OpenAI Models Solve Decades-Old Math Open Questions Without Fatigue

OpenAI模型不知疲倦破解数十年数学难题 OpenAI最新的推理系统使技术性证明和组合论证实现自动化，推动人类数学研究转向宏观方向把控与概念组织。 OpenAI研究员Mark Sellke和Mehtaab Sawhney深入剖析了Astra等推理模型取得的重大突破，包括攻克球填充界长期未决的难题以及构造非sofic群。他们解释称，AI系统通过执行繁杂的技术细节、对搜索树进行剪枝，并在不带认知偏差的情况下主动回溯，展现出了真正的数学推理能力。这些模型并非要取代人类数学家，而是正在将瓶颈从定理证明转移至高阶框架构建、研究普及化与文献综合。 Mehtaab Sawhney 分享了促使他转向 AI 数学研究的“转化时刻”：GPT-5 仅用五分钟就精准检索到埃尔德什（Erdős）未决问题的关键文献，快速厘清了难题的可解性路径。 (2:43) 在高维球堆积问题上，OpenAI 的 Astra 模型攻克了长期停滞的 Cohn–Elkies 线性规划界限，通过构造显式辅助函数证明了其最佳渐近上界精确达到 (e^{2π} + o(1))^{-d} ≈ 2^{-0.604d}。 (23:09) Astra 成功构造出数学界追寻数十年的“非索菲克群”（non-sofic group），通过代数组合论技巧排除了以往学者未能解决的结构性障碍，给出了仅约 15 页的精简反例证明。 (44:40) 研究人员观察到模型的推理链展现出真实的人类专家级探索与回溯机制，AI 在接管繁重计算与证明细节的同时，也将极大降低前沿数学理解与跨学科应用的门槛。 (57:32)

OpenAI researchers Mark Sellke and Mehtaab Sawhney argue that advanced reasoning models are catalyzing a renaissance of reachable results in mathematics by automating laborious technical proofs, navigating search spaces without fatigue or cognitive bias, and producing surprisingly concise solutions to open problems. Cognitive Advantages in Search and Backtracking (4:21) Human mathematicians frequently stall when confronting tedious technical bookkeeping—such as fine-grained epsilon-delta bounds—or abandon viable proof strategies because earlier failed attempts cognitively pollute their intuition. Sellke and Sawhney observe that AI models circumvent this friction by executing technical calculations reliably and restarting fresh generations with unpolluted context windows. Rather than relying on lucky random sampling, inspection of model reasoning traces shows genuine error identification, hypothesis testing, and backtracking. The researchers explain that these reasoning behaviors emerge from general-purpose reinforcement learning rather than mathematical literature alone, which they note is a flawed training set because published papers hide the messy, iterative failures of human discovery. Resolving High-Dimensional Sphere Packing Bounds (23:09) Applying OpenAI's Astra model to long-standing geometric optimization problems yielded major progress in sphere packing and error-correcting codes. For decades, the leading upper bound on sphere packing density in general dimensions stood at approximately 2^(-0.599d), proven in 1978 by Kabatyansky and Levenshtein. Astra analyzed the Cohn–Elkies linear programming framework to establish an optimal asymptotic bound scaling as (e^(2π) + o(1))^(-d) ≈ 2^(-0.604d), constructing an explicit auxiliary function using complex analysis alongside a matching impossibility proof. For spherical and binary error-correcting codes, the model derived improved bounds by leveraging group representation theory. Sellke and Sawhney note that this was the only interactive result in Astra's 10-problem evaluation set, where researchers iteratively prompted the model to push its algebraic symmetry arguments deeper. Concise Counterexamples and Non-Sofic Groups In group theory, mathematicians long questioned whether every infinite group is sofic—meaning its structure can be locally approximated by finite permutation groups—a problem linked to Gottschalk's 1970s surjunctive conjecture. While a recent disproof of the related Aldous-Lions conjecture spanned roughly 250 pages and required quantum complexity theory, Astra resolved the open question by constructing a non-sofic group within pure group theory in approximately 15 pages. Sellke emphasizes that contrary to expectations of unreadable, bloated AI outputs, the model produced an elegant, dense combinatorial argument. Astra succeeded by identifying a specific algebraic condition that eliminated a structural combinatorial obstruction that previous human researchers had been unable to rule out. Shifting Mathematical Workflows and Democratizing Expertise (57:32) The researchers note that reasoning models are transforming how mathematics is absorbed and produced. Rather than spending hours deciphering dense papers on arXiv, researchers can use models to immediately extract overarching proof strategies. Sawhney defines mathematical taste as the utilitarian capacity to make sound pruning judgments that solve problems faster; as models assume the burden of routine proof steps, human mathematicians can shift their focus toward high-level direction, conceptual synthesis, and exposition. Host Lisha Li and Sellke add that automated mathematical reasoning democratizes advanced techniques for adjacent domains, including theoretical physics and applied engineering. By providing on-demand mathematical problem-solving, AI lowers the historical barrier of needing specialized human mathematicians to execute complex applied proofs. While AI models significantly lower the barrier to verifying technical sub-problems and navigating complex literature, Sellke stresses that mathematics has an uncapped difficulty ceiling, meaning landmark problems like P versus NP will likely remain open for a long time. The models serve not to eliminate human insight, but to automate routine technical barriers so researchers can pursue deeper questions.

OpenAI研究员Mark Sellke与Mehtaab Sawhney指出，先进AI推理模型正在推动数学界迎来“可及成果的复兴”；系统不仅能高精度处理繁琐的技术性推导，还能在不受认知偏差干扰的情况下自主回溯探索，并在高维球体填充与非索菲克群等长期悬而未决的课题中取得突破。 推理机制：系统化回溯与无偏探索 (4:21) 在传统数学研究中，研究人员常因漫长繁琐的技术细节（例如管理边界或验证ε-δ论证）而精疲力竭，并在面对不确定的收益风险比时过早放弃可行路径。Mark Sellke与Mehtaab Sawhney指出，AI推理模型能够不知疲倦地精确执行这些底层推导。更重要的是，推理轨迹表明模型并非单纯进行随机采样，而是具备识别错误并主动执行回溯的能力。 Mark解释称，人类数学家在探索错误路径后，直觉往往会受到失败尝试的“污染”，难以重置思维；而AI模型在新的上下文窗口或并行生成中能够保持完全客观的认知白板，重新审视被人类过早否决的方向。此外，针对正规数学论文仅展示精炼成品而隐去探索过程的缺陷，OpenAI所构建的通用推理架构使假设检验与自我纠错等元能力自发涌现，而非依赖专门为数学编码的形式化启发式规则。 高维球体填充与线性规划界突破 (16:20) 球体填充问题探讨如何在d维空间中实现不重叠单位球体的最大堆积密度。虽然1维（平凡线段）、2维（六边形蜂窝点阵，1960年代证明）与3维（开普勒猜想，Thomas Hales通过复杂计算机辅助证明）已获解决，且Maryna Viazovska在2017年凭借E8点阵和Leech点阵攻克了8维与24维的情况，但一般高维情形长期缺乏紧致的上界。自1978年苏联数学家Kabatyansky与Levenshtein给出约2^(-0.599d)的上界以来，数学界在此方向停滞数十年。 针对Henry Cohn与Noam Elkies提出的基于傅里叶变换的线性规划（LP）框架，OpenAI的模型Astra直接解决了该无限维优化问题。Astra证明了在该LP框架下能够达到的最优上界渐近收敛于(e^(2π) + o(1))^(-d) ≈ 2^(-0.604d)。模型不仅显式构造了达到该界的辅助函数，还给出了匹配的不可能性证明，证实在此框架内无法取得更优结果，且整套证明仅需数页优雅的复分析论证。 对称性表示与非索菲克群的构造 在将填充问题拓展至曲面及离散超立方体的球面码与二进制纠错码时，Astra通过深度运用对称性与表示论改进了渐近界限。该突破也是测试集中唯一涉及人机交互的案例：在研究人员提示其“进一步推进”后，模型调动更深层次的表示论工具，导出了小球极限下的全空间猜测值，并最终完成了完整证明。 在抽象代数领域，索菲克群（sofic groups）指全局结构可被有限置换群局部逼近的群。Gottschalk在1970年代提出是否存在非索菲克群的问题；此前概率论中更强的Aldous-Lions猜想曾耗费长达250页的复杂量子复杂度理论论文才得以证伪。相比之下，Astra在群论内部直接构造了一个非索菲克群作为明确反例。该证明篇幅仅约15页，巧妙整合了既有文献，并通过引入特定的额外代数事实消除了此前学术界无法排除的组合学障碍；数学家Andreas Thom等人已基于该成果展开了后续研究。 数学品味、范式转移与工具平民化 Mehtaab Sawhney以实用主义视角将数学中的“品味”定义为能够做出更好判断以加速解决难题的能力，其核心在于对搜索树的高效剪枝与方法工具的敏锐直觉。当前AI模型仍呈现出高度的任务导向性，即在达成明确目标后便停止深入，因此高层级的外部监督模型或人类专家的引导仍然不可或缺。 对于日常学术研究，Mehtaab表示AI已极大加速了文献消化过程，使研究人员能在数分钟内提取预印本的核心证明策略。Lisha Li与Mark Sellke进一步指出，随着证明常规引理的技术瓶颈被模型分担，人类数学家的核心价值正向框架构建、概念阐释与理论整合转移。此外，先进数学工具的掌握门槛将显著降低，使得理论物理及其他应用领域的非专业研究者无需依赖顶尖纯数学家即可直接应用高阶数学理论。 Mark Sellke强调，由于数学问题的难度上限几乎无限，即使AI能力呈指数级提升，诸如P与NP等深层根本谜题短期内仍难以被彻底攻克。AI并不会终结数学，而是接管机械而繁重的基础证明工作，使研究者得以从枯燥推导中解脱出来，聚焦于宏观科学图景的构建与探索。

## Background 该视频由知名风险投资机构 a16z 于 2026 年 9 月 8 日发布在其官方 YouTube 频道上。内容为 a16z 基础架构合伙人 Lisha Li 与 OpenAI 的两位数学家兼研究员 Mark Sellke 和 Mehtaab Sawhney 展开深度对谈，探讨 AI 在数学推理方面的迅猛进展、前沿模型解决数十年数学难题的底层机制，以及这波技术浪潮对数学研究范式带来的深远影响。 ## TL;DR 对谈围绕 OpenAI 模型（特别是 Astra 项目成果）在严肃纯数学领域的实质性突破展开，重点拆解了高维球体填充（Sphere Packing）与非 Sofic 群（Non-sofic group）存在性等硬核数学成果。嘉宾分享了从一线数学家转型至 AI 前沿实验室的“皈依时刻”，深入分析了模型在长程推理轨迹中展现出的理性回溯、动态概率校准与剪枝能力，打破了机器只会暴力穷举或随机抽样的刻板印象。此外，讨论还剖析了任务导向模型与科研“品味”的边界、未来 AI 双层分工协作架构，以及当 AI 接管繁重证明后，人类数学家向知识编织、理论阐释与群体理解转移的新角色。 ## Atmosphere & style 整场对谈兼具顶尖理论数学家的求真严谨与科技前沿见证者的纯粹兴奋，三人既有共同的学术渊源，又有扎实的专业切磋，交流节奏自然流畅。从拆解经典的几何与抽象代数难题，到复盘模型给出的短小精妙反例，言谈间充满了对数年未解之谜终获攻克的释怀与惊叹。听感上既像是一场高密度的学术研讨流水席，又如同一幅正在徐徐展开的数学与机器共生图景，兼具思辨深度与对科学复兴的真挚乐观。 ## Bulletin - [23:09](https://www.youtube.com/watch?v=1JvyLGd2Sfs&t=1389s) OpenAI 的 Astra 模型在球体填充（Sphere Packing）的高维 Cohn-Elkies 线性规划渐近界上取得突破，不仅证明了渐近上界收敛至优雅的解析等式，还证明了该框架下不可能给出更优的界，终结了数十年的数值猜想。 - [44:38](https://www.youtube.com/watch?v=1JvyLGd2Sfs&t=2678s) 在群论难题中，Astra 模型仅用约 15 页纯群论代数证明直接构造出反例，攻克了悬决已久的“非索菲克群（Non-Sofic Group）存在性”难题，打破了此前类似猜想证伪需要数百页量子复杂性巨著的范式。 - [06:00](https://www.youtube.com/watch?v=1JvyLGd2Sfs&t=360s) 前沿推理模型的突破并非仅凭暴力穷举，其公开的思维链表明模型展现了类似人类顶尖专家的“数学品味”，能够动态校准路径成功概率、果断剪枝，并不受情绪和上下文污染地执行有效回溯。 - [11:44](https://www.youtube.com/watch?v=1JvyLGd2Sfs&t=704s) 数学论文和教科书由于清洗了探索过程与失败尝试，被证明是极差的训练集，当前 AI 的数学能力更源自跨领域的长程通用逻辑推理涌现，而非单纯的符号形式化语料记忆。 ## Key takeaways - [00:00](https://www.youtube.com/watch?v=1JvyLGd2Sfs&t=0s) 人类数学家攻坚常受限于体力和挫败感，容易在思维死胡同中产生上下文污染；AI 模型具备不知疲倦的绝对符号执行力，能以纯净上下文探索不同路径。 - [02:43](https://www.youtube.com/watch?v=1JvyLGd2Sfs&t=163s) GPT-5 仅用约 5 分钟便为埃尔德什开源猜想库中标记为未决的难题精准定位文献并给出可行证明路径，展现出连接知识孤岛与扫除学术盲区的强大能力。 - [06:00](https://www.youtube.com/watch?v=1JvyLGd2Sfs&t=360s) 做数学研究本质上是在与未知“对赌”，模型在面对不确定性时展现出敏锐的数学判断力，能在广袤的解题空间中做出极佳下注，并将算力集中在少数关键分支。 - [09:34](https://www.youtube.com/watch?v=1JvyLGd2Sfs&t=574s) 模型在推导中能动态理性更新路径成功概率，而人类在遭遇几次受挫后常过度悲观地彻底抛弃正确路径；此外并行开启新 Session 相当于派遣多位无偏见代理人同时探索。 - [11:44](https://www.youtube.com/watch?v=1JvyLGd2Sfs&t=704s) 经典数学教材（如 Baby Rudin）为追求精炼隐去了思考过程中的痛苦挣扎，数学成果的文字记录本身缺乏直觉课程，但通用推理模型凭借对底层逻辑结构的理解跨越了语料缺陷。 - [18:35](https://www.youtube.com/watch?v=1JvyLGd2Sfs&t=1115s) 高维几何极度反直觉，球体体积随维度急剧衰减，在开普勒猜想耗费数百年产出数百页繁复证明后，仅在 8 维和 24 维分别依托 E8 格与 Leech 格找到了完美精确解。 - [23:09](https://www.youtube.com/watch?v=1JvyLGd2Sfs&t=1389s) Astra 模型借助复分析工具自主构造了高维极限测试函数，彻底攻克了 Cohn-Elkies 线性规划界在高维下的严格等式极限，并将经典上界由 2^(-0.599d) 推进到约 2^(-0.604d)。 - [32:41](https://www.youtube.com/watch?v=1JvyLGd2Sfs&t=1961s) 在纠错码与球面码的界限改进中，模型转向了深度的表示论工具；在人类数学家追问“能否推得更远”后，模型推演出的极限竟然与全空间球体填充理论精确闭环。 - [36:17](https://www.youtube.com/watch?v=1JvyLGd2Sfs&t=2177s) 大语言模型具有强烈的“任务导向性”，一旦达成指定目标便容易停下脚步；当前阶段人类数学家的宏观视野与品味充当了关键的外挂指路框架（Harness）。 - [40:01](https://www.youtube.com/watch?v=1JvyLGd2Sfs&t=2401s) 从功利主义角度看，做出更好判断以更快解决难题本身就是“品味”的衡量标准；未来或可采用“主管 AI（负责品味与纠偏）+ 下属 AI（负责长程攻坚）”的多智能体分工架构。 - [44:38](https://www.youtube.com/watch?v=1JvyLGd2Sfs&t=2678s) 数学界长期期盼所有群皆能被有限群逼近（即均为索菲克群），以便将有限群理论向无限群推广，但 Astra 成功通过精妙组合论证证明了非索菲克群的存在。 - [51:55](https://www.youtube.com/watch?v=1JvyLGd2Sfs&t=3115s) 不同于前人证伪奥尔德斯-莱昂斯猜想时搭建的 250 页量子复杂性巨著，Astra 解决非索菲克群的证明仅约 15 页且完全保留在纯代数语境下，粉碎了前人无法排除的“代数共谋”。 - [56:41](https://www.youtube.com/watch?v=1JvyLGd2Sfs&t=3401s) 顶级数学家团队（如 Thom 等人）迅速跟进 Astra 的成果拓展出新数学，显示出科学研究的新型分工：AI 负责源头破局与核心反例，人类数学家负责泛化拓展与理论深化。 - [58:28](https://www.youtube.com/watch?v=1JvyLGd2Sfs&t=3508s) AI 在制造海量数学成果的同时，也充当了极佳的吸收工具；研究者只需将 PDF 喂入模型即可几分钟内提炼核心证明策略，有效化解了未来的学术信息过载危机。 - [01:00:09](https://www.youtube.com/watch?v=1JvyLGd2Sfs&t=3609s) 数学研究的核心瓶颈正从“推导严密证明细节”转向“知识编织、构建思想框架与群体理解”，数学家将从解题者转变为知识的策展人与翻译家。 - [01:02:43](https://www.youtube.com/watch?v=1JvyLGd2Sfs&t=3763s) 即便 AI 推理能力指数级飞跃，诸如 P vs NP 这类硬核谜题依然是难以撼动的天花板；而模型的普及将赋能物理与工程等跨领域学者，催生人人可参与的应用数学复兴。 ## Detailed walkthrough ### [00:00](https://www.youtube.com/watch?v=1JvyLGd2Sfs&t=0s) 引言与数学研究面临的瓶颈 在数学研究中，职业数学家的日常往往伴随着长期的挫败感：心中产生一个灵感，觉得可能行得通，于是投入数小时、数天甚至数周去尝试；然而一旦推导受阻，精力耗尽，很多时候就只能被迫放弃。相较之下，人工智能模型展现出了截然不同的特质——只要给出明确的指令，大语言模型就会不知疲倦、极其顽强地沿着设定的路径执行下去。这种不知疲倦的执行力，正在开启一段解决那些“人类触手可及却因精力有限而未能攻克”的前沿数学难题的新浪潮。 与此同时，这种突破也引发了深层的讨论与疑问：模型到底是在以一种疯狂的方式盲目乱猜，还是具备了真正的数学直觉？尽管 AI 在数学领域的推理能力正以指数级速度提升，但数学难题本身的复杂度上限极高，即便模型能力持续飞跃，诸如 P 对 NP（P vs NP）这类世纪难题可能依然遥不可及。然而，数学界正在逐渐形成共识：AI 显然已经不仅仅是在做平凡的浅层推导，而是在实质性地参与并推动严肃的数学发现。 ### [00:50](https://www.youtube.com/watch?v=1JvyLGd2Sfs&t=50s) 从一线数学家到 OpenAI：Mark 与 Mehtaab 的转型之路 a16z 基础架构合伙人 Lisha Li 与两位来自 OpenAI 的数学家 Mark Sellke 和 Mehtaab Sawhney 展开对谈。Lisha 提到自己十多年前就离开了纯数学界，而 Mark 与 Mehtaab 都是近期活跃在数学研究前沿的学者。三人之间还有着学术渊源——著名华人数学家、麻省理工学院（MIT）数学系教授赵宇飞（Yufei Zhao）曾是两人的导师或合作者。在加入 OpenAI 之前，Mark 和 Mehtaab 彼此相识已久，甚至还曾共同合著过一篇学术论文。 谈及为何从高校的象牙塔全职跨界加入 OpenAI，Mark 回忆道，转折点发生在去年夏天，当时他亲眼见证了 AI 系统在国际数学奥林匹克竞赛（IMO）中斩获金牌水平的突破。这一标志性事件让他深受震撼，迫切想弄清楚背后的技术架构和实现原理。不久之后的秋天，Mark 先行入职 OpenAI，并为 Mehtaab 开通了当时内部正在开发的 GPT-5 账号。Mehtaab 在实际测试并上手把玩这些先进模型后，立刻被其展现出的惊人潜力所折服，迅速决定一同加入。 ### [02:43](https://www.youtube.com/watch?v=1JvyLGd2Sfs&t=163s) “皈依时刻”：GPT-5 与 Erdős 开源猜想的检索破局 Mehtaab 分享了彻底改变他对大模型认知的“皈依时刻”。作为一名组合数学方向的研究者，保罗·埃尔德什（Paul Erdős）提出的海量数学猜想一直是他日常关注的核心。近年来，学界将埃尔德什留下的未决问题整理成了一个专门的在线数据库网站，Mehtaab 经常会去翻阅。然而，浏览该网站时常令人极其抓狂：许多题目虽然赫然标记为“Open”（未解决），但由于数学文献过于浩瀚繁复、检索极其困难，你根本无法确认它是否在几十年间的某个冷门分支里其实已被某位学者悄然攻破。 有一次，Mehtaab 和几位数学家朋友在网站上看中了一个标记为未决的埃尔德什问题，几个人坐在一起研讨推导了数个小时，依然摸不清该问题的深浅，也无法判断它是否能在现有工具下解决。抱着试一试的心态，Mehtaab 将这个问题输入了内部的 GPT-5 模型。令人震惊的是，仅仅过了大约 5 分钟，模型不仅直接精准挖出了相关文献索引，甚至明确指出了：“该问题是可达的，具体的切入推导路径应该是这样的……”当 Mehtaab 顺着模型的解答理清思路并激动地告诉 Mark 后，两人又陆续对网站上的类似问题进行了深入挖掘，一口气找出了 10 多个此前人类学者未曾注意到的对应案例。这一刻让他们意识到，模型在连接学术脉络与扫除盲区方面蕴含着颠覆性的力量。 ```mermaid flowchart TD accTitle: Erdős 猜想传统排查与 GPT-5 对比流程 accDescr: 对比传统数学家查阅文献与使用 GPT-5 验证未决猜想的效率差异 subgraph Traditional["传统人工验证流程"] A["浏览 Erdős 猜想库"] --> B["问题标为 Open 未决"] B --> C["数学家团队研讨数小时"] C --> D["难以穷尽冷门文献 / 放弃"] end subgraph ModelAssisted["GPT-5 介入流程"] E["输入 Erdős 难题至模型"] --> F["5 分钟内完成全域文献关联"] F --> G["定位参考文献并给出证明路线"] G --> H["快速验证 10+ 个未决或边缘问题"] end ``` ### [04:21](https://www.youtube.com/watch?v=1JvyLGd2Sfs&t=261s) 从全域检索到深度执行：AI 数学推理的质变 Lisha 指出，早期大众对 AI 的理解往往停留在“信息检索器”或“跨领域跳跃连接者”，因为人类的大脑容量有限，穷尽跨领域的知识网络确实更适合机器。但随着近期 Astra 项目等一系列成果的发布，AI 的表现显然已经深入到了更底层、更硬核的数学推理层面，展现出了与人类职业数学家高度相似的思维方式。 Mark 分析了 AI 模型与人类学者之间在微观层面的优劣势对比： 1. **全域知识储备（Search & Coverage）**：对所有已知数学分支的普遍熟悉度仍然是 AI 的核心优势，这也决定了它目前最擅长解决哪类交叉型问题。 2. **符号与严谨细节的绝对执行力（Flawless Execution）**：在人类构思出一个宏大证明框架后，通常需要处理海量极其繁琐的边界条件和技术性细节，例如各种“$\epsilon$（Epsilon）是否必须严格小于 $\delta$（Delta）”的极限定理推演。人类在处理这类枯燥细枝末节时极易疲倦、迷失甚至出错，而模型在理顺逻辑链条、精准敲定每一个微小不等式方面的执行力几乎无懈可击。 ### [06:00](https://www.youtube.com/watch?v=1JvyLGd2Sfs&t=360s) 单位距离问题与数学“品味”：在不确定性中果断下注 围绕埃尔德什著名的“单位距离问题（Unit Distance Problem）”，Lisha 探讨了 AI 的角色是仅仅承担了庞大而枯燥的计算工程，还是注入了某种真正的“创新灵感”。在很多困难问题中，证明的初步方向或许早有人隐约提及，但通向成功的道路极其漫长且布满荆棘。普通学者如果在一套方案上尝试了数周仍无突破，出于投入产出比（Risk/Reward）的现实考量往往会选择割肉退出；只有少数像安德鲁·怀尔斯（Andrew Wiles）那样甘愿闭关十年的学者才可能走到底。 Mehtaab 对此进行了深入拆解。在单位距离猜想这扇门前，历史上已经有无数顶尖数学家尝试过极其相近的构造思路，但最终都被其错综复杂、极度微妙的技术细节所阻退。做数学研究在本质上就像是在和问题本身“对赌”： > “你常常会想：也许我应该试试这个方法，但这看起来希望渺茫，根本不值得我耗费青春。然而在好几个前沿问题中，模型凭借着丰富的先验知识与极佳的数学品味，做出了最正确的下注。” 从 OpenAI 随 Astra 成果公开的精简思维链（Summarized Chain of Thought）中可以清晰看到，模型绝非盲目尝试所有可能性的暴力穷举机器；它能像一位经验老道的数学大师一样，利用高度敏锐的“数学品味（Mathematical Judgment）”，在广袤的可能性空间中果断剪枝（Prune the search tree），只把算力集中在极少数最有可能成功的关键分支上。 此外，Mark 补充了人类大脑认知层面的另一大天然缺陷——**“上下文污染（Context Pollution）”**。当人类数学家沿着一条错误的道路苦苦挣扎了几个月后，原有的错误思维定式会与核心直觉深度纠缠，极难从脑海中抹去重新出发；人类无法在思维走入死胡同后，像计算机一样瞬间克隆出一个“上周还没被带偏的自己”，直接清空上下文窗口另辟蹊径。而大语言模型可以随时卸下思维包袱，以纯净的视角在全新路径上高速迭代。 ### [09:34](https://www.youtube.com/watch?v=1JvyLGd2Sfs&t=574s) 推理轨迹中的回溯与概率评估：是幸运抽样还是真正的数学思维？ 一旦拥有大致的正确方向，借助 AI 来补全和搞定具体的技术细节变得极其容易。主持人 Lisha Li 提出一个核心疑问：当模型在推理轨迹（reasoning trace）中做出选择、尝试新路径或者发生回溯（backtracking）时，它究竟只是随机抽样中撞了大运（lucky sample），还是真正在像人类数学家一样思考？尤其是人类很容易在某条路走不通时被受挫的情绪所污染，而模型是否能够回到起点并且不被之前的错误上下文所带偏？ Mark Sellke 和 Mehtaab Sawhney 明确指出，模型在解题过程中确实会频繁犯错，但它会折返回去重新审视并做出修正。它的行为显得极其精于计算且具有惊人的正确性。相较于人类数学家，模型在概率评估上表现出显著的优势： - 人类数学家往往在某个策略第一次受挫时，就会在潜意识中剧烈下调该路径可行的概率，反复碰壁几次后便彻底放弃； - 模型则远比人类更擅长动态校准与更新一条路径成功的概率，不会因一两次失败就盲目抛弃可行路径； - 哪怕不考虑单次会话中的回溯机制，用户也可以低成本并行开启新的模型会话（session），这相当于同时派遣大量不受挫折偏见影响的代理人在解题空间中探索。 人类数学家在卡住时常常需要找同行倾诉灵感，由另一位同事换个角度做出来；而模型现在以极高的效率独自复现了这种“交流—调整—重试”的心智闭环。 ```mermaid flowchart TD accTitle: Human vs Model Mathematical Reasoning Flow accDescr: Comparison of how human mathematicians and reasoning models handle dead ends and probability updates. subgraph Human["人类数学家的探索瓶颈"] H1["提出数学路径"] --> H2["遭遇死胡同 / 计算错误"] H2 --> H3["过度下调可行性概率<br>（受挫折心理污染）"] H3 --> H4["过早抛弃潜在正确路线<br>需要同行讨论重启思路"] end subgraph Model["前沿推理模型的行为模式"] M1["模型生成推理轨迹"] --> M2["发现推导矛盾 / 失败"] M2 --> M3["理性回溯 (Backtracking)<br>客观更新路径成功概率"] M3 --> M4["保持无偏上下文继续攻坚<br>或并行多个 Session 探索不同分支"] end ``` --- ### [11:44](https://www.youtube.com/watch?v=1JvyLGd2Sfs&t=704s) 为什么数学论文和教材是极差的训练集？反直觉的通用推理能力涌现 关于模型究竟如何学会这种思维，对话转向了底层训练语料的特殊性。数学论文甚至教科书，从先验上看其实是训练真正数学思维的“极差数据集”（a very poor training set）。教科书为了展现严谨和优雅，往往把中间所有走弯路、失败和探索的过程清洗得一干二净。Lisha 举例提到，许多人告诫“不要用 Rudin 的书学实分析”（Baby Rudin / Walter Rudin 的经典教材），就是因为它太干净、太精炼，完全掩盖了数学家为了定义实数、构建现代分析框架而经历的痛苦挣扎。正常论文写作的初衷从来不是为了教学或还原思维流水账，因此数学成果的文物（artifacts）中本身缺乏“如何像数学家一样思考”的显式课程。 针对为何模型仍能展现出数学专家的思维模式，Mark 解释道： - OpenAI 一直在全方位探索通用推理模型（General-purpose reasoning models），致力于让模型能够跨领域进行更深、更长期的逻辑推演； - 诸如“回溯”、“重置思考”等特质并不是专属于数学的特异算法，而是高质量通用推理在面对复杂问题时的自然沉淀与表现； - 与传统的自动化形式验证（Auto-formalization，如完全依赖 Lean 语料）不同，这种纯依靠推理链涌现出的能力更贴近人类专家的日常直觉。虽然代码语料因其强语法结构利于长上下文学习，但代码往往也缺乏高级语义层面的“为什么要这样写”；数学推理的涌现证明模型穿透了表层语料的局限，触及了逻辑底层的结构。 正是因为这种思维过程极为真实，团队决定公开这些成果摘要后的思维链（summarized chains of thought）。如果不公开思考痕迹，外界看到凭空出现的精巧证明，只会怀疑模型是不是用了某种奇怪的穷举暴力法或者不可理喻的黑盒撞大运；而一旦审视轨迹，就会发现其行文“极其像一位极具天赋的人类专家草稿”，甚至如同亲密合作者在深夜邮件中毫无保留倾泻而出的头脑风暴笔记。 --- ### [16:20](https://www.youtube.com/watch?v=1JvyLGd2Sfs&t=980s) 探索 Astra 十大问题集：球体填充问题（Sphere Packing）的迷人魅力 谈及 OpenAI 将前沿数学推理系统应用于一组精心挑选的难题（即包含 10 个问题的 Astra 测试集）时，Mehtaab 和 Mark 确认他们深度参与了题目的甄选与验证。当被问及各自最喜欢的题目时，Mark 毫不犹豫地提到了球体填充问题（Sphere Packing Problem）。 这一问题的表述极为简明朴素：在 $d$ 维欧几里得空间中，如何最紧密、高密度地摆放一堆半径为 1 的互不重叠的球体？ Mark 在白板上边画边讲解了不同维度下的直观情况与背后的极端复杂性： - **一维情况（$d=1$）**：问题毫无悬念。一维下的“单位球”就是单位线段，直接在实数轴上首尾相连即可填满整个空间，填充密度为 100%； - **二维情况（$d=2$）**：即硬币摆放问题。众所周知最优解是蜂窝状的六边形晶格（Hexagonal Lattice）。然而 Lisha 追问这在数学上是否显而易见时，Mark 直言这绝非显然——尽管人类凭直觉和自然观察早已知晓，但严谨的数学证明直到 20 世纪 60 年代才被彻底敲定，即便存在短证明，论证过程也相当不易； - **数学的无知与自然的智慧**：Mark 打趣道，关于二维六边形晶格的最强直觉其实来自蜜蜂建造蜂巢的本能，“如果存在比六边形更高效的填充方式，蜜蜂在演化中早就采取其他方案了，因为演化是讲究极致效率的”； - **更高维度的迷雾**：除了这种来自大自然的直觉，人类对该问题的理解其实极其有限。维度一旦升至三维（$d=3$），即开普勒猜想所涉及的水果摊叠橙子问题，其证明的复杂程度和耗费的岁月便呈现指数级上升。 ### [18:35](https://www.youtube.com/watch?v=1JvyLGd2Sfs&t=1115s) 低维球堆积的奇迹与高维几何的怪诞 在日常杂货店里堆叠橙子看似简单，但要在数学上严格证明这种面心立方堆积是最优的，却耗费了人类数百年的时间。Thomas Hales 在 2000 年代初给出了著名的证明，但这个证明长达数百页，充斥着大量复杂的线性规划计算与极其精细且繁琐的几何推导，被数学界公认为是一个“极其丑陋的证明”。然而，当维度上升到 8 维和 24 维时，几何学却展现出了截然不同的优雅面貌：2016 年至 2017 年间，数学界证明了在这两个特殊维度下，分别存在着来源于其他代数与数论领域的奇妙结构——$E_8$ 格（$E_8$ lattice）与 Leech 格（Leech lattice）。这两个格具有极高的对称性与不可思议的紧密程度，构成了最优的球堆积结构。除了 1、2、3、8、24 这五个维度之外，人类至今未能确定任何其他维度的确切最优堆积方式。 在高维空间中，人类对最优堆积密度的认知极为匮乏。设 $\Delta_d$ 为 $d$ 维空间中最密球堆积所占据的空间体积比例。直觉上，存在一个相对初等但基础的平凡下界 $2^{-d}$：假设你在空间中贪心地放置互不重叠的单位球，直到再也塞不进任何一个球为止；此时若将每一个球的半径放大一倍（体积膨胀 $2^d$ 倍），这些膨胀后的球必然会覆盖整个空间，否则未被覆盖的空间中心就还能再塞入一个单位球，因此原始球的密度至少为 $2^{-d}$。然而在高维空间下，球体体积的分布行为极其反直觉：如果你在一个高维单位超立方体的各个角上各放一个相切的超球，随着维度升高，这些球所占的体积会急剧趋向于零，以至于立方体正中央竟然能凭空塞进一个半径极大的全新球体。 由于这种奇异的高维几何效应，球堆积密度随维度呈指数级急剧衰减。但在很长一段时间里，数学家能够给出的最佳上界却停留在 1970 年代两位苏联数学家 Kabatiansky 与 Levenshtein 的工作上，其给出的衰减界约为 $2^{-0.599d}$。当主持人追问这个古怪的常数从何而来时，嘉宾笑着解释它源自一个“极其丑陋的极值优化问题”。更富戏剧性的是，由于当时苏联纸张短缺，那篇奠基性论文总共只有一到两页长，推导过程高度简略，几乎找不到任何展开的细节，这让后来的研究者既惊叹又头疼。 - **三维开普勒猜想**：由 Thomas Hales 解决，最短的计算机辅助验证证明长达数百页，过程极其繁琐。 - **神奇的 8 维与 24 维**：分别对应模形式与特殊对称性格结构 $E_8$ 格与 Leech 格，是极少数已知精确最优堆积解的高维特例。 - **贪心堆积与膨胀论证**：通过无法再塞入球的极大堆积与双倍半径扩张，严谨确立了密度下界 $\Delta_d \ge 2^{-d}$。 - **Kabatiansky-Levenshtein 经典上界**：70 年代给出的 $2^{-0.599d}$ 上界统治了该领域数十年，其源于复杂的球形编码与极值几何优化。 --- ### [23:09](https://www.youtube.com/watch?v=1JvyLGd2Sfs&t=1389s) Cohn-Elkies 线性规划界与 Astra 的理论突破 球堆积问题的现代转机源自 Henry Cohn 和 Noam Elkies 提出的线性规划（LP）松弛方法。该方法将无限维几何堆积问题转化为在凸函数空间上的线性优化：在 $d$ 维欧氏空间中构造一个径向测试函数 $f: \mathbb{R}^d \to \mathbb{R}$，只要满足当向量模长 $|x| \ge 1$ 时 $f(x) \le 0$，且其傅里叶变换在全空间恒非负（$\widehat{f}(x) \ge 0$），那么球堆积密度的上界就可以被 $f(0)/\widehat{f}(0)$ 乘以半径为 $1/2$ 的球体积精确约束。Maryna Viazovska 正是通过巧妙利用模形式与积分变换，在 8 维与 24 维中构造出了神奇的显式函数 $f$，使得该线性规划上界与 $E_8$ 格及 Leech 格的密度严丝合缝地重合，并因此荣获了 2022 年菲尔兹奖。 ```mermaid flowchart TD accTitle: Cohn-Elkies 线性规划界机制 accDescr: 展示从构造测试函数到推导球堆积密度上界的数学松弛流程 subgraph Conditions["函数约束条件"] C1["空间截断条件:<br/>当 |x| ≥ 1 时，f(x) ≤ 0"] C2["频域非负条件:<br/>傅里叶变换 f̂(x) ≥ 0"] end subgraph Bound["LP 界导出"] B1["Cohn-Elkies 线性规划不等式"] B2["Δ_d ≤ [f(0) / f̂(0)] · Vol(B_d(1/2))"] end subgraph Impact["突破与进展"] I1["低维奇迹: Viazovska 构造模形式函数<br/>完全匹配 8 维与 24 维格子"] I2["高维极限: Astra 发现解析极值函数<br/>给出 ~ 2^(-0.6d) 的紧致等式"] end Conditions --> B1 B1 --> B2 B2 --> I1 B2 --> I2 ``` 然而，当维度 $d$ 趋于无穷大时，这个极其简洁的线性规划框架究竟能给出怎样的渐近界，长期以来一直是一个巨大的谜团。此前，Henry Cohn 等人曾通过极高精度的数值实验推测该渐近界可能具有优美的形式（与 $e^{-2\pi}$ 渐近相关，数值上接近 $2^{-0.61d}$ 或 $2^{-0.604d}$ 左右），但所有人都不知道其背后的理论根源。而在 OpenAI 的 Astra 模型实验中，研究人员仅给出了“分析高维空间中该线性规划界”的开放式任务，模型竟自主推导并构造出了极限测试函数 $f$，一举证明了渐近上界不仅小于且严格等于这个优雅的渐近值。 更令人惊叹的是，模型不仅证明了上界，还进一步证明了在 Cohn-Elkies 框架下不存在任何能够给出更好界的函数 $f$，从而彻底确立了该框架在高维下的严格等式极限。这一成果不仅解释了为何几十年前数值计算会得到特定的衰减常数，还将经典的 $2^{-0.599d}$ 理论上界实质性推进到了约 $2^{-0.604d}$。嘉宾在回顾这一突破时颇有感触地坦言，自己读博士期间曾花整整六个月苦思冥想这个问题，最后却毫无斩获；而如今 AI 模型在没有任何人类偏见提示的情况下，凭借对高维分析与优化的深层推理，仅靠一个宏观目标就完全厘清了这一数学死结，这一表现堪称不可思议。 - **Cohn-Elkies LP 界的核心架构**：利用傅里叶变换的正定性与空间支集限制，将离散几何球堆积转化为实分析与泛函分析的极值问题。 - **菲尔兹奖级别突破的基础**：Viazovska 巧妙构造出的解析测试函数构成了该理论在偶数低维中最巅峰的应用。 - **Astra 的解析证明与紧致性**：模型自主构造了极值函数并证明了最优化性质，确立了 $e^{-2\pi + o(1)}$ 形式的高维渐近等式。 - **数值直觉的理论闭环**：终结了仅凭精巧数值模拟猜测极值界限的历史，为数十年来悬而未决的高维渐近分析提供了完整且严密的数学解释。 ### [28:32](https://www.youtube.com/watch?v=1JvyLGd2Sfs&t=1712s) 优雅解答的冲击：为何此前无人想到？ 在回顾 Astra 模型解决的第一个问题（球体填充及线性规划界限，LP bound）时，研究者们感叹之前人类在攻克该极限上几乎毫无进展，因而亲眼看到模型证明“线性规划确实无法做得更好”令人格外愉悦。Mark Sellke 指出，此前已有好几位顶尖数学家尝试过这个问题，但模型的解法却出人意料地精炼短小——仅仅用了几页复分析（complex analysis），便精准切中了核心路径。 这种解法展现了数学研究中最令人心折的一种美感：“一旦你看到了答案，就会忍不住惊呼：‘天哪，为什么我之前就没想过这样做？’”。虽然研究者自嘲其实心知肚明自己为何没能想到，但看到模型能以极其自然、恰到好处的工具切中要害，依然是一次极富乐趣与震撼的体验。这也是该问题在整个十题集合中格外令人喜爱的原因。 ### [29:19](https://www.youtube.com/watch?v=1JvyLGd2Sfs&t=1759s) 从球体填充到纠错码：球面码与二进制码 紧接着，讨论转向 Astra 攻克的十个问题中的第二个，它与球体填充有着极深的内在联系——即球面码（spherical codes）与二进制码（binary codes）。Mark Sellke 解释，球面码本质上就是“弯曲表面上的球体填充”，即把球体堆积在另一个更高维球体的表面上。而之所以称之为“码”，根源在于信息论中的纠错码（error-correcting codes），其几何图景相当于在高维超立方体（hypercube）的点集上寻找分布。 纠错码的核心目标是保障不可靠信道上的通信准确性。例如发送一串比特流时，若其中某些位发生跳变损坏，系统需要借助冗余设计进行纠错解码以恢复原始信息，这就像人类语言具备容错能力、即使有几个打字错误也能读懂一样。在数学上，定长二进制字符串就是超立方体上的顶点；为了防止信号混淆，人们需要在词典中挑选彼此“距离足够远”的码字。如果要求任意两点在超立方体上不相邻，即可挑选汉明距离（Hamming distance）较大的顶点集（例如所有数位和为偶数的顶点），从而保证在发生单比特错误时能够被检测甚至纠正。 ```mermaid flowchart TD accTitle: 从纠错码到几何填充的联系 accDescr: 展示了超立方体上的二进制码、球面上的球面码以及欧氏空间球体填充之间的几何与数学联系。 subgraph Discrete["离散空间几何"] A["超立方体上的二进制码<br/>(Binary Codes)"] B["汉明距离与纠错能力<br/>(高维信道容量极限)"] A --> B end subgraph Continuous["连续曲面几何"] C["球面码 (Spherical Codes)<br/>(高维球面表面的球体堆积)"] end subgraph Euclidean["全空间几何"] D["欧氏空间球体填充<br/>(Full-Space Sphere Packing)"] end A -. 类似几何极值问题.-> C C -- "小球半径趋于零<br/>(曲率影响消失)" --> D ``` - **高维信道极限**：在极高维度下，面对固定的错误率（如 1% 误码率），消息究竟需要拉长多少才能完全容忍错误，这是信息论中最基础的极限速率（rate）问题。 - **几何与曲率的过渡**：当球面码中填充的小球半径极其微小时，大球面的曲率便可忽略不计，球面码问题在极限情形下直接退化为常规欧几里得全空间中的球体填充问题。 ### [32:41](https://www.youtube.com/watch?v=1JvyLGd2Sfs&t=1961s) 对称性与表示论：突破经典 KL 界限 在这类码率界限问题中，前人最著名的经典成果来自卡巴季扬斯基-列文斯坦（Kabatiansky-Levenshtein, 即 KL）等人基于线性规划提出的界限。Astra 模型成功打破了长期停滞的局面，针对球面码和二进制超立方体码均给出了更优的理论界。 令人惊奇的是，模型在不同问题中所展现的技术手段截然不同： - 在全空间球体填充的线性规划分析中，模型主要运用了纯粹的**复分析**工具； - 在球面与超立方体的情形中，模型却转向了深度的**表示论**（representation theory）。 由于球面和立方体自身具有极高的对称性，证明的核心切入点正是充分榨取这些对称结构。经典方法虽也借助了一定对称性，但 Astra 的突破在于“极其重度地倚赖表示论”，以远比以往更加精细复杂的方式将代数对称性引入估计。更妙的是，若在模型由表示论推导出的公式中取“小球半径极限”，恰好能够部分还原出全空间球体填充中的关键估计常数，揭示了两类问题背后严密的数学统一性。 ### [35:10](https://www.youtube.com/watch?v=1JvyLGd2Sfs&t=2110s) 模型的交互与推进：从“浅尝辄止”到逼出猜想 Mehtaab Sawhney 与 Mark Sellke 透露，十个问题中绝大多数都是直接将题目抛给模型后由其自主端到端解决的，唯独这个问题经历了一次极具启发性的**人机交互**： 1. 研究团队最初只提示模型去尝试改进纠错码的常数界（模型只需将前人界限中的指数项常数提升一点即可完成任务）； 2. 模型给出了一个初步改进，并运用了一定程度的表示论工具； 3. 数学家具备敏锐的鉴赏力，追问了一句：“你能把这个方法推得更远吗？极限会发生什么？”； 4. 受到追问后，模型拿出了复杂度骤增的高深表示论证明，而将这套机制推演到极致所得到的极限值，竟然直接对齐了全空间球体填充的理论猜想值； 5. 看到这一联系后，研究人员随即让模型直接去分析全空间的情形，最终补全了整个理论拼图。 这一过程表明，两个问题在模型层面的突破绝非巧合，而是人类数学家的洞察与模型强大的符号推演能力结合的产物。 ### [36:17](https://www.youtube.com/watch?v=1JvyLGd2Sfs&t=2177s) 任务导向还是品味缺失？Harness 与模型自主性的边界 Lisha Li 由此引发了一场深入的讨论：人类仅仅给出了一个极度朴素的提示——“你能再推进一步吗？”，但这背后依赖的却是资深数学家对技术潜力的“品味与判断”（judgment）。这究竟是当前模型能力不足、必须依赖人类外挂框架（harness）与引导，还是未来随着模型规模扩展就能自主跨越的阶段？ Mark 解释道，模型之所以在一开始没有穷尽潜力，并非“能力达不到”（capabilities issue），而在于大语言模型有着极强的**任务导向性**（task-oriented）： - 当你给它一个具体的指标任务（例如将某个渐近界改善一个常数因子），一旦它达成了目标，它就会“感到心满意足”，停下脚步，而不会自发决定“让我看看能不能把整座理论大厦再推倒重建一层”。 - “它只是当时不想费那个劲罢了（it kind of didn't feel like it at the time）”，只要人类再次发问，它立刻能展示出深不见底的推演纵深。 这种现象带来了关于“品味（taste）”与“上下文视野”的哲学思辨：模型目前缺乏的是高维度的宏观判断力——意识到某种表示论方法不仅能完成当前小任务，还能通向更大的数学统一。人类数学家正是察觉到“这套工具潜力远未耗尽”，才充当了关键的指路人。未来模型是会继续作为强大的局部任务执行者、由人类来搭建 harness 与方向引导，还是能逐渐演化出自主探索更高阶数学审美与研究方向的品味，成为了摆在 AI 数学研究面前最核心的悬念。 ### [38:09](https://www.youtube.com/watch?v=1JvyLGd2Sfs&t=2289s) 复合问题的连续推进与“品味”的本质 讨论首先聚焦于模型解决复杂数学问题时的演进机制。Lisha Li 指出，过去在利用评估或提示工程框架（Harness）时，往往需要对模型进行高密度的引导与约束；然而随着模型能力的迅速飞跃，人类所需施加的外部控制正在减少。解决一道更困难的数学难题，本质上等于连续解决许多较小、难度稍低的子问题。模型能够攻克更高难度的问题，正说明它已经有能力在一个“单一连续单元”（single continuous unit）中承担越来越庞大的推理工作量。对于数学证明而言，几乎没有任何高深问题能凭单一一枚灵感火花直接迎刃而解，往往需要多块不同的拼图相互咬合、协同呼应。 模型并非在一瞬间就凭空抽取出所有关键想法，它必须经历“观察构件 A 如何与构件 B 产生交集”的过程。这种拼装与串联本身就是在独立解决子问题。Lisha 提出一个核心议题：当研究者指示模型“朝这个方向继续深入推进”时，这种提示在量级上其实与模型内部自行解决的一系列子步骤别无二致。这引发了一个更宏观的疑问——模型是否正在展现出真正的科研“品味”（taste）？在评估模型从初级研究员向独立科学家跃迁时，人们往往纠结于它是否只会蛮力试错，还是真正具备了感知“哪些是好问题、哪些是死胡同”的判断力。 - 解决难题的本质是多个子想法的动态交互，模型在长程单体上下文中协调不同组件的能力正显著提升。 - 外部提示逐步从密集干预转向极简指导，说明模型自主完成从灵感到多步拼装的全过程比例大幅增加。 --- ### [40:01](https://www.youtube.com/watch?v=1JvyLGd2Sfs&t=2401s) 功利主义视角下的“品味”与分工协作架构 面对“品味究竟是什么”的哲学追问，Mehtaab Sawhney 给出了极其务实的功利主义（utilitarian）解释：“如果你能通过做出更好的判断，从而更快地解决问题，我认为这就是衡量品味的最佳综合指标。”以此标准衡量，模型能够解决难度更高的问题，在定义上就意味着它具备了更好的品味。虽然当前以任务为导向的模型偶尔会出现令人遗憾的症状——例如它明确意识到自己完成了一项突破，却因为人类没有明确要求穷追猛打而未将结果推向极限，但 Mehtaab 认为相比整体的技术突破速度，这类缺陷微不足道。 Mark Sellke 则从认知负载与系统设计的角度提出假想：让同一个模型既执行具体的长程繁重推理，又同时展现高层级的审美与品味，容易引发上下文污染。更合理的设想或许是构建分工协作系统：由一个专注于“品味与方向”的模型充当主管 AI，另一个模型作为下属负责在漫长时间跨度内死磕攻坚。两人随后指出，这与人类数学家的工作状态非常相似： - 人在单打独斗时容易钻牛角尖（pigeonholed），此时只要有一位同行探过头来问一句“你在干什么？”，花上十秒钟抽身纵览全局，往往就能打破僵局。 - 这种多智能体协作或双层模型架构，实际上人工复刻了人类顶级学术合作中极具成效的动态机制。 - Mark 补充指出，品味的很大一部分还体现在“对自身或某种特定方法究竟擅长解决何种问题有着敏锐嗅觉”，这种知己知彼的敏锐感（having a nose for what to pursue），往往是长久高效完成硬核任务的自然副产品。 ```mermaid flowchart TD accTitle: 模型分工协作与数学品味架构 accDescr: 展示监督模型负责审美取向与方向把控，执行模型负责长程计算攻坚，两者协同避免思维僵化的架构流程。 subgraph Supervisor["监督决策层 (品味与全局判断)"] A["高层模型 (Supervisor AI)"] --> B["判断方法可行性与问题价值"] B --> C["关键节点抽离视角：防止钻牛角尖"] end subgraph Worker["长程执行层 (求解与子命题验证)"] D["基底执行模型 (Underling AI)"] --> E["高维搜索空间剪枝"] E --> F["单体长程推理与子模块拼接"] end C -- "方向纠偏与顶层指导" --> D F -- "中间突破与计算反馈" --> A ``` --- ### [44:38](https://www.youtube.com/watch?v=1JvyLGd2Sfs&t=2678s) 群论基础与 Astra 对非 Sofic 群（Non-Sofic Group）的存在性突破 对话随后切入 OpenAI 的 Astra 模型在纯数学领域最重磅的代表性成果之一：证明非 Sofic 群（Non-sofic group）的存在性。为了向听众解释这一成就的分量，Mark Sellke 首先回顾了群论（Group Theory）的基本定义： 1. **群的构成**：一个集合及其配备的乘法运算，它是数学家刻画“对称性”（symmetry）的语言。若 $g$ 与 $h$ 是群内元素，则其复合乘积 $gh$ 仍是良定义的群内元素。 2. **基本公理**：满足结合律（associativity）、拥有单位元（identity），且对每个元素 $g$ 都存在唯一的逆元（inverse）。 3. **几何与代数意象**：群运算抽象自各种操作的复合，比如数的相乘、矩阵乘法，或是旋转刚体。例如正方形的所有旋转操作构成仅含 4 个元素的有限群；而平面的圆盘旋转则构成拥有不可数无限元素的连续群。 随后讨论聚焦于无穷群的近似问题。数学家研究的 Sofic 群是一类特殊的有限群或可数无限群（元素数量与整数集同等），其核心定义在于：**它在某种意义上能够被有限群充分逼近**。在过去几十年中，数学界始终不知道宇宙中是否存在“无法被有限群逼近的群”，即非 Sofic 群。许多数学家甚至暗自期盼着一种“奇迹”——希望所有群都是 Sofic 群，因为如果一切无限群都能被有限群近似，学者们就能将在有限群上成立的大量优良性质与定理体系直接推广推广出去。然而，Astra 模型彻底打破了这一幻想，通过严密构造证明了：**非 Sofic 群确实存在**。这一反例的发现不仅平息了长期悬而未决的理论猜测，也展示了模型在抽象代数结构探索中深入人类盲区的惊人威力。 ### [48:20](https://www.youtube.com/watch?v=1JvyLGd2Sfs&t=2900s) 索菲克群与高特沙尔克猜想的起源 围绕非索菲克群（non-sofic group）的探索，其根源可以追溯到动力系统与抽象代数中的经典猜想。数学中有一个概念叫做“亚满射群”（surjunctive group），描述的是群上动力系统的一种拓扑性质。早在 20 世纪 70 年代，数学家高特沙尔克（Walter Gottschalk）就提出了著名的猜想：是否每一个群都是亚满射群？ 后续研究发现了一个核心事实：任何索菲克群（sofic group）都必然是亚满射群。其背后的经典范式是“先在有限群上证明该性质，然后利用近似逼近推广到无穷情形”。正因为索菲克性提供了一种用有限对象逼近无限结构的绝佳途径，数学界才迫切想要弄清——究竟是否存在无法被如此逼近的“非索菲克群”。 - **高特沙尔克猜想（Gottschalk's Conjecture）**：20 世纪 70 年代提出，探讨是否所有群都具备亚满射性质。 - **逼近范式**：若一个性质在有限群上成立，且目标群能被有限群局部良好逼近（即索菲克性），该性质便可传递至无穷群。 --- ### [49:09](https://www.youtube.com/watch?v=1JvyLGd2Sfs&t=2949s) 凯莱图、整数环逼近与奥尔德斯-莱昂斯猜想 为了跳脱繁复的技术细节、直观解释索菲克逼近的本质，Mark 以最基本的整数集 $\mathbb{Z}$ 为例展开说明。若将整数构造成凯莱图（Cayley graph），只需把每个整数与其相邻的前后两个数相连，就会形成一条无限延伸的双向直线。这种结构可以用有限图进行局部逼近：取模 $n$ 剩余类群 $\mathbb{Z}/n\mathbb{Z}$，其凯莱图是一个巨大的圆环。对于圆环上的任意一点及其近邻，局部微观视角下的拓扑结构与无穷直线完全相同；唯有走到非常远的地方，才能察觉它在大尺度上弯曲闭合成了一个圆。因为 $\mathbb{Z}$ 和 $\mathbb{Z}/n\mathbb{Z}$ 在加法下都是群，这种“大环局部逼近直线”的方式正是整数群作为索菲克群的最生动缩影。 在概率论领域，戴维·奥尔德斯（David Aldous）和罗素·莱昂斯（Russell Lyons）提出了一个更为宏大的推广猜想（即奥尔德斯-莱昂斯猜想，Aldous-Lyons conjecture）。该猜想断言：任何具有良好性质（即单模性，unimodularity）的无限单模随机图，都可以被大型有限图所局部逼近。普通的群凯莱图是确定性的，而奥尔德斯-莱昂斯猜想允许网络具有随机性，因而是一个条件更宽泛、形式更强的大猜想。 ```mermaid flowchart TD accTitle: Sofic Approximation and Aldous-Lyons Comparison accDescr: Diagram illustrating the Cayley graph of integers approximated by modulo circles and its relation to the Aldous-Lyons conjecture. subgraph SoficConcept["群的索菲克逼近（确定性凯莱图）"] Z["整数群 Z (无穷直线)"] Zn["模 n 剩余类群 Z/nZ (大圆环)"] Zn -->|"局部结构一致，全局闭合逼近"| Z end subgraph Conjectures["猜想强弱关联"] AL["奥尔德斯-莱昂斯猜想 (随机单模图逼近)"] Sofic["所有群均为索菲克群猜想 (确定性凯莱图逼近)"] AL -->|"涵盖范围更广（强猜想）"| Sofic end ``` - **凯莱图对应**：群选定生成元集合后，可唯一确定一个连通网络；整数群对应的就是最简单的网格直线。 - **猜想强度对比**：奥尔德斯-莱昂斯猜想包含随机图族，其适用范围超越纯代数结构；否定更强的猜想，并不自动解决群论中关于索菲克性的狭义命题。 --- ### [51:55](https://www.youtube.com/watch?v=1JvyLGd2Sfs&t=3115s) 250 页量子复杂性巨著 vs AI 的 15 页纯代数证明 大约两年前，奥尔德斯-莱昂斯猜想被国际团队攻破并证伪。但那次证伪是一场极度繁重的技术“体力与智力巡礼”（tour de force）：论文长达 250 页，建立在另一篇 200 页的巨著之上，深度横跨了量子复杂性理论（如纠缠与 MIP*=RE 框架）并搭建了极其复杂的理论桥梁，Mark 坦言“当时几乎没有多少人能够彻底读懂”。更关键的是，证伪奥尔德斯-莱昂斯猜想在逻辑上弱于否定“所有群皆为索菲克群”，并未彻底终结群论本身的悬案。 出人意料的是，OpenAI 的 Astra 模型对“非索菲克群存在性”的直接证明完全走了一条不同的道路。它的证明文本极其精炼，大约只有 15 页，并且“完全留在了纯群论的领地内”，没有引入任何复杂的量子复杂性跳跃。它虽然站在了库恩（Gábor Kun）和托姆（Andreas Thom）等数学家既有研究成果的肩膀上，但本质上是一篇逻辑清晰、结构极为典雅纯正的标准数学论文。 - **人类经典突破的厚重感**：基于量子复杂性的奥尔德斯-莱昂斯猜想反例篇幅达数百科页，构筑了跨学科的高难度深渊。 - **Astra 的反例风格**：篇幅仅约 15 页，完全在经典群论语境下推演完成，论证干净利落。 --- ### [54:07](https://www.youtube.com/watch?v=1JvyLGd2Sfs&t=3247s) 拆解“巧合阴谋”：Astra 证明中的精妙组合学 针对这一突破的内部机理，Lisha 追问究竟是什么让这处“可数无穷”坚决抗拒有限对象的逼近，以及团队在事后分析（postmortem）中向 Astra 提问得到了什么解答。Mehtaab 与 Mark 解释道，证明某对象是“非索菲克群”的最大难点在于，索菲克性质极其抽象飘渺，你必须证明它“无法被任何可能存在的有限群逼近”，这要求给出极其明确的结构性阻碍（combinatorial obstruction）。 Astra 的解法在本质上是一场“非常非常微妙的组合学论证”。在库恩和托姆等前人学者的论文中，他们其实已经圈定了怀疑对象并摸到了正确边缘，但始终无法完全排除某种极端的病态情形（即几位数学家隐式提及的“某种怪异的代数共谋”/ conspiracy）。Astra 在深度阅读与推演中抓住了这个死穴，引入了一个额外的代数事实，彻底粉碎了这种共谋出现的可能性，跑通了最艰难的临门一脚。 更为戏剧性的是，研究人员原本以为 AI 解决深奥数学问题时，会像蛮力搜索一样吐出长达几千页、人类根本无法审阅的“天书”，但事实恰恰相反：“目前只有人类才能写出两百页的巨篇，AI 给出的反例证明反而异常短小精悍且充满美感。” - **组合学结构障碍**：索菲克定义难以正面捕捉，模型的核心贡献在于找到并严格论证了限制逼近的离散障碍。 - **击碎“代数共谋”**：前人隐约察觉到但无法证明不存在的特殊例外，被 Astra 借助精准的代数约束彻底排除。 - **反直觉的审美范式**：AI 当前产出的深刻定理大多证明篇幅紧凑，推翻了“机器证明必然冗长繁琐”的固有偏见。 --- ### [56:41](https://www.youtube.com/watch?v=1JvyLGd2Sfs&t=3401s) 从机器发现到人类拓展：数学界的新分工 在 Astra 给出非索菲克群的论文后，数学界立刻产生了良性共振。先前被 Astra 借鉴思路的原创数学家（包括托姆等人）迅速跟进，以此为基础发表了后续论文，进一步拓展了这套思想，构造出了更多的非索菲克群实例。Mark 强调，这正是他们团队最期望看到的局面——OpenAI 的研究者并不想亲自扑进每一个领域去写海量的后续跟进论文，只要模型能激发人类顶级数学家跟进、延伸出更多新数学，就是最理想的科学协作。 Lisha 借此将话题引向数学界对 AI 的接纳现状与未来伦理。目前学术界正处于转变的十字路口：绝大多数一线数学家已经清醒地意识到 AI 正在产出极其深刻的非平凡成果，在日常科研工作流中拒绝使用 AI 将沦为一种纯粹的竞争劣势。然而，新的焦虑也随之爆发：论文署名权该如何分配？AI 能否作为共同作者？人类同行评审该如何应对几何级暴增、急需检验的 AI 新成果？整座数学大厦正面临着前所未有的范式重塑。 - **学界即时跟进**：库恩和托姆已基于 Astra 的突破开展后续泛化研究，证实了机器成果对主流学界的牵引力。 - **研究分工转型**：研究机构提供破局性的源头发现与短小精妙证明，领域数学家负责后续的理论深化与横向展开。 - **署名与验证困境**：当 AI 成为不可或缺的推导主力，合作属性界定、学术归属机制及审稿承载力均面临巨大制度挑战。 ### [58:28](https://www.youtube.com/watch?v=1JvyLGd2Sfs&t=3508s) AI 既制造信息过载，又成为消化数学知识的解药 Mehtaab Sawhney 指出，模型既然具备生成高深数学推理的能力，反过来它也能成为人类理解复杂数学的高效工具。以他个人的日常经验为例，当他在 arXiv 上浏览论文并试图搞懂某项证明时，过去可能需要花大量时间精读引言和繁复细节，而现在只需将论文的 PDF 直接丢进心仪的模型中，让它迅速提炼出粗略的证明策略与核心脉络。 尽管大家担忧 AI 可能会以前所未有的速度产出呈指数级增长的数学成果，从而引发认知过载，但模型在降低理解门槛、帮助吸收知识方面同样表现出惊人的效率。Mark Sellke 坦言，虽然目前与模型反复交互推导有时仍存在些许波折，但利用模型去理解一项数学成果的速度，已经远远超过传统人工阅读模式。AI 在制造“数学产出过剩”问题的同时，恰好也提供了解决这一困境的关键钥匙。 - **快速吸收论文**：研究者无需逐行死磕几十页的论文，直接让模型提取整体架构和核心证明直觉（Proof Strategy）。 - **缓解信息过载**：AI 带来海量成果的同时，赋予了学者以极高吞吐量快速消化复杂知识的杠杆。 ```mermaid flowchart LR accTitle: AI in Math Discovery and Absorption accDescr: Flowchart illustrating how AI accelerates both mathematical discovery and human comprehension. subgraph Generation ["AI 生成端"] A["指数级增长的数学论文与证明"] end subgraph Absorption ["AI 消化端"] B["大模型提炼核心证明策略"] C["跨学科与大众更易吸收"] end A -->|"PDF 喂入大模型"| B B -->|"快速拆解逻辑"| C ``` --- ### [01:00:09](https://www.youtube.com/watch?v=1JvyLGd2Sfs&t=3609s) 瓶颈转移：从“证明定理”走向“知识编织与群体理解” 谈到数学研究形态的转变，Mark Sellke 剖析了数学这门学科在过去与未来的底层逻辑差异。传统数学研究中，提出一个成果固然需要有人发现它，但更重要的是人们去吸收、内化它，并将其锚定在人类现有的认知图景之中。在过去的年代里，“给出严格证明”本身就是极度艰难的头号瓶颈，以至于只要某人成功证明了定理，他自然而然就会深入理解它，并自动承担起向整个学界解释、维护和推广该理论的责任。 然而随着大模型逐渐接管证明这项繁重而耗时的核心瓶颈，数学知识组织的约束条件发生了根本性位移。Lisha Li 提出这是否会让数学变得更具经验主义色彩，而 Mehtaab Sawhney 强调，未来的数学价值将更加偏向“构建恰当的思想框架”，以及“如何将机器发现的深奥结论通俗而透彻地解释给其他人类”。原本在数学共同体中隐性存在的交流、整合与教育价值，正变得愈发显性化和不可或缺。 - **研究瓶颈迁移**：从“能否算出证明细节”，转变为“能否理解其含义并将其置于正确的理论框架中”。 - **数学家的角色重塑**：从单纯的解题者兼定理维护者，转变为知识的翻译家、策展人与群体理解的搭建者。 - **群体认知的显性化**：数学不仅是符号的成立，更是人与人之间共享理解的艺术。 --- ### [01:02:43](https://www.youtube.com/watch?v=1JvyLGd2Sfs&t=3763s) 数学天花板、未解之谜与人人可参与的数学复兴 Mark Sellke 指出，数学这门学科最迷人的一点在于“问题难度的天花板极高”。即便 AI 的数学推理能力以指数级飞速跃升，诸如 P vs NP 这类硬核核心命题，人类与 AI 依然很有可能在很长时间内都无法攻克。这意味着整个数学领域可能会迎来分流：日常性、常规性的推导被模型大幅自动化，而人类与学界的研究重心将更加纯粹地附着在那些最深邃、最重大的终极谜团上。 这种转变给数学家带来了极大的解脱与精神慰藉。Mehtaab Sawhney 感叹，自己职业生涯中曾花费数月甚至数年苦苦思索某些问题却毫无头绪，如今能看到其中一部分谜题有望被 AI 解答，这本身就是纯粹的快乐。Lisha Li 则将此形容为一场“数学成果与理解的文艺复兴”，让非全职数学家也能深度参与数学探索。在实际应用场景中，不论是理论物理还是工程应用，跨领域学者再也无需四处寻找某细分方向的顶尖专家，即可借助模型运用最尖端的数学工具，大幅加速整个现实世界的应用数学演进。 - **顶层谜题的坚固性**：即便常规数学工作被自动化，如 P vs NP 等超高难度谜题依旧是人类智力的试金石。 - **解开心结的快乐**：数学家能从长年悬而未决的思考死胡同中解脱，通过模型见证答案的揭晓。 - **打破学科壁垒**：无需成为数学专业从业者或依赖稀缺的数学家同行，模型便能协助工程师和物理学家无缝调用精深数学成果。 ## Notable quotes > [06:00](https://www.youtube.com/watch?v=1JvyLGd2Sfs&t=360s) 你常常会想：也许我应该试试这个方法，但这看起来希望渺茫，根本不值得我耗费青春。然而在好几个前沿问题中，模型凭借着丰富的先验知识与极佳的数学品味，做出了最正确的下注。 — Mehtaab Sawhney > [16:20](https://www.youtube.com/watch?v=1JvyLGd2Sfs&t=980s) 如果存在比六边形更高效的填充方式，蜜蜂在演化中早就采取其他方案了，因为演化是讲究极致效率的 — Mark Sellke > [28:32](https://www.youtube.com/watch?v=1JvyLGd2Sfs&t=1712s) 一旦你看到了答案，就会忍不住惊呼：‘天哪，为什么我之前就没想过这样做？’ — Mark Sellke > [36:17](https://www.youtube.com/watch?v=1JvyLGd2Sfs&t=2177s) 它只是当时不想费那个劲罢了 — Mark Sellke > [40:01](https://www.youtube.com/watch?v=1JvyLGd2Sfs&t=2401s) 如果你能通过做出更好的判断，从而更快地解决问题，我认为这就是衡量品味的最佳综合指标。 — Mehtaab Sawhney > [54:07](https://www.youtube.com/watch?v=1JvyLGd2Sfs&t=3247s) 目前只有人类才能写出两百页的巨篇，AI 给出的反例证明反而异常短小精悍且充满美感。 — Mark Sellke > [02:43](https://www.youtube.com/watch?v=1JvyLGd2Sfs&t=163s) 该问题是可达的，具体的切入推导路径应该是这样的…… — Mehtaab Sawhney > [35:10](https://www.youtube.com/watch?v=1JvyLGd2Sfs&t=2110s) 你能把这个方法推得更远吗？极限会发生什么？ — Mark Sellke / Mehtaab Sawhney ## Mentioned OpenAI, a16z, Lisha Li, Mehtaab Sawhney, Mark Sellke, Yufei Zhao (赵宇飞), Paul Erdős (保罗·埃尔德什), GPT-5, Astra, Andrew Wiles (安德鲁·怀尔斯), Walter Rudin, Lean, Thomas Hales, Maryna Viazovska, Henry Cohn, Noam Elkies, Kabatiansky, Levenshtein, Walter Gottschalk, David Aldous, Russell Lyons, Gábor Kun, Andreas Thom, IMO (国际数学奥林匹克竞赛), Sphere Packing Problem (球体填充问题), Kepler Conjecture (开普勒猜想), E8 Lattice, Leech Lattice, Cohn-Elkies Linear Programming Bound, Spherical Codes (球面码), Binary Codes (二进制码), Non-Sofic Group (非索菲克群), Gottschalk's Conjecture (高特沙尔克猜想), Aldous-Lyons Conjecture (奥尔德斯-莱昂斯猜想), MIP*=RE, P vs NP ## Video description a16z Infra Partner Lisha Li sits down with OpenAI mathematicians Mehtaab Sawhney and Mark Sellke to discuss how quickly AI’s mathematical capabilities are advancing, what recent results reveal about model reasoning, and what happens when AI begins making progress on problems mathematicians have struggled with for decades. Mehtaab and Mark unpack several recent results from OpenAI’s models, including advances in sphere packing and the construction of a non-sofic group. They explain why the surprising part isn’t simply that models can search more possibilities or work longer than humans: in many cases, the reasoning traces look remarkably similar to the work of an expert mathematician, including choosing promising approaches, backtracking when they fail, and combining ideas from across the literature. They also explore what this means for mathematics itself: how the role of human taste and judgment may change, whether AI could produce far more mathematics than humans can absorb, and why models that accelerate discovery may also make sophisticated results easier to understand. Timestamps: 00:00 - Intro 00:50 - From Practicing Mathematician to OpenAI: Meet Mark & Mehtaab 02:43 - Why GPT-5 Was the Conversion Moment 04:21 - Beyond Search & Connections: How Recent Progress Goes Deeper 09:51 - Reasoning Traces: Is It Lucky Sampling or Actual Backtracking? 11:44 - Why Math Papers Are a Bad Training Set for Real Mathematics 16:20 - The Astra 10-Problem Set: Favorites & Deep Dives 36:17 - The Harness vs the Model: What Actually Matters? 40:01 - What Even Is "Taste" in a Model? 57:32 - How Should the Math Community Adopt AI? 01:00:01 - Empirical vs Theoretical Math & the Positive Vision Resources: Follow Lisha Li on X: https://x.com/lishali88 Follow Mehtaab Sawhney on X: https://x.com/mehtaab_sawhney Follow Mark Sellke on X: https://x.com/MarkSellke Stay Updated: If you enjoyed this episode, be sure to like, subscribe, and share with your friends! Find a16z on X: https://twitter.com/a16z Find a16z on LinkedIn: https://www.linkedin.com/company/a16z Listen to the a16z Show on Spotify: https://open.spotify.com/show/5bC65RDvs3oxnLyqqvkUYX Listen to the a16z Show on Apple Podcasts: https://podcasts.apple.com/us/podcast/a16z-podcast/id842818711 Follow our host: https://x.com/eriktorenberg Please note that the content here is for informational purposes only; should NOT be taken as legal, business, tax, or investment advice or be used to evaluate any investment or security; and is not directed at any investors or potential investors in any a16z fund. a16z and its affiliates may maintain investments in the companies discussed. For more details please see http://a16z.com/disclosures. [Source](https://www.youtube.com/watch?v=1JvyLGd2Sfs)

[6:00] Mehtaab Sawhney: 你常常会想：也许我应该试试这个方法，但这看起来希望渺茫，根本不值得我耗费青春。然而在好几个前沿问题中，模型凭借着丰富的先验知识与极佳的数学品味，做出了最正确的下注。 [16:20] Mark Sellke: 如果存在比六边形更高效的填充方式，蜜蜂在演化中早就采取其他方案了，因为演化是讲究极致效率的 [28:32] Mark Sellke: 一旦你看到了答案，就会忍不住惊呼：‘天哪，为什么我之前就没想过这样做？’ [36:17] Mark Sellke: 它只是当时不想费那个劲罢了 [40:01] Mehtaab Sawhney: 如果你能通过做出更好的判断，从而更快地解决问题，我认为这就是衡量品味的最佳综合指标。 [54:07] Mark Sellke: 目前只有人类才能写出两百页的巨篇，AI 给出的反例证明反而异常短小精悍且充满美感。 [2:43] Mehtaab Sawhney: 该问题是可达的，具体的切入推导路径应该是这样的…… [35:10] Mark Sellke / Mehtaab Sawhney: 你能把这个方法推得更远吗？极限会发生什么？

[2:43] Mehtaab Sawhney: I spent a few hours... plugged it into GPT-5 and like five minutes later it found a reference... it was just very nice to be told, 'Yes, this is in reach, here's how you do it.' [9:51] Mark Sellke: As a human, if you have some wrong path you go down for a while, it can be hard to rewire your brain to start over... your context window is a little polluted. But it's very easy to do this with an AI. [23:09] Mehtaab Sawhney: One of them is just like you see it and you're like, 'Oh man, why didn't I think of this?'... It was quite nice to see it, and it was fun to see. [40:01] Mehtaab Sawhney: I tend to be pretty utilitarian in my view of taste... if you're able to solve problems faster by making better judgments, I think that's the best general proxy I have for taste. [54:07] Mark Sellke: only humans can generate 200-page proofs right now [64:29] Mark Sellke: sorry mathematicians [64:29] Mark Sellke: It'd be nice for the world if applied mathematics went a lot faster.

