and white subject. There’s a right answer, there’s a wrong answer. When you start talking about beauty, it sounds very subjective. There’s the old line about beauty being in the eye of the beholder. And it seems a little surprising that mathematicians would be so obsessed with beauty, given that it’s so hard to define what beauty means.
And I guess I’d like to know first of all, what would it look like for math to be not beautiful?
WILLIAMS: So a beautiful piece of math is short, clean, elegant, maybe a statement that connects two mathematical objects that you didn’t expect to be connected. That is something that I would find more likely to be beautiful than if the theorem takes paragraphs or pages to even get to the statement.
On the other hand, one could come up with a true theorem or a true fact that seems very arbitrary or specific. Maybe you ask like, “Okay, what if I take 73 random lines in the plane, and then I look at the, I don’t know, I look at some curve going through these four intersection points, what’s gonna happen? You know, that’s less likely to be considered beautiful than a statement like there are infinitely many primes. You know, a statement that is short and simple, clean, and feels more sort-of universal is going to be considered, considered more beautiful than the other kind of statement.
STROGATZ: So let me see if I follow you. The aesthetic you’re talking about tends to like economy of means or minimalism. You like universality. You don’t want a lot of very highly specific things. To the extent that it’s general, that’s a plus. To the extent that it’s compact, that helps. But I was hearing some disrespect towards certain types of numbers in favor of other types of numbers.
WILLIAMS: Well, I am a combinatorialist after all.
STROGATZ: Okay.
WILLIAMS: But, but it was more about questions where the answer is a theorem that holds true for infinitely many cases, as opposed to a question with a single answer, which is some real specific number.
STROGATZ: So in terms of, um, like if we were doing a self-help manual for mathematicians, and you suppose you’re studying something and it’s coming out not beautiful according to these and maybe some other criteria, what do you do? Like, are there moves that you can make to push yourself on a more beautiful track?
WILLIAMS: I guess I would try to tweak the question. If it seems like I’m arriving at the answer, but it’s simply not elegant, doesn’t seem to be interesting, then I would try to tweak the question. I mean, I might in an extreme situation just abandon that effort, but if I feel like there should be something interesting nearby, then I would try to adjust the question.
STROGATZ: And that’s the nice gambit because in a lot of fields you’re sort of stuck with the question. But in math, you mentioned the freedom to think about whatever they want, and so you’re saying you can leverage that freedom to maybe change the question.
Maybe your ugly answer is a sign that your question isn’t quite right. Okay. So now I think I understand what you meant by that.
WILLIAMS: Yeah. I would imagine that that people in music or art could relate to this kind of statement. Maybe there’s a theme that they’re trying to play with in composing a piece of music and it doesn’t have the right harmonies. I’m sure there’s a lot of tweaking of the framework or tweaking of the approach that goes into art that results in a better end product.
STROGATZ: Well, so I wanna dive into that now, this set of ideas related to a thing called the Grassmannian and the positive Grassmannian, which for a long time were a playground for pure mathematicians.
So okay, what the heck is this thing? For people who have never heard of the Grassmannian, start us off. How should we think about it in the simplest cases?
WILLIAMS: Yeah. So the Grassmannian, it’s a geometric object, and it’s a bit like a library for keeping track of simpler objects. So, for example, the Grassmannian of one-planes in two-space is the set of all lines through the origin in the x-y plane.
And so we’re, we’re thinking of this as a sort-of one mathematical object that’s keeping track of all of these lines. And so you can imagine all these lines passing through the origin in two-dimensional space, and then the positive part of this Grassmannian would be the set of those lines that go through the positive orthant.
STROGATZ: Let me think about these lines. If I were looking at a compass, there’s a line that goes north and south.
WILLIAMS: Yes.
STROGATZ: I think of that as one line. So north and south are both part of the same line.
WILLIAMS: Yes, exactly.
STROGATZ: And east and west are also part of the same line.
WILLIAMS: That’s right. And then there’s sort of infinitely many lines that we can get by tilting either of those lines.
STROGATZ: Yeah. So I’m thinking about infinitely many different lines, and you’re telling me there’s a way I could think about them as somehow described by a single shape?
