数学前沿

本文讨论了为AI输出指定数学家的不利之处，包括延迟或不平等访问、剥夺了其他数学家的机会、选择偏见、借用声誉以及这种做法的不必要性。文章建议AI实验室应遵循特定的发布路径，并支持人类对AI生成产品的理解。

陶哲轩博客介绍了oai-math-revised项目，这是一个由数学社区共同参与改进OpenAI数学写作的平台。该项目旨在解决OpenAI数学文档中难以理解和引用不完整的问题，邀请学生、博士后、教授和爱好者共同参与修订工作，并可通过指定邮箱申请成为管理员或审稿人。

国际高等研究院（IAS）已开放下一年度会员项目的申请页面。今年，数学学院特别提出，为了减轻申请者的压力，申请者可以选择仅基于已完成的研究提交研究陈述。建议申请者聚焦于几个精选项目（通常最多三个），而非广泛的研究列表。陶哲轩教授还期望其他部门和机构在发布职位广告时也能采取类似的指导原则。

本文探讨了黎曼ζ函数非平凡零点在临界线上的比例问题，特别是简单临界零点的下界。作者回顾了从Selberg(1942)到Pratt等人(2020)的历史进展，介绍了Montgomery(1973)在假设黎曼猜想下证明至少67%的零点是简单的，以及近期BGST-B团队和Lamzouri(2026)在不假设黎曼猜想的情况下证明至少67.25%的零点是简单且临界的结果。这些突破通过配对相关方法实现了对零点性质的更深入理解。

这是 Tristan Buckmaster 在陶哲轩博客发表的客座文章。作者借“数学是年轻人的游戏”这一旧说法，指出它将在另一种意义上成真：数学正经历史上最快的变化，研究生和博士后将塑造其未来。文章以 Deep Blue 击败 Kasparov 为类比，认为当前 AI 数学证明的算力门槛虽高，但成本下降后可能普及；证明的“认证”不等于理解，数学的持久价值在于人类理解。 作者还提出几个具体主张：AI 公司挖掘用户数据、与客户竞争存在利益冲突，应推动机构集体协议、零数据保留和更严格监管，并投资开放权重基础设施；在“证明丰裕”时代，应改革信用体系，奖励阐释、工具贡献和思想方法，而非仅看谁先证明。文末展望数学与物理的重新靠近，以及纯数学与应用数学界限的模糊。

陶哲轩提出一项实验性AI数学挑战，旨在利用第十判别式界限来探索排除Landau-Siegel零点的论证。该挑战鼓励使用AI工具，从现有或新颖的论证中提取第十判别式的有效上界，以促进对数论问题的理解和解决。

For a given prime , the Legendre symbol is defined as follows: if is a quadratic residue modulo , then ; if is a quadratic non-residue modulo , then ; and if divides , then . Let be an integer that is not a perfect square. (One could require to be a fundamental discriminant if […]

陶哲轩博客刊登的这篇客座博文宣布启动 Bourgain Project，一个由社区驱动的倡议，目标是系统消化 Jean Bourgain 留下的 500 多篇论文，梳理贯穿其研究的共同线索与统一思想，并让这些以艰深著称的成果对更广泛的数学界更易接近。项目同时希望探索传统期刊论文之外的数学呈现形式，鼓励贡献者以任意形式分享论文解读、组队协作，并通过社区 Zulip 频道讨论。Bourgain 的工作横跨调和分析、数论、偏微分方程与泛函分析等多个领域。

[This is a guest post by Francesco Maggi. This blog post was initially written in a different file format and converted using AI. — T.] Large language models (LLMs) and the research systems built around them (and commonly known as “artificial intelligence”) have demonstrated remarkable abilities in generating mathematical proofs. While the quality and limits […]

[This is a guest post by Shaoming Guo – T.] Several math colleagues and I are preparing to launch a new journal. As we considered which values we wanted the journal to uphold, we realized that there may be many other values worth exploring as a community, potentially through the creation of additional journals. We […]

As readers of this blog have no doubt noticed, I have recently turned over the majority of content on this blog to guest posts from other authors, either within the mathematics community or adjacent to it. In this post I record some content coming from my own talks and interviews in this time period. “The […]

As announced a few weeks ago, the SAIR Open Math OS is now soft launched. It offers services and web interfaces similar to frontier AI models, though currently we are hosting existing open math models (DeepSeek V4.1 Flash, Kimi K3, and GLM 5.3 Flash) which are roughly a year or so behind the current frontier […]

Jeremy Avigad, Kevin Buzzard, Johan Commelin, Rémy Degenne, Kim Morrison, Oliver Nash and Adam Topaz have written a blog post over on Kevin Buzzard’s blog entitled “How to Formalize All of Mathematics”, regarding recent proposals to automatically formalize the entirety of the mathematical literature. The issues regarding formal mathematics are parallel to those regarding informal […]

If you sprinkle sand on a vibrating plate it forms the most beautiful patterns. We explore why that happens, what it has to do with mathematical pioneer Sophie Germain, and how the question opens up a huge range of maths.

文章从克拉德尼（Chladni）1808年在巴黎展示的沙粒振动图形讲起，回顾索菲·热尔曼（Sophie Germain）如何以错误但开创性的边界条件为弹性板振动方程奠基，以及后来由Kirchhoff–Love板理论修正。作者随后把话题引向Isaac Newton Institute的“Geometric spectral theory and applications”项目：热斑猜想（hotspot conjecture）预言板在趋于热平衡时最热和最冷点趋向边缘，但已被证明对某些形状不成立，该项目在更多形状和高维情形上取得进展。文中还解释了Laplacian、驻波与节线（nodal lines）为何把振动膜与热流、数值分析、数论、数学台球和极小曲面联系起来。

The sound of a plucked string depends on how long it is, so you can hear the length of a string. But can you hear the shape of a drum?

The sound of a plucked string depends on how long it is, so you can hear the length of a string. But can you hear the shape of a drum?

[This is a guest post by Rikab Gambhir. This blog post was initially written in a different file format and converted using AI. — T.] First, they came for the artists, and I did not speak out, because I was not an artist. Then, they came for the mathematicians, and I did not speak out, […]

A new press release from the Simons Foundation, calling for workshops and other initiatives centered on mathematics in the era of AI.

To celebrate Ada Lovelace day, we've spoken to some of the brilliant women honoured at the International Congress of Mathematicians.