数学前沿

这篇文章报道了一个新的自动研究项目启动，目标是广义Hodge猜想中被对合（involution）固定的六次四维形（sextic fourfolds）这一特例。该特例由Voisin分离出来，Diamond的归约工作使其变得可有效计算。文章还提到，Hodge猜想近期因有实验室声称取得“重大进展”而受到关注，其中OpenAI提出的有理Hodge猜想K3曲面乘积的所谓证明因错误而被撤回。

数学 X/Twitter 圈的知名作者 nihilbounded 撰写了一篇随笔，回顾 AI 进入数学领域的近期历史——至少是 X 上呈现的那段历史，并尝试在扎实的历史脉络下理解当前的进展过程。Proofs and Prompts 推荐这篇稍长的文章，认为值得一读。

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。

Justin Moore [has written](https://justintatchmoore.github.io/homepage/10-6-2026.html) about [OpenAI’s claimed proof](https://github.com/openai/math/blob/main/preprints/Thompsons-group-F-is-nonamenable-September-23-2026/paper.pdf) that Thompson’s group $F$ is nonamenable, which had been a major open problem in Geometric Group Theory. Moore’s essay appeared on 08 October, and recounts his personal experiences with this problem and the human connections formed. A human-digested proof of OpenAI’s announcement was written by [Kim and Lodha](https://arxiv.org/pdf/2610.18360v1), and uploaded to the arXiv on 13 October.

[This post](https://federicoaschieri.substack.com/p/the-future-of-mathematics-part-i), on Aschieri’s substack ‘[Understanding Intelligence](https://federicoaschieri.substack.com/)‘, this is the first part of a longer essay on ‘The future of Mathematics’. In it, Aschieri explores the conceptual framework that could underpin Mathematics in the future. It originally appeared on 08 October, shortly after OpenAI’s release.

Spearheaded by Fields medallist Shing-Tung Yau, [the new institute](https://www.scmp.com/news/china/science/article/3370492/leading-chinese-mathematicians-unite-behind-use-ai-research), called the ‘The Yau Laboratory Alliance for Mathematics and Artificial Intelligence’, will aim to create a network of more than 1000 researchers.

丘成桐 1982 年在讲义《 Seminar on Differential Geometry 》中的第 57 问题提出：寻找一个非平凡的充分条件，使二维完备负曲率流形能够等距嵌入三维欧氏空间。丘成桐特别指出，这类条件可能与曲率在无穷远处的衰减速度有关。洪家兴 1993 年证明了只要高斯曲率在无穷远处单调下降且代数衰减速率比 -2 快，二维完备负曲率流形能够光滑等距浸入到三维欧氏空间。这是洪家兴院士的代表性成果之一。但以上条件不是最优，洪家兴在 1993 年的论文中也希望能放宽条件，并私下指出总曲率有限是一个最优条件。自 1993 年迄今，洪的条件一致没有得到任何改进。黄飞敏和合作者自 2013 年开始研究，历经 11 年，直到 2024 年才得到了几乎最优条件：总曲率有限及允许曲率适当振荡，去掉了曲率单调下降条件，从而在丘成桐第 57 问题的研究上取得了重大突破。 相关成果发表在《数学进展》（ Advance in Mathematics ）上。 Publication: Advance in Mathematics Authors: Wentao Cao Academy for Multidisciplinary Studies, Capital Normal University, Beijing, 100048, China Email: cwtmath@cnu.edu.c

[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.

Meta, whose AI division recently released a popular [personal agent](https://ai.meta.com/muse/), has released their new frontier model [Muse Wonder](https://research.meta.ai/blog/muse-wonder?utm_source=ai_meta_site&utm_medium=web&utm_campaign=homepage_learn_more_10122026&utm_content=homepage_learn_more_10122026). It seems impressive on benchmarks, but we’ll need to wait for real-world usage. As is par for course, they have been touting their models’ [Mathematics credentials](https://research.meta.ai/blog/solving-open-research-problems-together), and to be fair to them – they managed to follow Anthropic and OpenAI in [offending the Mathematical community](https://www.youtube.com/watch?v=8eG99DjBTCA) (at least in part). Google have also thrown their hat back into the ring, dropping their new model [Gemini 4 Argon](https://blog.google/innovation-and-ai/models-and-research/gemini-models/introducing-gemini-4-argon/). Same story – great benchmarks, no real-world performance yet, and Google hailing it as a model that can solve ‘the hardest problems’. Apparently, the hardest problems nowadays are how to get people to stop being abusive to their models, and Anthropic have been giving their models the [option to end such interactions](https://www.anthropic.com/news/2026-usage-policy-update). Speaking of Anthropic, if rumours are to be believed, there might be some more new models this week.