WILLIAMS: Yes. And if you like, if you’re thinking about this set of all lines through the origin in two-dimensional space, each one passes through the sort-of northern half of your compass. And so you could actually just keep track of each of those lines by the intersection point with a circle.
And then if you just think about this positive semicircle at the top, then each point on that sort-of positive semicircle will be specifying uniquely a line that goes through the origin.
STROGATZ: Which on a compass would just have little markings on it that say north, northeast, we could even have north-north-north-northeast.
WILLIAMS: Yes.
STROGATZ: I mean, that gets cumbersome to do it that way. But you’re saying there’s infinitely many points on this northern half circle, and then the totality of all those points makes what you called a semicircle. But do I consider… What’s going on with the equator or the east and western points?
WILLIAMS: That’s, that’s right. We should only include one of them because there’s only one … there’s one line that passes through both the west and the east-most points. Yeah. So we should only keep one of them and throw the other point away.
STROGATZ: As far as the collection of all these lines, am I right in thinking that it’s like a line segment that goes from east, but I don’t include east, all the way around the top to west, but I do include west?
WILLIAMS: Exactly.
STROGATZ: So is it like a line segment with one end point but not the other endpoint?
WILLIAMS: Exactly, yes. That would be the Grassmannian of 1-planes in 2-space. We, that’s how we can think about it.
STROGATZ: You could think of it as being like a half-open interval.
WILLIAMS: Yes.
STROGATZ: Huh. Okay. That’s a kind of a weird object. I mean, it feels kind of unsymmetrical, a half-open interval, like I’m, I’m dying to close that other endpoint, but I shouldn’t. Not if I want the Grassmannian.
WILLIAMS: Well, we sort of identify those two points, so it’s almost like we could think about sort of gluing those together.
STROGATZ: Okay. And then for the positive Grassmannian, you only want the positive orthant.
WILLIAMS: Right. So first the Grassmannian of lines in three-dimensional space in this slightly higher dimensional setting would look more or less like the northern hemisphere. And then if we restrict to the positive part of the Grassmannian, then we’re just looking at the intersection of that hemisphere with the positive orthant. And so it’s actually gonna look like a triangle, like a kind of a curvy triangle.
STROGATZ: Okay. And this word “orthant,” which isn’t totally familiar, is the 3D version of quadrant.
WILLIAMS: Exactly.
STROGATZ: Right. There’s eight of them. So that’s why you’re saying “orthant.”
WILLIAMS: Yeah. So that positive orthant would be where the X, Y, and Z coordinates are all positive or non-negative.
STROGATZ: Okay. And you say it looks sort of like a curvy triangle. And so at this point, if people are still with us, why would anyone think about this object? This doesn’t seem like an obvious thing to think about.
WILLIAMS: Yeah, yeah. It wasn’t an obvious thing to think about, but back in the 1900s, mathematicians were studying certain kinds of matrices called totally positive matrices, and they had nice properties. They had some connections to different systems like oscillation.
And then in the late 1990s, early 2000s, Lusztig and Postnikov realized that there was a way to sort of generalize this notion of totally positive matrices to an object that lived inside the Grassmannian. And so it was just sort of a purely interesting mathematical idea to try to study total positivity, not just for matrices anymore, but for geometric objects like the Grassmannian.
So, there’s all kinds of sets of matrices that describe motions and symmetries for the real world.
STROGATZ: And they come up in quantum theory. They’re used all the time now in artificial intelligence, but even inside of math, as you say, they’re, they can act like machines that do things to other mathematical objects.
WILLIAMS: Yes, in math, one of the common themes is that we study not just the mathematical objects, but also the relationships between the objects. And matrices can give us a way to create or to analyze relationships between different mathematical objects.
STROGATZ: Now, one of the results that you are known for was this positive Grassmannian that we talked about, you looked at in a combinatorial way, in the very general case. So, tell us a little bit of the flavor of what you did there.