[0:00] often as a practicing mathematician you [0:01] have an idea and then you kind of think [0:04] it might work then you try for a few [0:06] hours a few weeks and at some point you [0:08] give up [0:08] >> whereas for GPT like okay a human told [0:11] me to do this let's let's just do this [0:13] and so that's why we're sort of in this [0:14] renaissance of like reachable results [0:16] >> this is the best part about this problem [0:18] which is really nobody has any idea [0:20] >> is the model just guessing in some [0:21] insane way [0:22] >> it doesn't seem like there's a limit so [0:24] far but it doesn't have that context yet [0:27] >> it'd be nice for the world if applied [0:28] mathematics went a lot faster The [0:29] ceiling for difficulty of a math problem [0:31] is pretty high. Even if AI continues [0:33] getting like exponentially better at [0:35] math, plausible will never solve [0:37] something like P versus [0:38] >> what's the ideal way that this is being [0:40] taken up by the math community? Probably [0:42] most at this point are like, okay, AI is [0:44] obviously doing some non-trivial stuff. [0:46] >> Um, so [0:50] >> well, thank you guys for coming. This is [0:53] really exciting because I think math has [0:55] been moving so fast uh with AI. I'd just [0:57] love to, you know, get to both [0:59] practicing mathematicians and who work [1:01] at OpenAI to chat on uh some of these [1:04] results. Um so, you know, we have with [1:07] us Mark Selki and uh Matab Swani. We're [1:11] connected actually because EU um was [1:14] actually your adviser. And so both of [1:15] you guys um have worked much more deeply [1:18] in math since I've like quit many many [1:20] like over a decade ago. Um, and so this [1:22] is very exciting to kind of hear a [1:25] download of your thoughts on how OpenAI [1:27] has been sort of approaching this and [1:29] also just like where you think math is [1:32] going with the incredibly rapid advance [1:35] of uh of how AI has been helping. Um so [1:39] yeah I don't know maybe uh we can start [1:41] off with some very basic questions of [1:43] like you know you guys what what do you [1:46] do to the extent that you can of course [1:48] share um and uh and how did you come [1:52] from you know being a practicing [1:54] mathematician to working at open AAI. [1:57] >> Yeah I mean I guess we we both you know [2:01] broadly got excited last year when the [2:03] model started to really take off in [2:05] math. [2:06] >> Yeah. Um, so I uh I joined a little bit [2:09] uh before Matab. I saw the IMO gold [2:11] medal last summer basically and I I [2:13] thought, you know, this is this is [2:14] amazing. You know, I I want to see what [2:17] the heck they did. Let me let me go see. [2:20] Um and then uh yeah, I guess in the fall [2:23] we um [2:25] Mark gave me a G Mark gave me a GP5 [2:28] account and then I started playing with [2:29] models and very quickly became convinced [2:32] that yeah, it was extremely exciting to [2:33] play with them. Yeah, [2:35] >> you two were collaborating before this. [2:37] >> Yeah, you've known each other for a [2:38] while. [2:39] >> Yeah, [2:39] >> we had like one paper we actually wrote [2:41] jointly. [2:42] >> Yeah. [2:42] >> Yeah. [2:43] >> So, GPG5 was your conversion. [2:46] >> What was the magic that sort of I don't [2:48] know as you're doing like what question [2:50] do you throw at it? [2:51] >> What process? [2:52] >> Yeah. So, I think actually Yeah. So, I [2:54] think how this started was u at least [2:56] for me the starting moment was something [2:58] like there's a collection of problems [3:00] called um so Paul Erdish is a very [3:02] famous mathematician. posed a bunch of [3:04] problems and so they've now all been [3:06] collected on this site and I so I [3:08] specifically worked in combinotaurics [3:10] and a lot of these questions are among [3:12] the most important. So it's always fun [3:14] to flick through the site but one thing [3:16] that often happened to me that was [3:17] extremely frustrating was I would look [3:19] at a question see that it's marked as [3:21] open and then not actually know if it's [3:23] correct not actually know if it had was [3:26] still unsolved because the literature is [3:28] often quite hard to search. Um, [3:30] >> yeah. [3:30] >> And I one instance I just plugged it [3:34] into GPD5 and like five minutes later it [3:36] found a reference and and this was aa [3:38] this was a case where a few few of my [3:40] friends had actually started thinking [3:41] about the problem on the site. I was [3:42] talking with them and I mean we had [3:44] spent a few hours didn't seem wasn't [3:47] clear if the problem was within reach [3:48] and it was just very nice okay to be [3:50] told yes this is in reach here's how you [3:51] do it and yeah GPD5 told me this and [3:53] then I told Mark about this and this [3:55] yeah this is sort of yeah this for me [3:57] was quite a surprising moment. Yeah. And [4:00] then we we looked more into it and we [4:02] found like 10 more cases sort of like [4:04] this [4:06] >> at the time. I feel like you know being [4:09] better at maybe making connections [4:11] between [4:12] wide or even just like as you're saying [4:14] like the the search for whether there's [4:16] been a result or a related thing earlier [4:18] is just like kind of humanely hard but [4:20] maybe better for machine. But I imagine [4:22] as the progress has happened in the last [4:25] year what has been impressive has kind [4:28] of reached beyond that and um maybe you [4:32] know through talking about it more [4:34] abstractly or if it's more natural to [4:36] talk about it through like one of the [4:37] problems that has been recently [4:38] announced through you know Astra you can [4:40] kind of enlighten me as to like how um [4:43] how the recent progress has been a lot [4:45] more than just like you know searching [4:47] through more areas making these [4:49] connections between the field and [4:50] perhaps just like actually deeper more [4:52] mathematical reasoning that's similar to [4:54] a working mathematician. [4:57] >> Yeah. I mean [4:59] I I I think this like search point of [5:03] being you know familiar with everything [5:05] is still definitely like a relative [5:07] strength that yeah [5:08] >> maybe informs like the types of problems [5:11] that AI is solving. Now [5:13] >> um [5:15] I think there's there there are some [5:17] other relative strengths and weaknesses. [5:19] Um, another relative strength that's [5:22] pretty noticeable is just like it's very [5:25] good at executing on some like idea once [5:28] it once it has it like you know when [5:30] whenever you have an idea there's like [5:34] there's there's usually some amount of [5:36] you know getting everything lined up [5:38] like you know is epsilon like smaller [5:41] than delta this this kind of of thing [5:43] you have to get everything correct and [5:45] like for a human you know you it's easy [5:49] to get lost in these kinds of details [5:50] and the AIS just kind of always nail [5:52] these kinds of arguments I find. [5:54] >> Yeah. Is it usually just I mean I feel [5:57] like you guys will know more detail on [5:58] this but like for the unit distance [6:00] problem it was just like the approach [6:03] there was definitely contributions from [6:04] you know the open AI but like the [6:06] approach perhaps was suggested even you [6:10] know originally by Erdos and then it's [6:13] just that the actual reasoning was a [6:16] very very like momentous um like um feat [6:22] and so for a human you're like well I [6:24] only have a limit amount of time and if [6:26] after so many you know steps it is still [6:29] not clear I mean maybe you're like [6:30] Andrew Wilds and you actually spend 10 [6:32] years alone and do something but like [6:33] it's not clear then it's just it it's it [6:37] doesn't become like the riskreward is [6:39] not good enough whereas for GPT like [6:43] okay I'll like a human told me to do [6:45] this let's just do this and so that's [6:46] why we're sort of in this renaissance of [6:48] like reachable uh results um does that [6:52] track and did you feel like with the [6:54] Astro results is that sort of like where [6:56] the strengths have been primarily or [6:58] there's an extra ingredient or magic [7:01] here. [7:04] >> I I feel I think the unit distance [7:06] examples about it's quite telling in the [7:09] sense of maybe the exact construction [7:12] you can kind of you can make it look [7:13] very similar to what people had tried [7:15] before. But I think what what you can [7:18] often I mean often as a practicing [7:19] mathematician you you have an idea and [7:21] then you kind of think it might work [7:24] then you try for a few hours a few days [7:26] a few weeks and at some point you give [7:28] up and then a not so uncommon experience [7:31] is that you find out a year or two later [7:33] that somebody else got the idea to work [7:34] that you thought that didn't work. So [7:36] somehow getting an idea to work is it it [7:38] can even be a large portion of the [7:40] battle. And um I think what the [7:42] especially in the case of the unit [7:43] distance conjecture there's just a lot [7:45] of extraordinarily finicky details and [7:48] um very often when you're doing [7:50] mathematics it's it's you're kind of [7:52] gambling against the problem. You're [7:53] like maybe I should try this approach [7:55] but it seems really unlikely and just [7:56] not worth my time. And the model I think [7:59] in several of these cases both by [8:01] combining what it knew and sort of [8:02] having good taste kind of made the [8:05] correct bet. And you can kind of see [8:06] this in um the the summarized chain of [8:09] thought we released. can sort of look at [8:11] it and it's reasoning like a [8:12] mathematician and because it knows a few [8:14] very correct bits, it makes the right [8:16] decisions and is eventually able to [8:19] prune the search tree. It's it's not [8:20] really trying everything. It tries a lot [8:22] of different things. It's extremely [8:23] dogged, but it kind of [8:25] >> I mean it can't try every idea. It has [8:27] to try a limited set of ideas and it's [8:29] able to kind of use its knowledge plus [8:31] good good mathematical judgment and find [8:34] the right path to go along. So I mean [8:35] that for me was like because this was a [8:37] problem which a lot of people had [8:39] thought about and yeah [8:40] >> I mean the idea the fact that the idea [8:42] is not so foreign probably indicates [8:43] that a lot of people had tried it or at [8:45] least a few very serious mathematicians [8:46] have tried and I think that's what made [8:48] it really interesting to see. I I think [8:50] something else that's uh that like I [8:54] feel when I see these proofs is like [8:56] like if I have an idea and I'm trying to [8:59] execute like it might be that I have [9:01] some like wrong plan for like how to get [9:05] things to work. And um [9:09] like as a human if you have some like [9:11] wrong path you go down for a while it [9:13] can be hard to like rewire your brain to [9:15] like start over and like try a different [9:18] path. like you're you're kind of the [9:20] initial idea is kind of linked in your [9:21] brain with these other things that ended [9:23] up not working. You know, it's sort of [9:24] like your context window is like like a [9:27] little polluted and like you know you [9:29] can't just make another clone of [9:30] yourself from like last week and say you [9:32] know don't do this try something else [9:34] build your intuition another direction [9:36] >> but you know it's very easy to do this [9:37] with an AI. So, I think this is another [9:40] reason that it's like [9:42] >> um [9:44] it like like getting the details right [9:46] once you have some good general [9:48] direction is like much less of a barrier [9:50] all of a sudden. [9:51] >> And when you say it's much easier to do [9:53] with AI, it's like it's not actually [9:55] being directed with human interference [9:57] too. It's like as you were saying in the [9:58] reasoning traces, it's like making these [9:59] choices. Maybe it backtracks, but then [10:01] it's able to not be distracted by maybe [10:04] like the context in which it's thinking [10:06] about the problem like via these [10:07] machinery and it's like go go back to [10:09] like do you see it kind of go back as [10:11] well or is it just like making good [10:12] choices like is it a lucky sample or is [10:14] it like actually reasoning like a [10:16] mathematician where okay it doesn't it [10:17] you know doesn't do well in this path [10:19] but it goes back but then it doesn't let [10:21] that pollute. [10:22] >> No, I mean it definitely makes mistakes [10:23] and then it goes back and thinks about [10:25] it. I think it's somehow very [10:26] calculating very correct. I mean, yeah, [10:29] >> as human mathematicians, you're not [10:30] always perfect at making these decisions [10:32] and like [10:34] >> like the first time something doesn't [10:35] work, you you automatically kind of [10:37] downgrade how likely this approach is to [10:39] work and you keep doing this a few [10:40] times. The model somehow is much better [10:42] able to like it seems for several of the [10:45] solutions we've seen somehow seem it [10:47] seems much better able to update the sol [10:50] like how likely the path is to work like [10:52] yeah versus rejecting a path versus a [10:54] human doing it. Um [10:56] >> but but I think even if it weren't the [10:57] fact that you could just start another [10:59] model session over means that like you [11:00] know it's always going to be the case [11:03] that [11:04] >> got it here. So in some sense it is like [11:06] still leveraging the fact that you could [11:08] like run kind of parallel you know [11:09] agents on the problem. Um but if it were [11:12] kind of backtracking then does make it [11:13] seem much more like a human you know [11:15] mathematician and and perhaps it is kind [11:17] of doing some of that stuff too because [11:18] like obviously like we have to like make [11:20] mistakes in in order to like even gain [11:22] intuition for like why that solution [11:24] space is like not you know not in the in [11:27] the set of paths that it could be in. [11:29] Um, [11:30] >> I mean, I think this kind of thing [11:31] happens with humans, too, where like if [11:33] you get stuck on some approach, you [11:35] might tell another human your kind of [11:36] general idea and then they'll come back [11:38] and like figure out how to get it to [11:39] work and you know, it's just [11:40] >> yeah, [11:41] >> it takes more time to do this like with [11:43] humans. I wonder I mean you know maybe [11:46] this gets to extent that you can [11:48] actually talk about sort of like [11:49] obviously don't talk about the training [11:50] recipes or whatever but like it's it's [11:53] interesting that if you're just studying [11:54] for instance for math papers it's like a [11:56] very poor training set like a priority [11:58] for math because I mean maybe math [12:01] textbooks are even a pure example of [12:03] this. It's like really bad at actually [12:06] reconstructing the motivation for why [12:08] things were you know it's like don't I [12:10] mean maybe some people like it but don't [12:11] learn real analysis from Ruden. it just [12:14] like it's very clean already and crisp. [12:16] Um, and I think that that's bad because [12:18] it doesn't show the struggle that made [12:21] us formulate um, definitions in a [12:24] certain way. Like why do we even need to [12:26] have real numbers be defined in this [12:28] like super abstract way etc. And so you [12:32] know I think papers also I mean unless [12:34] you're most people don't write papers [12:36] with the the [12:38] context of I need to educate somebody to [12:41] to be a mathematician and so like the [12:42] actual maybe curriculum of like learning [12:44] math is not inherent in like a lot of [12:46] our artifacts as mathematicians. So [12:49] maybe another way to ask this question [12:51] is if the reasoning tracers are actually [12:53] producing things that it's like okay [12:54] this is actually more close to [12:55] mathematical thought like how does that [12:57] arise? [12:59] Um I mean I yeah I guess OpenAI has been [13:04] like the pioneer of reasoning models and [13:07] yeah you know teaching AI to reason in [13:08] this way. Um so you know we're we're [13:10] doing a lot of work at kind of kind of [13:14] all possible directions on on you know [13:16] teaching models to reason better and for [13:18] longer and and you know in all kinds of [13:20] different domains. Um yeah, I mean I [13:23] think that [13:25] I I think we're training general purpose [13:27] reasoning models and if yeah kind of one [13:30] a lot of these behaviors that we're [13:32] describing mathematically like [13:33] backtracking or kind of starting again. [13:35] I mean these are these are not really [13:36] specific to mathemat I mean we're seeing [13:38] them specifically in mathematics in [13:40] these examples but kind of they're [13:41] general purpose tools for reasoning and [13:43] I think if you work hard at reasoning [13:45] you should see these patterns [13:46] eventually. Mhm. [13:47] >> So it's this emergent because it's I [13:49] mean I do think that's why the open AI [13:52] approach um was so I mean it's like it [13:55] doesn't rely on you you know doing auto [13:59] formalization in order to like guide the [14:01] reasoning and I think that's like [14:02] obviously more like us um but it's just [14:04] like so not obvious that if you're just [14:06] like training on say a corpus of like [14:08] math proofs maybe autoformalized and [14:11] lean that you get um uh get the sort of [14:15] like projection of like how to think [14:17] well like put another way like with [14:19] maybe maybe if we think about it with [14:21] code like code is such a good corpus to [14:23] train on because it's one of the few um [14:26] data sets that has such large context [14:28] you just like I mean maybe you see this [14:30] kind of with books but they're less [14:32] structurally interconnected there's just [14:34] like less structure there I think um [14:37] it's safe to say like on average in a in [14:39] a book compared to like a piece of code [14:41] um and so like with math papers [14:45] I feel like But maybe what what we're [14:48] still bad at with coding models is stuff [14:50] that that data set doesn't contain which [14:52] is like kind of the semantics like the [14:54] syntax is there but there's a little bit [14:56] of like the higher level semantics of [14:59] what produced like why do I have to [15:00] write it this way is not I'm kind of [15:02] getting to too much in the philosophical [15:04] but it is just like really interesting [15:05] how it's still emergent that it's doing [15:07] good mathematics and we'll probably get [15:09] into this in more detail if you guys you [15:11] know wanted to talk in more detail about [15:13] some of the problems which is just like [15:14] it's it's not just doing like the [15:16] expected like we'll push the brute force [15:19] thing like you clearly are impressed [15:21] with some of the reasoning traces and [15:23] it's just not obvious that's gleaned [15:24] from you know what we would imagine [15:26] would be the easy training set here. [15:28] >> Yeah, absolutely. I mean, I think this [15:31] kind of thing is one reason we decided [15:34] it was important to release like these [15:36] summarized chains of thought for these [15:37] kinds of results because [15:39] >> if you if you've never seen these and [15:42] you just see all these proofs coming [15:43] out, you're kind of [15:44] >> you're not sure what