WILLIAMS: Yeah, absolutely. Back when I was a grad student or postdoc, I think I had a conversation with some other mathematicians about, you know, just what combinatorics is as a field. and one thing that we discussed at that dinner was that one can think of combinatorics, not necessarily just as a field, but as an attitude.
You know, we can go through life, uh, with a combinatorial attitude and take a combinatorial approach to different problems. And the positive Grassmannian can be divided into pieces of different dimensions. An analogy I like to use is that of the cube, say the three-dimensional cube. If a combinatorialist looks at it, they may come away saying, “Well, it has six two-dimensional faces,” these squares on the different sides, “and it has 12 one-dimensional edges, and it also has eight zero-dimensional pieces,” the eight vertices.
And so you can associate these numbers, six, 12, and eight to a cube. That’s what a combinatorialist might do. Now, there are infinitely many positive Grassmanians, and they can have arbitrarily high dimension, but the first problem that I worked on in graduate school was coming up with an explicit formula for how many pieces there are of each dimension.
So, I wrote down a polynomial that for any k and n tells you how many pieces there are of each dimension in that positive Grassmannian.
STROGATZ: Okay, so let’s now make a little swerve from this pretty abstract realm of matrices and Grassmannians and positive Grassmannians to the much more mundane world of traffic and waves on the ocean and proteins being made inside of cells, because it turns out all those things can be viewed as part of one story.
WILLIAMS: That’s right. There have been sort of three different areas that I’ve had personal experience with where the positive Grassmannian got connected. During my postdoc, I learned that another mathematician, Sylvie Corteel, had written a paper which said that my polynomials that were counting pieces of the positive Grassmannian according to dimension were also computing probabilities in a model that had been introduced to study translation in protein synthesis and was also used as a model for traffic flow.
I was floored when I saw this paper to think that my polynomials had to do with a sort of, quote-unquote, “real world.” Her result was quite beautiful. Basically, she was saying that my polynomials were giving the probability that in a lattice with n sites or in a road with space for n cars, there are exactly k cars present. That’s what my polynomials were computing.
So great, so we know the probability that in a road with space for n cars, there’s exactly k present. Well, what if we want to know the probability that the cars are present in positions one, four, five, eight? You know, what if you want to know all the probabilities that any given configuration of cars is there? And so that was the natural question to ask. That actually kicked off a decades-long collaboration with Sylvie. I mean, we’ve written a number of papers together by now.
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[STROGATZ: So, there’s a lot to unpack there, Janna. Did that wash over you?
LEVIN: Well, I mean, I grasp some of it, right? This idea that you can cut some mathematical object into pieces, find some mathematical rule for the number of pieces occurring of a certain variety. She was saying dimensionality. So having heard all of that, which is very intriguing, just give me the bird’s-eye view of what a Grassmannian is.
STROGATZ: Okay, fair enough. Right, it’s not a concept we run into every day. So here’s what it is. There’s a technical way we could define it, but before I give you that, can I give you ‘What’s it gonna do for us?’ How is it helpful? So, it’s a really nice meta-concept. It’s a shape that tells us about other shapes.
LEVIN: Hmm.
STROGATZ: It’s a shape that can be used as a library or a catalog for other kinds of structures or shapes.
LEVIN: So it’s one shape or it’s a family of shapes?
STROGATZ: It’s a family of shapes. There’s different Grassmannians. I mean, here’s the technical definition, which may work for you, but I don’t wanna linger on it too long because I don’t think it’s the most helpful way to think about it.
Technically, it has to do with thinking about all the different ways that k-dimensional spaces, linear spaces, like a two-dimensional space would be a plane, a one-dimensional space would be a line, a three-dimensional space is what we’re used to for ordinary 3D space. Yeah. So you’re trying to think about the totality of all k-dimensional linear spaces through the origin of n-dimensional space.
LEVIN: Okay.
STROGATZ: So there’s two parameters, k and n. Now, the simplest case would be think about lines through the origin. That’d be one-dimensional spaces through the origin in a plane.
LEVIN: So n is two, k is one.
STROGATZ: Right. That would be the 1-2 Grassmannian or something like that.
LEVIN: Oh, I see. Okay.
STROGATZ: Okay? So there’s, infinitely many, a whole continuum of lines, but if you wanted to parameterize them in our language, if you wanted to catalog them, you could do it by saying, “What’s their compass direction?” Like there’s the line that goes north-south, or there’s the line that goes north-northeast and south-southwest or something like that, right? So if I listed all of those possible lines, it could be the whole upper semicircle. So that’s a shape.
LEVIN: And so people study different Grassmannians, some high-dimensional space and some lower dimensional.
STROGATZ: Exactly. Now, Lauren specializes in this piece of it that’s called the positive Grassmannian, which in our little example with lines through the origin in the plane would be like only considering the ones that have positive slope. And that turns out to have extra structure that makes it more helpful in lots of applications.
At this point, it seems like something that pure geometers would think about. This is about a shape that classifies other shapes. The spooky thing is that this pops up all over the place in real-world settings. So like she mentions, traffic flow. I want you to have not an image of cars motoring down the highway, ’cause she doesn’t really mean that kind of traffic.
Think of back when COVID was rampant and we had to stand in line at the checkout for the supermarket, and you had to stay six feet behind the person in front of you, right? So imagine you had something like 10 spots available that you could stand on. That would be like our n. And now people start arriving to get in line and also people at the front of the line can leave, and the rules of the game are that whatever spot you’re on, you have some probability of moving forward one spot, except not if someone’s standing there.
There’s a constraint. If you let this whole thing run for a long time with people arriving at random and leaving at random, and moving forward one spot at random when they can, you could classify all the possible ways that these 10 spots could be occupied by four people, let’s say. That would be the k. It turns out the 4,10 Grassmannian tells me something about the likelihood of seeing a particular number of people in this queue.
LEVIN: Now, I’m curious. I can imagine during COVID, as you said, having to solve this problem, right? It’s a problem that has to be solved because we now have distribution centers for vaccines, and this is happening or something like that. How does somebody notice that the polynomial that they’ve generated to answer this practical question happens to be the same as a polynomial a very abstract mathematician has found for a Grassmannian on the positive with positive Grass… I mean, how do they even notice this correlation?
STROGATZ: That might be the unique genius of Lauren Williams and her collaborator, Sylvie Corteel. And it’s not just about the queues. If you think about ribosomes moving down an mRNA molecule as they’re doing protein synthesis, it also pops up in that setting. You see what I’m getting at? This is a really fun, diverse set of applications all mysteriously falling under the heading of the positive Grassmannian.
But after the break, Lauren Williams will walk us through why this phenomenon might be happening, why it’s happening so pervasively, and also how artificial intelligence may or may not take over mathematics.
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STROGATZ: Welcome back to The Joy of Why. We’re speaking with Harvard mathematician Lauren Williams about algebraic combinatorics and the positive Grassmannian.
STROGATZ: I would love to ask about why these connections to the much more mundane world happen. I know that no one knows.
WILLIAMS: Well, you know, what’s really interesting to me is that you know, with the model of traffic flow, it’s this model of particles that repel each other. And with the shallow water waves, these are waves that are sort of coming together and interacting. And with the scattering aptitudes, it’s about particles that are being thrown together and interacting. And somehow it’s always about particles or waves that are being flung together and then they sort of repel in some way.
And you know, the coordinates one uses for the Grassmannian are Plücker coordinates. And if you have your k by n matrix, you think of this as a, as a list of column vectors. Well, if two, two vectors get so close that they’re actually on top of each other, your Plücker coordinate vanishes. So there’s something in the nature of the Plücker coordinates on the Grassmannian that build in this repelling property.
And so I’ve always wondered if it could be possible to connect these three different settings, whether it’s the particles repelling each other, or the waves, or the scattering aptitudes. But somehow I think it all comes down to Plücker coordinates on the Grassmannian.
STROGATZ: That’s nice. That’s a very, very nice answer. And I mean, it feels like a more refined way of saying what you said at first. ’Cause I’ve seen diagrams like when you look at the particles in the Feynman diagrams, veering towards each other and then bouncing off. Or if you look at the diagrams of the water waves where you’re just looking at, I don’t know what, the crests or something, there’s ways of drawing the pictures that it almost looks like you’re drawing the same picture over and over.