it means like like [15:46] is the model just guessing in some [15:47] insane way like is it thinking in some [15:50] totally foreign like what what's going [15:52] on? But but actually it it's it's [15:54] reasoning kind of shockingly like an [15:56] expert human would. [15:58] >> Yeah. Yeah. [16:00] Yeah, it's it's very much like reading a [16:03] colleagueu's like notes. I mean, it's a [16:04] little more disorganized in some way. [16:06] They kind of like especially you work [16:07] close enough with the collaborators, [16:09] sometimes you'll just see them like [16:10] spill out their thoughts in an email to [16:12] you and it kind of it feels like reading [16:14] a lot of those chained together. So, [16:15] it's yeah, it's it's very it's quite [16:19] surprising the first few times. Um, were [16:21] you two sort of very involved in [16:24] choosing the problems to to release in [16:26] this like last 10 problem set that [16:28] Astros applied to? [16:30] >> Um, [16:31] >> which was your favorite? [16:34] >> Yeah, we were definitely involved. Um, [16:36] do you want to start on? [16:37] >> Yeah, I mean packing maybe. [16:38] >> Yeah, I guess. Yeah. So, I guess my [16:40] personal favorite among these problems [16:41] is the following. It's it's extremely [16:44] simple question which is just like it's [16:46] just about how efficiently can you put a [16:48] bunch my circles are not very good and [16:50] they're not all the same size but [16:51] >> but we're assuming they are. [16:53] >> Yeah. So the question is just like how [16:55] dense can you place a bunch of um so you [16:58] have a bunch of spheres uh you have a [16:59] bunch of spheres of radius one in [17:02] dimensions. [17:05] So the question is how densely can they [17:07] pack and so yeah so so in two dimensions [17:12] it's kind of like the so so d equals one [17:17] this is not an interesting question kind [17:18] of it's just the real line and yeah you [17:20] can cut it up and uh a sphere in [17:23] dimension one is just a unit segment so [17:24] okay you can cover everything um so in d [17:27] equals 2 it's kind of the picture that [17:30] you know that everybody loves it's just [17:32] like uh it's just a bunch of sphere [17:35] which sort of form like a hexagonal [17:37] lattice. [17:38] >> Hopefully I've drawn it well enough that [17:40] I can draw the hexagon. [17:41] >> Kind of betraying my naite on this [17:43] problem. Is that like obvious? Is like a [17:44] very elegant proof that it's lattice. [17:47] >> Yeah, it's not so obvious that this [17:49] should work. It was only proven in the [17:50] 60s, I think. Um there's a short [17:53] argument, but it's not it's not so easy. [17:56] >> Yeah. Um [17:57] >> where's the intuition? Like what is kind [17:58] of like the machinery of the argument? [18:01] >> I mean it kind of like [18:02] >> I mean it definitely looks like it [18:03] should work. That's why we're [18:04] >> Yeah. So I think [18:05] >> but so did [18:06] >> Yeah. I think this is the best part [18:07] about this problem which is really [18:08] nobody has any idea this problem. So [18:12] yeah so I mean honestly the best [18:14] intuition I have for this is that like [18:16] bees do this and if there was a more [18:18] efficient way then probably bees would [18:19] pack honeycomb some other way. [18:21] >> Evolution is efficient. [18:22] >> Yeah. [18:23] >> I think beyond that like I don't have a [18:25] great argument. I mean, and I think how [18:27] little we know is demonstrated by the [18:29] fact so, okay, D equals three. The the [18:32] answer is just like um it's what it's [18:35] how you pack like oranges in a grocery [18:37] store. [18:38] >> And this was only this was proved by [18:40] Hails sometime in the 2000s and like and [18:44] we don't have a short proof of this like [18:45] I think the shortest proof is like a few [18:46] hundred pages. Um [18:48] >> and what area does it like draw from in [18:50] order to [18:50] >> so it's a lot of linear programming [18:52] arguments and it's very delicate like [18:55] geometry. It's It's quite ugly actually. [18:58] >> It's like [18:59] >> this is like a famously ugly argument. [19:01] >> Oh no. [19:02] >> And then the two most famous um results [19:04] are d= 8 and 24 [19:07] um [19:08] >> 8 and 24. There must be some like weird [19:10] subspace. [19:12] >> Yeah. Yeah. Exactly. So this was done in [19:15] >> Yeah. So this was done in 2017. [19:17] >> Sounds slightly prettier though. I hope. [19:20] So it so the reason it works out in [19:22] these two very special dimensions is [19:24] that so this is called a lattice [19:25] packing. So it's like kind of very [19:27] regular and it turns out in these two [19:29] dimensions um there are two very special [19:31] lattises. They're called the E8 and [19:33] leech lattice and they're very nice and [19:37] they're like unusually dense like kind [19:38] of [19:39] >> um they're just very very pretty [19:41] structures coming from other areas of [19:42] math and it turns out that they're the [19:44] optimal structures [19:45] >> but they're still like regular. Yeah, [19:47] they're very regular. But I mean beyond [19:50] this, so we don't know any more exact [19:51] dimensions. We know these five [19:52] dimensions and we kind of don't know [19:54] anything else. Um, and I mean like to [19:58] give an indication of how little we [19:59] know. So there are two very surprising [20:01] things about this. So you can define [20:04] like delta d to be like the densest [20:06] sphere packed in dimensions. So um [20:09] there's kind of an easy lower bound of [20:11] like 2 to the minusd. Um [20:15] basic. Yeah, this is [20:17] >> this is not so hard to show. Basically, [20:19] um any packing where you can't put in [20:22] another sphere has to have this density. [20:24] So, okay, it's not ridiculously small. [20:26] And we know that it has to decay [20:28] exponentially. So, it has to decay um [20:31] like it grows like 1 minus C for some at [20:35] least for some constant. So, in large [20:37] dimensions, you can only cover cover [20:38] like a vanishingly small portion. Um but [20:41] we know like basically nothing else. and [20:43] so forth. [20:44] >> That's just because like the high [20:44] dimensional sphere like thing where it [20:47] occupies um it just like Yeah. Yeah. The [20:49] volume behavior is weird. [20:51] >> Yeah. So basically I mean basically they [20:53] don't want to touch next to each other. [20:55] I I don't know if I don't think there's [20:57] a particularly short way to see that [20:59] that it's like exponentially small but [21:01] it's known to be exponentially small. [21:02] And for a long long time the best bound [21:05] was something like this funny number [21:07] like 2us.599 [21:10] uh d. This was proved by two [21:12] mathematicians um in the 70s [21:16] capski. [21:18] >> Okay, that's a weird number. Where's [21:20] that spit out of? [21:22] >> It's not it's like cominatorial or [21:24] >> it's a it's the answer to some extremely [21:26] ugly optimization problem. [21:28] >> There's there's like a nice underlying [21:30] strategy. [21:30] >> Okay. [21:33] >> Yeah. Yeah, I'll say one last thing [21:34] about this. Yeah. So, yeah, I mean these [21:36] were these two Russian mathematicians in [21:38] the 70s. Yeah, it's actually very hard [21:40] to find their paper. It's like one page. [21:43] It's like two pages long. Yeah, they [21:45] don't write very many details because [21:46] paper was flared. But um yeah, [21:49] >> two negative D is just like the square [21:51] lattice like the dumb one or is that [21:54] >> so? Yeah, it's actually not so easy to [21:56] So the argument for this is as follows. [21:58] Basically imagine that you have a set of [22:00] spheres and I tell you can construct a [22:02] set of spheres so that like you can't [22:04] put down another sphere because [22:06] >> if you could put down an extra sphere, [22:08] you just keep putting it down. So you [22:10] have a set of spheres so that there's no [22:11] other sphere which you can put down. [22:14] >> Um [22:14] >> that's your it's like almost like a [22:16] >> just Yeah. Just take any such packing. [22:18] >> Mhm. Okay. Yeah. Yeah. [22:19] >> And I claim that this has to cover at [22:20] least 2 to the minus d fraction. The [22:23] reason is that like if you blew up each [22:25] of these spheres by a factor of two then [22:27] like they have to cover every point in [22:28] space. [22:31] >> And the reason is otherwise you could [22:32] put down if there was any empty space [22:34] you could put down a sphere there. [22:35] >> Yeah. So I guess if you take the usual [22:38] lettuce, there are actually like more [22:39] places you can put things kind of [22:41] diagonally. Yeah. [22:42] >> Okay. Yeah. Yeah. So that's actually not [22:43] it's like a worse bound. [22:44] >> Yeah. You can just keep plopping things [22:46] in. [22:46] >> Yeah. Yeah. This is [22:47] >> So this is like more Okay. Okay. [22:49] >> Yeah. This is related to this like [22:50] really funny fact where you like put a [22:52] sphere on every point on. If you take a [22:54] cube in high dimensions, you put a [22:55] sphere on every point. [22:56] >> It's like vanishingly small. [22:58] >> It's vanishingly small. It's so small [22:59] you can put another sphere in the middle [23:01] >> and it fits. Um [23:02] >> high dimensional sphere behavior is [23:04] weird. [23:04] >> Yeah. very weird. Um and so okay uh so [23:09] the great part is that um so the model [23:12] shows the following. So I'll write two [23:15] things. So this is Astra I guess [23:18] probably the right way to refer to this. [23:20] >> Um so okay I'm going to write something [23:23] called the LP bound. I'm not I'll [23:25] explain this in a second. And it shows [23:27] that it's smaller than uh this very nice [23:30] number. Uh do you want to say equals? Uh [23:33] yeah, it's equal essentially [23:35] uh [23:37] e to the 2 pi [23:40] plus little01 to the d. And if you you [23:43] can work out what this number is, it's [23:45] like [23:45] >> it's like roughly something like 2 to [23:47] the minus.6 [23:51] >> close. [23:52] >> Yeah, it's surprising. [23:53] >> But this one, you know, e to you're [23:55] like, oh, maybe there's some like nicer [23:56] kind of like structure there that that [23:58] that fell out. It's at most This is like [24:01] roughly something like 2 to the minus.61 [24:05] dot dot dot D. [24:07] >> That's the numeric. [24:08] >> I thought it was 604. [24:10] >> Great. [24:10] >> Um yeah, this shows mine. Yeah. Um so [24:14] okay, so there are a couple of things. [24:16] So first, what is this LP of D? So [24:18] VZOs's work actually builds on some [24:19] earlier work. Um it turns out that [24:21] there's a way to attack sphere packing [24:23] via what's called a linear programming [24:25] ground. So LP just stands for linear [24:27] programming. So Conan Elkis [24:30] um [24:32] uh gave an approach for super packing [24:34] based on linear programming. Uh so it's [24:36] like a it's like a linear optimization [24:38] problem over a convex set but it's all [24:40] kind of infinite dimensional here. [24:46] >> And basically what this reduces down to [24:48] is you try to understand the follow. So [24:50] what you try to show is you basically [24:53] you construct a function f. So this is [24:56] in dimensions and it's mapping to r um [25:00] and it has the following properties. So [25:03] first um f ofx so this is a function in [25:06] dimensions. So it's always less than [25:08] zero if like the size of x is bigger [25:12] than one and you second have that the [25:14] fouryear transform of x this is always [25:17] non- negative. So this is just this is a [25:20] linear program because the for transform [25:22] is a linear operator and uh night race. [25:27] >> So you're taking just some arbitrary f [25:29] that satisfies this property. [25:30] >> Yeah. So you can take any f that [25:32] satisfies these properties. And what [25:34] they prove is that delta d is bounded by [25:37] um the ratio of the foyer transform at [25:40] zero to its uh to the um the the for the [25:45] for transform at zero of f of 0 to the [25:48] over the for transform of zero times the [25:51] volume of the ball of radius 1/2 in data [25:54] matches. Um and so okay this this proof [25:58] is not so short for experience [26:00] mathematician it's like it took half a [26:02] paragraph to prove it but it's a little [26:03] bit tricky. [26:04] >> Um and the point is um so it turns out [26:07] so this is a relaxation problem. There's [26:09] no guarantee that taking the optimal f [26:11] will give you a good bound on delta d. [26:14] Um but so what Viazosa um did and this [26:18] was sort of the key I mean a large part [26:20] of the reason she won a fuels medal in [26:22] um in 2020 was that or in 2022 was that [26:28] um she constructed a function in 8 and [26:30] 24 dimensions such that this upper bound [26:32] matches exactly these two very special [26:34] lises [26:35] >> and these are kind of miracles of nature [26:37] that both you can construct this [26:39] function and that it gives you the [26:40] optimal bound. Um but you can just this [26:43] is a very very natural problem. I mean [26:45] it's a function with a very two very [26:47] simple properties and you just want to [26:49] understand how this behaves in for large [26:52] dimensions d and that was a big mystery. [26:55] Um we there was a numeric paper by con [26:59] and several others which conjectured [27:01] that um just based on doing numeric that [27:03] this was the answer but they had no idea [27:06] why this would be the answer. M [27:08] >> and what the model shows is that [27:10] actually um the linear programming bound [27:13] in large dimensions has this extremely [27:15] nice asintoic behavior and the proof [27:18] kind of explains where this is coming [27:20] from and because you understand this LP [27:22] bound perfectly this actually just gives [27:24] a better bound on delta D. It turns out [27:26] that this old bound can be kind of [27:28] reinterpreted in this framework and what [27:31] the model does it shows you the best [27:32] possible bound you can get by this [27:34] framework. [27:34] >> So the model sort of made the connection [27:37] and Yes. What is the sort of like what [27:39] do yeah what I think so the model gives [27:42] a function f which so first it [27:45] constructs a function f which gives you [27:47] this bound and then it shows that [27:49] there's no function f which does any [27:51] better. So it's an equality which is [27:53] quite strong. So in like sort of we now [27:55] understand this this problem in high [27:58] dimensions very well [27:59] >> and that's pretty remarkable. Um and the [28:03] model was just kind of told like analyze [28:05] this linear program in high dimensions, [28:06] you know, go have fun. [28:08] >> Got it. [28:09] >> And and to give an indication of like [28:11] how little I think this conjecture was [28:13] based basically only by doing numeric um [28:16] extremely clever numeric but numeric and [28:18] so [28:19] >> okay [28:20] >> yeah you have to kind of figure out why [28:21] this is the right thing to aim for and [28:23] it does and that was pretty remarkable. [28:26] Yeah, I mean I I had actually thought [28:27] about this problem for about six months [28:29] at some point when I was a graduate [28:30] student and yeah just I remember making [28:32] like absolutely zero progress on it. So [28:34] it was very nice to be like explained [28:36] why it was yeah why it was true. Um so [28:39] this was that was a pleasant experience. [28:41] >> I think also in general it was it was [28:43] one of these solutions which I I knew [28:45] several people had tried the problem. [28:46] It's pretty remarkable because like the [28:48] the model solution especially for this [28:50] being like that the LP can't do better [28:52] than this was like quite short. It's [28:54] it's a few pages of like complex [28:56] analysis, but it's kind of exactly the [28:58] right approach. Like once you see it, [28:59] it's kind of it's like unbelievable like [29:01] why why hadn't somebody done this [29:03] before? It was it was like there are [29:04] many types of good mathematics, but I [29:06] think one of them is just like you see [29:07] it and you're like, oh man, why didn't I [29:09] think of this? And it was really fun and [29:10] it I mean I sort of knew why I didn't [29:12] think of it, but it was quite nice to [29:14] see it and it was fun to see. That's why [29:16] I like this problem a lot. [29:19] >> Yeah. So so this is the first of the 10 [29:21] problems that Astra saw. Mhm. [29:23] >> Uh but the second is actually uh closely [29:25] related. [29:26] >> Uh so this was sphere packing. The [29:28] second one is uh spherical and binary [29:30] codes. [29:31] >> Mhm. [29:31] >> Um so uh what's you should draw a code. [29:35] You drew a packing. It's it's going to [29:37] be the same picture. Okay. Sure. Sure. [29:41] >> Otherwise, we're going to have his [29:42] picture packing in our mind. [29:45] >> Yeah. A spherical code is literally just [29:46] a sphere packing but on another sphere. [29:50] Um, [29:57] >> yeah. So, I mean, a spherical code is [29:59] basically just a sphere packing on the [30:01] surface of another sphere. So, um, [30:06] yeah. [30:07] >> Yeah, it looks like a sphere. [30:10] >> Yeah. Okay. So, same same picture as [30:13] before, except you're kind of on a [30:15] curved surface. [30:15] >> Okay. Okay. [30:16] >> Okay. So, uh, why is it called a code? [30:19] Well, um you can [30:23] I I guess the the reason is because of [30:25] binary codes, which is again the same [30:28] sort of thing, but now it's on a cube. [30:30] Okay. Yeah, fine. Let me let me draw a [30:33] picture of a a cube and some like [30:36] simplest possible code on it. [30:39] So like when you're when you're like um [30:43] sending uh so so this is really like [30:46] about error correcting codes. [30:50] So, so what are error correcting codes? [30:53] So, you know, it's like I send you some [30:55] string of bits, right? Uh like [31:01] and [31:03] maybe I'm I'm like worried that some of [31:04] the bits I send you get corrupted, [31:06] right? So, so maybe like just because of [31:09] some errors in my system like this one [31:11] gets gets changed and we want some [31:14] communication protocol so that like you [31:17] can decode this like small amount of [31:20] error and like recover what I was trying [31:22] to tell you. Uh and you know like normal [31:25] English language kind of has this sort [31:26] of property, right? If I make a few [31:28] typos, you know, you're going to be able [31:29] to understand what I'm saying. But if I [31:31] if we have some like really brittle [31:32] communication scheme, it's not going to [31:34] work. So, um, uh, codes are kind of the [31:38] the way you you solve this. And [31:41] mathematically, it just means like, you [31:42] know, what's a binary string like this [31:44] of a fixed length? It's like a point on [31:46] some hyper cube. [31:47] >> Mhm. [31:48] >> And we want a dictionary of allowable [31:50] code words that are like separated from [31:53] each other. So, like in this case, if I [31:55] don't want any two to be adjacent, I [31:56] would kind of take uh these four [31:58] vertices, kind of the like even ones if [32:01] you sum up the the digits, right? and [32:04] and like okay I I guess okay in in this [32:08] case I guess if I if I have an error you [32:10] can't tell which one it's from but at [32:11] least you can tell it's like not uh at [32:13] least you can tell there was an error. [32:16] >> Oh I see. I see. Yeah. Yeah. Because [32:17] it's like kind of sparse um in the um [32:20] Yeah. It's like it's not two adjacent. [32:22] So