WILLIAMS: Right, right, right.
STROGATZ: Yeah. So, I mean, it would be weird, then again, maybe not so surprising that if at a very deep level we’re drawing this same picture over and over and nature is interpreting it, or math is interpreting it in different ways in different settings, but it’s kind of the same mechanism. But your, your thing with the Plücker coordinates, and the… does the zero mean that that’s the analog of repulsion. They won’t go through each other because of that zero?
WILLIAMS: Well, they could, but then there’s a sign change.
STROGATZ: Oh.
WILLIAMS: And then somehow, like with the shallow water wave stuff, I was thinking about why positivity comes into the picture. You know, to analyze these solutions, to analyze these water waves, you use Soliton solutions to the KP equation, and that involves a tau function in which you take the log of a certain function, and this function is built out of the Plücker coordinates in some way. And as long as the Plücker coordinates are all non-negative, you’re taking the log of something that’s always positive.
But if you lose this positivity, if you now are talking about all points in the Grassmannian and not just the positive part, you might at some point be taking the log of zero or something really close to zero. But then what happens is that your model for shallow water waves goes off to plus or minus infinity, which obviously does not represent the real world.
And so there’s something about, you know, if you want to stay in the real world, you have to stay away from this zero. And it means restricting to the positive Grassmannian. So, yeah.
STROGATZ: But if we were to just get a little sloppier, but I think maybe more understandable about it, is it that there are sort of a bank of possible patterns that can happen in our minds or in nature. And sometimes those patterns just, you know, if they’re fundamental enough, they will show up in many parts of our thought and in our observations.
So like there’s a certain family of patterns that you are swirling around and this positive Grassmannian story is encoding them, and they have different manifestations in math and in the world, but it’s kind of the same pattern over and over.
WILLIAMS: Yeah, maybe that’s right. Maybe that’s right. I mean, the Grassmannian is so universal, and then there’s something about positivity that just captures properties of the real world for some reason.
STROGATZ: This story is not over because then somehow you get entangled, naturally I use that word, with things happening in quantum physics specifically with things related to a very beautiful quantum field theory: N=4 supersymmetric Yang-Mills theory.
WILLIAMS: Yes, yes.
STROGATZ: If I’ve got that right. But anyway, Nima Arkani-Hamed and other collaborators are looking at this fantastic model and somehow you connect to them. You want to build that bridge for us?
WILLIAMS: Yes. So they started to realize that somehow the structure of the positive Grassmannian was helping to understand scattering amplitudes. So scattering amplitudes are basically probabilities that tell you what you might expect would happen if you throw a bunch of particles with given momentum together and more particles come out.
Well, I guess that sort of classical approach to scattering amplitudes was to use some complicated diagrams called Feynman diagrams. But the physicist Nima Arkani-Hamed and collaborators realized that there were more compact ways to understand these scattering amplitudes. And they involved a lot of the machinery of the positive Grassmannian. Um, yeah. And then this in turn led to a beautiful geometric object that they call the amplituhedron, whose volume computes scattering amplitudes.
STROGATZ: Before we start delving into the amplituhedron, if I’m saying that right, I, there was one question I had about something in, in doing a little background reading that you mentioned, these Feynman diagrams. It’s a wonderful technique for calculating the kinds of information that physicists need to try to match what they see in their experiments or to make predictions about future experiments. But it can be very arduous. There could be thousands of diagrams, sometimes even more that they have to calculate on computers.
And the crazy thing that seems to have come out in the amplituhedron story, as done by the physicists, is that the thousands of calculations can be reduced sometimes to one calculation.
That is, when you mention calculating a volume, it’s analogous to finding a volume of a shape. And it seems like a miracle. How could a thousand or a million things be replaced by one thing? And it reminded me of cancellations that I teach when I teach calculus. There’s something we teach and you probably have to teach calculus from time to time too. We talk, call it a telescoping series where there’s a series of terms and then on the inside there’s a lot of things being added and then subtracted again and added and subtracted and they all collapse. And I feel like from what I read, that in your picture you, ’cause we talked about positive as an adjective applied to the Grassmannian, that when you have this positivity extra thing thrown in there, it somehow gives rise to this kind of, it’s not the same cancellation as in a telescoping series, but it feels like it has that flavor.
A lot of internal cancellation simplifying a big messy thing to something much simpler. Am I on the right track with that? I mean, even morally, if not in detail.
WILLIAMS: So, there are many cancellations that occur when one goes from kind-of Feynman diagram expressions to the sort of most compact expressions that we know.
A big advance in this area was the recurrence of BCFW, Britto, Cachazo, Feng, and Witten, and they wrote down this beautiful and much more compact recurrence for computing scattering amplitudes. And then what was noticed a few years later by a physicist named Hodges was that in some special cases, if you take the recurrence and you express your amplitude as a sum of terms, it looked like the sum of terms was computing the volume of some geometric object by cutting it into pieces and adding up the volumes of those pieces.
So, this was an observation of Hodges in a few very special cases, and then he asked the question, “Is this true in general?” Can we write all of these scattering amplitudes as computing volumes of some geometric object by cutting them up into pieces and summing them up? So Nima Arkani-Hamed and Jaroslav Trnka invented/discovered the amplituhedron as the answer to this question.
So they defined this object, and it’s closely related to the positive Grassmannian, and they proposed in their 2013 paper that this was the answer to Hodges’s question. The volume of this object is indeed computing the scattering amplitudes in question.
STROGATZ: So, maybe we should close our discussion here by just going into a little bit of what you’ve been doing very recently, in connection with a project known as First Proof. Can you fill us in on what this project is about and what you’re trying to do with it?
WILLIAMS: Yeah. So First Proof is a project that we initiated in the fall, and the motivation and the idea was to try to come up with an objective measure of how good AI systems are at coming up with proofs of mathematical statements. There’s been a lot of noise in the media either sort-of hyping up the ability of AI or denigrating it, and we thought mathematicians themselves should try to figure out how best we can use AI in our own research, and in particular, to figure out how good AI is at coming up with proofs of statements.
But this is a very tricky thing to test because LLMs, AI models are extremely good at searching the literature. So, if you ask your favorite AI model to come up with a proof of a mathematical statement, if that statement and proof are on the internet somewhere, it’s gonna find it. So we wanted to know how good is it at coming up with new proofs that aren’t already out there.
And so what we decided we needed to do was take mathematical statements, lemmas, say from our own research, where we had proved the lemma or the statement, but we had not released the solution on the internet anywhere, and propose these kinds of statements as problems, as a challenge for AI systems. So, a group of 11 of us got together and produced these kinds of problems from our work and put them out on the internet in a paper on February 6 as a challenge for AI systems.
STROGATZ: That’s February 6th, 2026 for people in the future listening to this.
WILLIAMS: That’s, that’s right. Yes. And then what we did at the time was we wanted to sort-of make clear that we had solved these problems ourselves. We encrypted our solutions, we put the encrypted solutions on the internet, and then we said that we would release the key to the encryption, we’d release the solutions, publicly in one week’s time.
And so during that time, we were really gratified to see that there was just an incredible amount of interest, both from the mathematical community, like professional mathematicians or math afficionados, but also from the big AI companies, you know, jumping on the challenge and seeing what they could do.
STROGATZ: Yeah. ’cause these are not like the Olympiad problems or the high school math contest problems or anything like that. These are really research-level questions, but bite-sized.
WILLIAMS: That’s right.
STROGATZ: As you say, they’re lemmas, not the whole paper.
WILLIAMS: Right, right, right. So this was a new kind of challenge because as you said, most previous benchmarks consisted of problems with numerical answers, as opposed to answers that consisted of proofs. So with all of our problems, we made sure that we had proofs that were roughly five pages in length, or less.
STROGATZ: And how did the AIs do? Is it possible to assess?