that like is this is like when the [32:24] hamming distance is not [32:25] >> Yeah. Yeah. Right. Right. You want Yeah. [32:26] So you want like a large Hamming [32:28] distance between any distinct in your in [32:30] your dictionary. [32:31] >> And Yeah. I guess if you take two [32:33] opposite corners then if I have like a [32:35] single bit error I can always like [32:37] recover which point it was coming from [32:39] >> the one that it's definitely closest to. [32:41] >> Yeah. So there's there's kind of the you [32:43] know same question in both of these [32:44] cases like in a very highdimensional [32:47] setting what kind of rate can you get? [32:49] >> And like for binary codes it's really [32:52] like you know an extremely practical [32:53] question. It's it's sort of like if I [32:55] send you um like an n bit string and [32:58] there's like you know 1% error rate like [33:01] how much longer does my message have to [33:03] become to tolerate that amount of errors [33:06] >> right this like some fundamental [33:07] fundamental information theoretic limit [33:09] of like like uh you know send like [33:11] communication [33:13] um and um you know but but you can see [33:16] like like certainly this uh this [33:18] spherical case uh is like it looks very [33:21] much like sphere packing for example if [33:22] you like If you make all these little [33:24] spheres really small, uh, then like the [33:27] curvature of the big sphere is kind of [33:28] not going to matter so much and it it [33:30] looks like just packing spheres in full [33:32] space. [33:33] >> Um, and in fact, yeah, like these these [33:36] problems turned out to be very related. [33:39] Uh, so so for for these uh problems, [33:42] there was like there were similar bounds [33:43] coming from these uh KL authors and and [33:46] uh like there's like something for the [33:49] sphere and something for the cube, but [33:50] it's it's all kind of the same stuff. [33:52] Mhm. [33:52] >> And um uh our models found uh better [33:56] bounds for for these cases as well. And [33:59] uh like it you you I mean the the [34:04] techniques look pretty different [34:05] actually if you if you like write them [34:06] out. So uh so this uh this full space [34:09] analysis of this linear programming was [34:11] using like just complex analysis. [34:14] >> Um but if you uh like the the method for [34:18] for these cases we're using [34:19] representation theory. Oh, [34:21] >> like like both the sphere and the cube [34:23] have a lot of symmetry [34:24] >> and basically the um the idea of the [34:28] proof was to really leverage this [34:30] symmetry. [34:31] >> Uh like there's some amount of this in [34:33] the the previous like existing method [34:35] and and really the improvement is to [34:38] like lean into the representation theory [34:40] like really hard and and kind of make [34:41] the the algebraic symmetry like enter in [34:44] a more sophisticated way. Um and then it [34:47] like turns out that from the [34:48] representation theory formulas if you [34:50] kind of take this like like small sphere [34:53] limit in the spherical code case you you [34:56] recover like [34:57] >> part of this result and you recover this [34:59] this value. Um so uh like this result [35:02] isn't a special case. You kind of only [35:04] went one direction of the bound from [35:06] from looking at it from the code's point [35:07] of view. But like there's like a very [35:08] close connection. [35:10] >> Okay. [35:10] >> Yeah. [35:10] >> Yeah. Yeah. So you you guys let this [35:12] like run in parallel. So it's like kind [35:13] of discovering or because you're not [35:15] sort of like feeding it. [35:17] >> So so actually uh this was the one case [35:20] where there was some interactivity [35:21] involved. Interesting. So for for all of [35:24] so except for this pair it was just you [35:26] know we had some problems we we fed them [35:28] in and we um you know the model the [35:31] model came back with some solutions. [35:32] >> Um what happened here was actually [35:34] pretty interesting. So so we first asked [35:36] it to uh improve the bound for the [35:40] codes. [35:40] >> Mhm. And uh it came back with an [35:43] improvement that like used some amount [35:45] of representation theory. [35:47] >> And then uh we kind of asked it, hey, [35:50] can you like push this further like you [35:52] know what happens? And then it came back [35:53] with some like much more sophisticated [35:55] representation theory and uh like it [35:59] turned out that uh you got this uh [36:01] conjectured value for full space sphere [36:04] packing uh like out of that method by [36:07] pushing it as far as it can go. So then [36:09] we we kind of ask to directly analyze [36:11] the sky and try to complete the picture. [36:13] >> Okay. [36:14] >> Yeah. [36:15] >> So so the relationship like isn't a [36:17] coincidence. [36:17] >> Yeah. Yeah. [36:19] >> It's like interesting when you're saying [36:21] the first prompt which is you know maybe [36:23] so basic which is like can you push this [36:25] further? It does require some judgment [36:27] from mathematicians and but like [36:29] eventually you would imagine by scaling [36:31] the models you don't need to do that or [36:33] there's another view that the harness [36:34] actually does matter and this is kind of [36:36] part of the harness apparatus. Do you [36:38] guys have any views on that with your [36:39] working with Astra, especially [36:41] generations of models and how how much [36:43] you have to kind of input or how much [36:45] the harness matters versus not. [36:48] >> I mean, yeah, I I guess there have been [36:50] some like funny quirks like this that [36:53] just come from like exactly what you [36:54] asked the model to do basically like [36:57] like in in this case, what the model was [36:58] asked to do originally for codes was to [37:00] improve the bounds [37:02] >> by like some exponential factor. So it [37:04] it really like shows up in this like [37:06] leading constant up here. [37:07] >> Mhm. [37:08] >> Um and you know it it improved the [37:10] bounds and it it didn't try to push [37:13] things too much further. Like sometimes [37:14] you see it do but sometimes it just [37:16] doesn't bother but yeah you know you [37:17] just ask it again and it it goes [37:19] further. So it it wasn't like a [37:21] capabilities issue. It just [37:22] >> kind of didn't feel like it at the time. [37:24] >> Do you call that judgment or like what [37:26] is the because like there there is a [37:28] Yeah. What do you call that? [37:29] >> I think models tend to be pretty task [37:31] oriented. If you if you tell it to do a [37:34] task and it accomplishes the task, [37:36] >> it's pretty happy. [37:37] >> So yeah, like the task orientedness it's [37:39] like but do we expect that level to kind [37:42] of ascend up to [37:45] it's not that they will be less good at [37:47] being task oriented. It's like they'll [37:48] ascend to the level of like okay no [37:50] let's let's go in this direction. You'll [37:52] have the judgment because you guys had [37:53] the judgment. You're like okay this is [37:55] pretty promising. Looks like you're [37:56] using a lot of representation theory. It [37:58] doesn't seem like there's a limit so [38:01] far. Um, but it doesn't have that [38:03] context yet. But like I guess what I'm [38:05] trying to say is like this one it's hard [38:06] to maybe extra harder to extrapolate. [38:08] But from like previous generations when [38:09] you had to give it more maybe prompting [38:12] more of that harness work but eventually [38:14] you probably had to give it less. So it [38:15] probably gives you some confidence that [38:16] there's this like really fast ascension [38:19] and and do you see yeah like someow [38:22] somehow solving a harder math problem is [38:25] like you have to solve many smaller like [38:27] somewhat less hard math problems and the [38:29] fact that the math problems are getting [38:30] harder is kind of an indication that [38:31] you're the model is able to take on more [38:33] and more work in like a single [38:35] continuous unit. Um and I think that's [38:37] that's the thing that looks very [38:38] promising somehow like any of these [38:41] solutions it's not like one idea and [38:42] then you're kind of home free. You need [38:44] several you need several pieces to kind [38:46] of interact and talk to each other. And [38:49] >> the models I the model doesn't come up [38:52] with all the ideas at once, right? It [38:54] doesn't pull everything out of in in an [38:56] instance of kind of the fact that it [38:58] needs to sort of see how this piece [39:00] interacts with another piece. That's [39:01] kind of like solving a problem in itself [39:03] or piecing together many problems in [39:05] itself. It could just be that okay when [39:07] you're telling it okay push this even [39:09] further that was of the same order of [39:12] like magnitude as like all the smaller [39:14] things it's solving as well in in [39:16] between and so you don't think that's [39:17] kind of like a privilege direction it's [39:18] just sort of like hey let's give it like [39:20] one one more help or you actually think [39:21] that there's I guess what I'm trying to [39:23] get at a bigger question is like is [39:24] there a good sense of like [39:28] you know taste because like when people [39:29] talk about for instance how well the [39:32] models are getting at like um doing [39:35] research church for instance that's what [39:36] we you know we want a little bit of RSI [39:39] um and uh and sort of like there's [39:43] surprising things about um how that [39:45] improves and then there's like the oh [39:47] you know maybe right now it's at a level [39:48] of still like a junior researcher it's [39:50] like not really asking like the right [39:51] problems and so I'm just trying to get [39:53] like a maybe a sense of like where [39:55] you're seeing that progress through the [39:57] model advancements each generation [40:01] I mean what is taste even [40:04] >> yeah I I think I tend to be pretty [40:06] utilitarian in my view of taste and like [40:08] if you're able to solve problems faster [40:10] by making better judgments like I think [40:13] that's like the best like general proxy [40:15] I have for taste and somehow the fact [40:17] that solving harder problems means it [40:19] has kind of by definition means it has [40:20] better taste. I think there there are [40:23] these no yeah I think occasion [40:25] occasionally because they are task [40:26] oriented you do occasionally get these [40:28] these symptoms of like oh it clearly has [40:31] made a breakthrough it kind of [40:32] understands it's made a breakthrough and [40:34] then it doesn't kind of push all the way [40:35] to the limit because that's not what you [40:37] asked but that seems yeah that seems [40:41] seems rather minor compared to like the [40:43] set the the state of progress we've seen [40:45] so far [40:46] >> okay yeah I think it's a pretty clear [40:48] >> like I think it's it's like maybe you're [40:50] liable to get confused if you're trying [40:52] to like do a concrete long horizon task [40:56] and show taste kind of at the same time. [40:59] But like you know if if you if you have [41:03] like one model that's responsible for [41:05] taste and one model that's responsible [41:07] for going out and like you know working [41:09] for a long time at solving a hard [41:11] problem kind of as the as the like uh [41:14] you know [41:16] underling of the of the the supervising [41:19] AI. I I feel like that kind of going to [41:22] be fine currently. [41:24] >> Oh, interesting. Because that is like [41:25] saying that they these two things are [41:27] somewhat sep if not separate at least [41:30] they shouldn't kind of pollute each [41:31] other's contexts which is [41:34] a little bit I mean it could be [41:35] potentially like a strong stronger [41:37] statement um than I guess. Yeah. Know [41:40] it's just kind of interesting because [41:41] it's it it it might just be like to your [41:43] point it's you know let's take the [41:45] utilitarian answer is solving harder and [41:47] harder problems. is it's doing a lot [41:48] more than just like you know brute [41:50] forcing something. It's making choices. [41:52] It's like pruning you know a vastly [41:55] large space of possible paths into [41:58] something that's like really you know [42:00] it's both tractable but then ends up um [42:02] being like it's a diminishingly small [42:04] path within that space. Um but like [42:07] having like why would would be like a [42:09] separate model a se separate generation [42:12] or something that's a different version [42:14] of the model that would contribute to [42:16] taste or maybe that's totally like it's [42:18] too abstract doesn't make any sense you [42:20] know we should just let the actual this [42:23] might like the related question be like [42:25] you know what is what is the the thing [42:27] that gets us to a better version of [42:30] intelligence the harness and the model [42:31] is it just the model and it's like we we [42:33] see this in you know at least in um [42:36] applied AI or you know startups where [42:39] it's like it's a continual battle of [42:41] like you need the harness but then the [42:42] harness adapts very poorly to a new [42:45] model because sometimes like a very very [42:46] minimal harness is still the best way to [42:48] expose to the raw power of the model but [42:50] then now we also have these like [42:51] training regimes where we require the [42:53] harness to be you know trained with I [42:54] mean maybe part part of this is to keep [42:56] things more proprietary and harder to [42:58] harder for other people to use it but I [43:00] think partially it's it's maybe actually [43:02] that it helps um have more control on [43:04] like the reasoning traces you care [43:06] about. It's a long rambling way of [43:07] saying it's like, yeah, I I don't [43:09] actually like this is so interesting to [43:11] see how the models have gotten better at [43:13] math and um maybe something that's like [43:15] very abstract and hard to describe like [43:17] taste is a way to tease out like what is [43:19] actually necessary here. [43:22] I think my only like non-trivial thought [43:24] here is that like when you're working I [43:26] mean just when you're doing any task [43:28] occasionally you get pigeon holed and [43:29] you like work really hard and just [43:31] having a friend look over your shoulder [43:32] and be like [43:34] >> what are you doing and then just like [43:35] just having that one bit of like step [43:37] back for 10 seconds like this is often [43:39] very useful. [43:40] >> Yeah. Yeah. I mean, I see no reason why [43:42] humans would be so different than models [43:44] somehow. [43:45] >> Having or models would be so different [43:47] than humans. Having [43:48] >> having a few humans working together is [43:50] often more powerful than just having [43:51] one. [43:52] >> Yeah. It's like in this kind of [43:54] collaborative thing, you actually you [43:55] kind of Yeah. artificially created it, [43:57] but it's very similar and dynamic. [43:59] >> But I think a lot of taste is also like [44:02] having a sense of what problems you or [44:04] like some method you have in mind are [44:06] going to be good at solving. Mhm. [44:08] >> Like it's I mean certainly there's some [44:10] amount of like absolute aesthetic point, [44:13] right? But there there's also just like [44:16] you know having a nose for what what you [44:19] might want to pursue because you'll be [44:21] able to make progress. Um and you know I [44:25] think for that like there's you know you [44:28] would expect that as a side product of [44:30] being good at completing tasks you would [44:32] you would get there sort of right. Let [44:34] me know if we still want to do like a [44:35] section on sophetic groups because I [44:37] think you know up to you guys it's [44:38] definitely super interesting. [44:40] >> So maybe the the first question is what [44:42] is a group? Let's remind ourselves. So [44:46] uh a group [44:49] is [44:51] uh a set of elements with some [44:55] multiplication operation. [45:00] And uh basically this is how [45:03] mathematicians think about symmetry. [45:11] So, so you're like [45:14] basically like if G and H are elements [45:17] of your group, then GH [45:22] has some is some other well- definfined [45:24] uh element of your group and you have [45:28] like uh associativity [45:36] uh and you have an inverse [45:38] So for every G there's some inverse [45:42] uh and there's some like specific [45:45] element in the group that uh is is kind [45:48] of the identity. Uh okay so you know [45:51] it's it's some like abstraction of like [45:54] uh composing operations. So these could [45:56] be like numbers they could be like [45:57] multiplying matrices they could be like [45:59] like rotating something which is you [46:01] know a special case of multiplying [46:02] matrices. Um and uh a group is uh sofic. [46:12] What if [46:14] well there's some uh you know precise [46:17] definition uh but you know uh roughly uh [46:22] it means it [46:25] uh so so I should say like groups that [46:26] can be finite or infinite. So like you [46:30] know um [46:32] if if you have like a square like all [46:34] the rotations of it form a group with [46:36] like four elements. [46:37] >> If you have like a circle then the [46:38] rotations form a group with like [46:40] uncountably any many elements. [46:42] >> Um and uh so sophet groups are either [46:45] finite or countable. You should think of [46:47] them as being countably infinite. So [46:49] there's like the same number of elements [46:51] as like the integers. and uh if it's [46:55] sofic if in some sense uh it can be uh [46:58] approximated by finite groups. So we [47:00] didn't know if there was a non-sulfic [47:02] group. So so the the the result that uh [47:05] Astra proved [47:06] >> is simply that uh there exists a non-sic [47:12] group. [47:14] >> Yeah. And without like I mean we can you [47:16] know before going to to that proof it is [47:19] like you know it's like I feel like a [47:21] lot of the programs of math is like okay [47:22] we are such finite creatures let's see [47:24] how well our finite approximations are [47:26] you know uh do and in this case [47:28] especially for the countable case it's [47:30] like maybe you'll be relating it to like [47:33] the eldest Leon's thing. It's just like [47:34] it helps kind of anchor the picture of [47:36] like it seems like such a I mean it's a [47:40] nice result if it were true, but it it's [47:42] not and it seems almost like reasonable. [47:45] And so yeah, I I actually didn't um [47:48] didn't uh go I would love to hear the [47:50] explanation of like how it it found a [47:52] counter example. Yeah, I mean I I would [47:54] say that like you know the the hope that [47:57] there was no non-s so fit group so every [48:00] group has this kind of approximation [48:02] like maybe this is sort of like people [48:05] hoping that there's a miracle [48:07] >> because uh it turns out that groups like [48:09] this have a lot of nice properties um [48:12] because uh you can run certain proofs [48:15] for finite groups and then uh you know [48:17] kind of approximate them in whatever way [48:20] the definition of being sofic lets you [48:22] approximate them and get the result. So, [48:24] so like um there's this notion of being [48:26] a subjunctive group. So there's uh [48:30] there's some fact that uh any uh group [48:34] which is sofic is uh also surjunctive. [48:40] Uh surjunctive is some property of like [48:44] um dynamical systems on the group [48:46] >> and um I guess the original question was [48:49] whether every group is uh surjunctive. [48:51] This is some question of gotch from the [48:52] 70s. Uh and this uh this fact that um [48:57] follows this like pattern of prove it [48:59] for finite groups and then do this [49:00] approximation is is what motivated the [49:02] question about if there's a non-sulfic [49:04] group. [49:05] >> Mhm. Um [49:09] yeah may maybe I'll say a little bit [49:11] about this I'll just lions conjecture [49:13] and [49:13] >> yeah sure yeah yeah [49:14] >> so I I guess I had like heard of this a [49:16] little bit beforehand because uh there's [49:18] a related uh stronger conjecture in um [49:22] probability that was made popular by [49:24] Aldos and Lions this this conjecture [49:27] roughly what it says is like um any um [49:33] like infinite graph with some nice uh [49:37] property called uh unodularity. [49:46] Um [49:47] a [49:49] random in modular random graph uh can be [49:56] approximated [49:57] by uh large finite graphs. [50:03] So um maybe the the way to like explain [50:07] what these kinds of things are trying to [50:08] say without without getting into [50:10] technical weeds is to say what they mean [50:12] about the integers. [50:13] >> So so like how would I draw the integers [50:16] as a graph um [50:20] uh so this is called like hilly graph uh [50:22] you're just going to connect nearest [50:24] neighbors. Okay so there's some uh kind [50:27] of canonical way in which this is like [50:28] the graph that represents the integers. [50:31] Okay. And uh [50:34] there's some sense in which you can [50:35] approximate this by finite graphs. Uh [50:39] why? Well, um if you look at integers uh [50:43] mod n then you kind of get the same [50:47] picture [50:48] but uh [50:51] like you have like a big circle instead [50:53] of an infinite line. Um and the the [50:57] point is if you like look at any point [50:59] here and any point here like in nearby [51:03] things look the same. You have to go [51:04] like very far away to kind of see this [51:06] global geometric structure that you have [51:07] a circle and not a line. [51:09] >> Mhm. [51:09] >> Um and in fact uh the integers and [51:12] integers mod n are both groups just by [51:14] like adding numbers or adding numbers [51:16] mod n. [51:17] >> So uh these integers mod n are like uh [51:21] sofic approximations for the it full [51:23] integers. So like this approximation is [51:26] kind of why the integers are a sofic [51:28] group. [51:29] >> So um [51:32] so the the sufficity uh the the [51:35] statement that every group is sofic is [51:37] sort of a generalization of the fact [51:38] that you can do this approximation with [51:41] groups and this uh also lions conjecture [51:44] is kind of a broader conjecture that [51:46] like any network you can do this um and [51:50] it like you don't require as much [51:51] algebraic structure roughly. So it's [51:53] it's kind of a broader conjecture. [51:55] >> So so this conjecture was disproved uh [51:58] earlier um like two years ago and it was [52:01] kind of a really to force uh work like [52:05] it was like 250 pages building on [52:07] another 200 pages. It uses like like [52:09] quantum complexity theory. So it really [52:11] builds this like [52:12] >> you know very complicated bridge and [52:15] like you know I I think not very many [52:17] people could understand this. Uh right. [52:20] So since this is a stronger conjecture, [52:22] the disproof uh like is weaker than [52:24] disproving uh this uh statement that all [52:27] groups are sofic. [52:28] >> Um but it turns out that the direct [52:30] proof that all that there's a non-sopic [52:32] group was like much shorter and and [52:34] easier than this than this really uh [52:36] amazing disproof of the all this lions [52:38] conjecture. It's like like 15 pages [52:40] maybe and it doesn't have any of this [52:43] like very complicated connection with [52:45] quantum complexity. just kind of stays [52:47] in group theory land. Um I mean it uses [52:50] some important like [52:52] you know existing results by other [52:54] mathematicians like Kun and and Tom [52:56] but it's like it's it's like a it's like [52:59] a very reasonable normal kind of proof. [53:01] >> Yeah. And and kind of spelling maybe out [53:03] the obvious but like the the connection [53:06] between the sof group um statement is [53:08] just you take the kay graph and that's [53:10] the one that is like what they use or uh [53:12] for um eldest Leon. [53:14] >> Yeah. [53:14] >> Yeah. And so that's why it's like a you [53:16] know a subset of um [53:18] >> right so yeah basically what happens is [53:20] yeah so for a group you can take exactly [53:22] a kaly graph so you take some like uh [53:24] elements that like generate the group [53:26] and you kind of connect elements that [53:28] are that are [53:29] >> adjacent so so in this case like uh this [53:31] is a kaly graph of the integers [53:33] >> so so right so when you do that um from [53:35] a group you get like a deterministic [53:37] graph [53:38] >> right you just get like a single graph [53:40] the graph you fix some set of generators [53:42] so this conjecture is stronger basically [53:44] because um it allows a broader set of [53:47] graphs that aren't deterministic. It [53:49] allows them to be random but have some [53:51] extra uh you know um you know modularity [53:55] property that constrains exactly how it [53:57] can be random. [53:58] >> But yeah, basically that's that's the [54:00] difference. Um like uh here you kind of [54:03] have to give a deterministic network [54:04] instead of a random one. [54:07] >> Yeah. Anything kind of interesting [54:10] surprising about the result? I mean you [54:11] mentioned some things which is like it's [54:13] it's stayed within group theory um the [54:16] techniques. Um [54:19] >> I mean I I I think maybe it's like a a [54:21] nice example of this general pattern [54:24] that um theorems produced by AI have [54:27] generally been like like the proofs are [54:30] pretty short generally. They're like [54:33] >> um [54:34] >> like with the counter examples so far. [54:36] >> Yeah. Yeah, but I mean this one it's [54:38] like okay it's sort of a counter example [54:40] but like there's some you know there's [54:41] some like [54:42] >> stuff you have to to do to to analyze [54:45] things. The difficult part here is that [54:47] like [54:48] >> the property of being a Sophie group is [54:50] not so easy to get your hands on. So you [54:52] have to find like a concrete way of [54:54] saying like like producing a producing a [54:57] way of saying this group cannot be so [54:58] and like yeah [54:59] >> and the proof is actually it's very [55:01] short. It's like it's almost it's a [55:02] combinatorics argument, but it's like a [55:04] very very delicate combinatorics [55:05] argument and the model like somehow you [55:07] need to both have the right statement [55:09] and know what pieces in the literature [55:10] and then execute it correctly and that's [55:12] very nice. [55:13] >> Um like the difficulty of this problem [55:15] is that like it's just a really really [55:17] hard it's like very hard to get your [55:18] hands on like being approximated by any [55:20] possible finite. [55:22] >> I was going to say it's like what is [55:23] happening at that countable infinity [55:25] that's like resisting this [55:26] approximation? Like do you guys kind of [55:28] did it give a sense of like when do you [55:29] do postmortem when you're like okay [55:31] Astra explain to me like what is the [55:33] >> oh yeah we did that all right what was a [55:36] good explanation you got out of it [55:38] >> I think there's like some concrete like [55:40] cominatorial obstruction um basically [55:42] it's like [55:43] >> it's hard to explain but there's like [55:45] some concrete cominatorics obstruction [55:46] which if you read the previous papers [55:48] you realize that that's what they [55:50] couldn't rule out and Astra found a way [55:52] to kind of [55:52] >> say okay no no if you add this one extra [55:54] algebraic fact this this like weird [55:57] conspiracy can't happen. Um it's like [55:59] very clearly trying to rule out [56:00] conspiracy in um the sort of previous [56:03] authors had implicitly written about. [56:05] >> Yeah. [56:06] >> And those were the actual suspects [56:09] it turned out. So they were sort of on [56:11] the right track and then this did the [56:13] last mile of well whatever however you [56:16] quantify that. But but I I think it's [56:19] like like a year ago I I would have been [56:21] very surprised to learn that like all of [56:23] these AI proofs are like very short and [56:26] elegant. [56:27] >> Yeah. [56:27] >> Like they're you know you you're kind of [56:30] like afraid that they're going to like [56:32] generate all these thousand page things [56:33] and you're just like never going to be [56:35] able to understand it. But it's been [56:36] kind of the opposite. Yeah. Like only [56:38] humans can generate like [56:39] >> 200page proofs right now. [56:41] >> Yeah. Well, and also I was like asking [56:43] I'm like if you do that postmortem it [56:46] ends up usually engendering more [56:47] mathematics because when you do that [56:48] with humans like that's what you know [56:51] breeds new mathematics. So maybe maybe [56:53] if you kind of alter the prompt a little [56:55] bit and be like how would you you know [56:58] generalize this or something like yeah I [57:00] don't know if that's been a technique [57:01] for you guys to like have it explore and [57:03] exploit what it has already developed. [57:06] Well, well, there has been some uh there [57:08] has been followup on this already [57:10] actually by by and Tom who this was [57:12] built on. So they they like the [57:13] >> math community is coming on. [57:14] >> Yeah. Yeah. Yeah. Which is kind of what [57:16] what we're hoping, right? You know, we [57:17] don't want to be you know writing lots [57:20] of followup papers ourselves, but if if [57:22] there's some interesting followup that [57:24] you know it's like we're very very happy [57:26] that that there's some there's some [57:27] followup building out these ideas more [57:29] and giving like more examples of [57:31] non-sitic groups in this case. [57:32] >> Yeah. Well, actually maybe that's a [57:33] great segue into like how you know [57:35] what's the ideal way that this is being [57:38] taken up by the math community because I [57:39] feel like there's a spectrum of answers [57:41] from working mathematicians sense of [57:43] like [57:44] some you know probably most at this [57:46] point are like okay AI is obviously [57:48] doing some non-trivial stuff. Um it [57:50] would be a disadvantage not to admit [57:52] that in my workflow. Um, I've definitely [57:55] heard some stories where people of, you [57:58] know, [58:00] would find it hard to either take AI as [58:02] a co-author or like how do you even do [58:04] kind of attribution this way? But I [58:06] don't know like what um maybe to paint [58:08] the the more optimistic picture. So [58:10] you're saying you want the [58:11] mathematicians to be building on this [58:12] these results. It definitely generates a [58:16] lot more results to be verified. So it [58:19] you know it puts pressure in on the [58:21] community and the profession like how do [58:22] you how do you kind of expect the [58:24] evolution of kind of uptake and and [58:26] collaboration with mathematicians. [58:28] >> I mean given that the fact that the [58:30] models can produce sophisticated [58:31] mathematics means that they can help you [58:33] understand like sophisticated [58:34] mathematics. I mean like I don't know [58:36] occasionally I like I enjoy looking at [58:38] the archive and I want to understand [58:39] some proof and like I could read the [58:41] introduction but in practice it's just [58:43] much faster take the PDF put it into put [58:47] into my favorite model and then like get [58:49] a get an output of like what is the [58:51] rough proof strategy and somehow this [58:52] like [58:54] >> along I mean of course models are going [58:56] to help us produce exponentially more [58:58] mathematics but they also make it much [59:00] easier to absorb it. Um, yeah. And right [59:03] now, okay, it's still a bit of a [59:05] challenge back and forth, but I think [59:06] it's for me at least a much much faster [59:09] at understanding it's much much faster [59:11] to understand a piece of mathematics [59:12] with a model than without it. Um, so [59:16] it's helping solve the problem it [59:17] creates anyways. [59:18] >> Yeah, I feel like that at least and [59:20] it's, you know, I I don't I don't view [59:22] it as like creating much more problem, [59:23] but again, like I don't have such you [59:25] know, high stakes in like, okay, I'm I'm [59:27] going to get I'm not going to get [59:28] tenure, etc. So like I I I agree like [59:30] making it more accessible. Like if I'm [59:32] not spending so much time absorbing an [59:34] area, I can like put it into chat GPT [59:36] and then expect to I mean you guys have [59:38] an even more powerful model hopefully [59:40] releasing um for other people to enjoy [59:42] as well. But like it's um I think like [59:46] the the positive version of that is [59:47] actually more people can participate [59:49] mathematics. It's like people might be [59:50] coming with other intuitions and they [59:52] could actually maybe generate good [59:53] mathematics. Is that sort of like closer [59:56] to the vision of what you're hoping this [59:58] is, [59:59] you know, pushing towards? Um or like [60:02] what what things do you think to be wary [60:05] of to to kind of adapt fast enough to [60:08] take advantage of AI? [60:09] >> Yeah, I mean I I think certainly there [60:12] will be a lot of changes, right? Like I [60:14] I guess in math like there are there are [60:18] a lot of things that are kind of [60:20] important for for like a given result, [60:22] right? You need someone to come up with [60:24] it, but you also need people to [60:26] understand and absorb it and like, you [60:29] know, internalize it enough to to do [60:31] more with it and and like figure out [60:33] where it fits into like humanity's [60:35] understanding, right? And like uh a a [60:39] couple of years ago like the proving the [60:42] result was like so hard that kind of the [60:45] the other stuff was just kind of coming [60:47] along for the ride, right? you know, [60:49] like if you if you manage to like prove [60:51] this thing yourself, you're [60:52] automatically going to understand it [60:54] quite well. You're kind of responsible [60:56] for like maintaining it and in some [60:58] sense and like you know, explaining it [61:00] to other people. Um, and yeah, now this [61:03] kind of what was the main bottleneck [61:05] before is kind of um much less of a [61:09] bottleneck and you know, these other [61:11] kind of constraints uh come into play. [61:15] Um so it's [61:17] yeah the the like optimal um [61:22] structuring for you know organizing the [61:23] knowledge could could look rather [61:25] different. [61:26] >> Yeah. How does that look? I mean does [61:29] this make the field a lot more kind of [61:30] empirical? Will people do sort of the [61:33] hard like the first thing that was [61:35] scarce which is like all the reasoning [61:36] and then more I mean not that it's like [61:38] a bad thing to make it empirical but [61:40] it's almost like it functions as a very [61:41] different discipline. Um like a lot of [61:44] the fun stuff is understanding you know [61:46] um and so understanding communicating [61:48] maybe assembling having still the human [61:50] taste does that sort of remain [61:53] rarified and and that's how you know [61:56] current mathematicians need to adapt and [61:58] and reward you know uh contributions or [62:02] or is this too much of a caricature it's [62:03] like something else [62:05] >> I think certainly understanding how to [62:06] put as we get more and more mathematics [62:09] put it in like a proper framework and [62:10] sort of how sort of like being able to [62:14] explain it to other humans so that they [62:15] can also appreciate it. I mean somehow [62:17] implicitly we valued this but it was [62:19] usually because you were the person [62:20] proving the result that gave everybody [62:22] else the understanding but I think [62:23] increasingly it would be a function of [62:25] like you're sort of helping you're the [62:28] human who can sort of give this [62:30] understanding to other people and sort [62:31] of help them with it. Um I think that [62:33] more of that communal understanding will [62:36] I think become it was much more implicit [62:38] in how we view math in general but I [62:39] think it will be an increasingly more [62:41] explicit and valuable part of the [62:42] subject. [62:43] >> I mean a nice thing about math is that [62:45] um the the ceiling for difficulty of a [62:48] math problem is pretty high. So even if [62:52] um you know kind of even if AI get you [62:55] know continues getting like [62:56] exponentially better at math like it [62:58] might you know it's plausible we'll [63:01] never solve something like P versus NP [63:03] >> and it could be that like the the field [63:07] kind of [63:09] becomes more um [63:12] >> you know attached to like like these big [63:15] mysteries and and less to like smaller [63:18] mysteries that are more like routine. I [63:19] mean know. [63:21] >> Yeah. Yeah. I think that's a positive [63:23] vision of the [63:24] >> I mean also like I don't know there are [63:26] things I spent like months or years of [63:28] my life wondering about not getting to [63:29] know and hope [63:31] >> we get what a joy. [63:32] >> Yeah. Some portion of them I'll get to [63:34] know the answer to. I'm pretty happy [63:35] about that. [63:35] >> No, exactly. No, I'm I'm excited about [63:37] this like renaissance of results and [63:39] understanding and I feel like I mean [63:40] this is such an an infinite, you know, [63:43] field like no pun intended, but like [63:47] it's just like it's it's it's just [63:48] there's so much that you can actually [63:50] create here. Um so I mean especially for [63:54] somebody like me who's not going to have [63:55] the time to actually like practice [63:56] mathematics now there's like a lot more [63:58] that you can actually do in the activity [64:00] of math. So yeah. [64:02] >> Yeah. I think the the like the ability [64:04] of someone who's not working on math is [64:06] like their literal job all the time to [64:08] like understand what's going on and like [64:10] you know learn about some of the [64:12] mysteries they might have wondered about [64:13] will will go up quite a lot. Um also you [64:17] know if you're if you're like if you're [64:19] working on something that requires some [64:21] math [64:22] >> you know suddenly you you don't need to [64:24] like find a world expert on this topic [64:26] to to be able to you know use it in your [64:28] own work. you could [64:29] >> sorry mathematicians. [64:31] Yeah. Yeah. No, it's true. I mean, I [64:32] think there was just like a der of [64:34] actual like people who could could do [64:36] that and so I think this is helpful. [64:37] Maybe it's helpful for the theoretical [64:39] physics. Like we'll see. Um but a lot of [64:41] other applied areas as well. [64:43] >> It'd be nice for the world if applied [64:44] mathematics went a lot faster. [64:46] >> Yes. I mean I'm of that opinion. Well, [64:48] thank you guys for joining. This is a [64:50] lot of fun and I'm, you know, just so [64:52] excited for how much the models are [64:55] advancing. So maybe we'll have you guys [64:57] back soon. Yeah, thanks so much for [64:59] having us. [65:00] >> Yeah, thanks for having us.