WILLIAMS: Yeah, so we did our own private assessments at the time that we came up with these 10 questions. And, actually deciding the protocols around testing is also a tricky thing to do because you could give an AI model one shot to answer the question. You know, you could just give it the problem and see how it does. Or one could have an extended conversation with the model and try to coax it to give a better answer. But so in our private tests that we did beforehand, we just gave each AI model one shot to answer the question. We didn’t have any back and forth, and what we found at that time was that the models could solve two of our 10 questions.
STROGATZ: Oh, okay. That’s not bad. These are hard questions.
WILLIAMS: Yeah, yeah, yeah. No. Not bad. And during that week various individuals and also people with the companies were working on the problems and coming up with solutions. And if you sort-of put together the best efforts from all of the different people and groups who submitted answers, we did get perhaps correct solutions to six of the 10. But we are trying to shy away from making any formal statements about how people or groups did because we didn’t lay any ground rules. Since different people and different groups and different companies would’ve had different procedures, and different amounts of feedback, it’s hard to sort of compare how the models did.
STROGATZ: And so now you have very recently, it was only a few days before our conversation right now, you released what you’re calling, what are you calling it?
WILLIAMS: The second batch.
STROGATZ: The second batch.
WILLIAMS: Yes. First Proof is a baking pun. It’s about proofing the dough before you bake it. And so we put out, you know, our first batch of problems back in February and just a few days ago on March 14, 2026, on Pi Day, we put out an announcement that we will release a second batch of problems sometime later in the spring. They will similarly be sort of bite-sized problems from different areas of mathematics coming from research of mathematicians. But this time we mean for our problems to be a more formal benchmark. And we do intend to get the solutions graded at the end.
STROGATZ: Okay. Well, this’ll be interesting to see. Are there any discoveries about either of the things we really talked about the Grassmannian and its relatives, or this AI work, you most hope to see, say 10 years from now?
WILLIAMS: As far as the Grassmannian goes, I’m hopeful that maybe there’s even more exciting connections to other parts of the real world. And as far as the AI model go, it’s very hard for me to predict. You know, it feels like the ecosystem in which we’re doing math is being upended and we’re trying to figure out how best to adapt, how we can use these new tools. I would hope that 10 years from now, they would be sort of research partners, with a sort of higher level of reliability and confidence than we have at the moment.
STROGATZ: All right, and the last thing, is there something you could put your finger on that particularly is a source of joy for you as a mathematician? What brings you joy in your work?
WILLIAMS: I think it’s identifying connections between things I didn’t expect to be connected. You know, just finding these kinds of connections, whether it’s to the traffic flow, or to the shallow water waves, or to the scattering aptitudes. I have so many stories from my research where I might have a conversation with another mathematician and they show me some numbers of something that they were computing, and then I recognize them as having come up before. It’s always so exciting and intriguing. I mean, it’s this sort of mystery and then we have to do the detective work of figuring out how these objects are connected.
Yeah, so I think that’s the thing that I find most exciting. And then of course, the joy is when you realize, you make that connection. You understand, you have this realization of how they are secretly connected and how you can sort of make that rigorous.
STROGATZ: Well, very, very good. It’s really been fun. Thank you, Lauren.
WILLIAMS: Thank you, Steven
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LEVIN: Wow. So this is terrifying, right? Now, I really do wonder, hey, have I written the last of my very, very technical papers. But then I also remember the time that computers were first invented, and everybody was saying … that’s not… I don’t remember when computers were first invented. But you know what I’m saying. When they got cheaper, more readily available, large processing machines could do huge datasets, and the same kind of thing was said: “Well, now the technical people are obsolete.” I don’t know. What do you think?
STROGATZ: Well, um, I’m confused about it. I really am of two minds, and, you know, we still have CAPTCHA, that thing where you have to identify that you’re not a robot by doing some little image processing. Apparently, that’s still hard for the AIs. So yes, they’re very good at certain things, but there’s still a way to go. They also seem to lack common sense in a lot of domains. But still, back to your question though, I mean, will they make you and me and people like us obsolete? Because what we do isn’t exactly the realm of common sense. We’re in a… as my wife would be the first to tell you.
LEVIN: Right, exactly. And she’d be right.
STROGATZ: No, but you know, like a lot of the work we do involves very explicit rules. You could imagine we might be at risk more than the people who do plumbing or caregiving, who will be the last to be superseded by robots and AIs.
LEVIN: Well, just to play devil’s advocate, I think the idea of these machines as thought partners is closer to what I’m imagining is going to happen, because I still don’t see the machine asking the questions.
STROGATZ: Not yet, no. Do you think in the era when AI starts doing math alongside us, or maybe even instead of us, will beauty play the same role then? Like a guide to what you should think about, what questions you should ask, how to judge whether you’re on the right track with the theorems you can obtain.
LEVIN: Gosh, it’s a really… profound question, ’cause one of the roles beauty might be playing is rendering some very complex subject comprehensible to us. Which I really need because I don’t have infinite compute. So, I need to have a more aesthetic approach.
So, I mean, you could kind of say, in a way, nature already has all the answers. The whole game is discovering what nature already knows. So, if the AI just simply has this infinite list of things it knows, you know, if we don’t understand it, I don’t know that the game has changed that much. I don’t know. What do you think?
STROGATZ: I always wonder about is understanding overrated? So, here’s what I mean, that we might be confusing means and ends. Like, if the end is to predict nature, to be able to find formulas and theorems that are true, understanding may be a crutch. It helps us get good answers. It helps us get more control over the universe, but it’s not the game.
Like, if you’re trying to save someone’s life, you may have to come up with a medical therapy that you don’t understand that works. And so it’s not always so clear to me that understanding is the goal in itself.
But on the other hand, there are people who say it’s not science without understanding. It’s something less than science. It’s like a degradation of the human spirit. Why even do it if you’re not understanding? I don’t know what to think about that. I can see both sides of that argument.
But what’s really interesting in what Lauren Williams and her colleagues are doing is they are giving these secret problems from research-level math that haven’t been published, so the AI can’t look them up on the internet, and asking them how many of our 10 problems can you solve? It’s just an interesting benchmark, different methodology than we’re seeing elsewhere.
LEVIN: Yeah. Yeah, it’s amazing ’cause it means they’re not just regurgitating, culling a human response.
STROGATZ: So far they’re not mastering that. They’re not climbing the whole mountain.
LEVIN: But right now, in the context of what people are doing, is it possible to have a machine that says, “You know, here’s an interesting idea,” or, you know, “I’m bored today I’m going to try this,” or…
STROGATZ: We’ll really know that they’ve arrived when they’re a guest on The Joy of Why.
LEVIN: Yeah. When we have Claude on.
STROGATZ: Yeah, when we have Claude, and course, by then, maybe we won’t be the hosts anymore.
LEVIN: Yeah. Oh, man.
STROGATZ: But until then…
LEVIN: Until then.
STROGATZ: See you later, Janna.
LEVIN: If you’re enjoying The Joy of Why and you’re not already subscribed, hit the subscribe or follow button wherever you’re listening. You can also leave a review for the show. It helps people find this podcast. Find articles, newsletters, videos, and more at quantamagazine.org.
STROGATZ: The Joy of Why is a podcast from Quanta Magazine, an editorially independent publication supported by the Simons Foundation. Funding decisions by the Simons Foundation have no influence on the selection of topics, guests, or other editorial decisions in this podcast or in Quanta Magazine. The Joy of Why is produced by PRX Productions. The production team is Caitlin Faulds, Jade Abdul-Malik, Genevieve Sponsler, and Merritt Jacob. The executive producer of PRX Productions is Jocelyn Gonzales. Edwin Ochoa is our project manager.
From Quanta Magazine, Simon Frantz and Samir Patel provided editorial guidance, with support from Samuel Velasco, Simone Barr, and Michael Kanyongolo. Samir Patel is Quanta’s editor-in-chief. The episode art is by Chanelle Nibbelink, and our logo is by Jackie King and Kristina Armitage. Special thanks to Garth Avery at the Cornell Broadcast Studio.
I’m your host, Steve Strogatz. If you have any questions or comments, please email us at [email protected].