数学前沿

本文介绍了Danny Calegari的演讲，他分享了如何使用Claude工具进行数学可视化以及进行协作的案例，并提供了几个实用的应用场景。

本文由Lekili和Sheridan撰写的预印本详细解释了OpenAI声称的霍奇猜想案例证明中存在的著名‘符号错误’，并解释了这一错误导致其三篇论文被撤回的原因，指出可能需要新的输入才能修复这个问题。

近期发生了一系列网络安全事件，其中最引人注目的是PHP中的一个漏洞，影响了约70%的互联网。许多组织在修补系统时暂时关闭了系统。OpenAI发现了这个漏洞并与相关组织合作开发了补丁。这一事件可能对开放互联网的访问产生重大影响。

Tim Gowers在10月10日的一次演讲中分享了OpenAI发布300多个解决方案后，AI在他暑假期间所取得的成果。

本文探讨了机制可解释性领域，即研究神经网络思维和操作方式的领域。尽管该领域投入巨大且取得显著突破，如Anthropic的J-space发现，但该领域仍处于起步阶段，被视为降低AI存在风险的关键。

微软研究院Xinzhi Zhang与合作者证明了1995年提出的Kannan-Lovász-Simonovits（KLS）猜想：对对数凹分布，最佳平坦切割与任意形状切割的差距总被一个与维度无关的常数界定。研究者用AI智能体探索了约100条候选路径，由人筛选方向，最终通过递归细化与误差有界性论证完成证明；同日另有两组独立证明与Lean 4形式化验证发布。

本文是 Proofs and Prompts 对数学圈 X（原 Twitter）用户 nihilunbounded 所发长文的推荐。该文以社交媒体上的第一手观察为线索，回顾了 AI 进入数学研究的历史脉络，并对当前 AI 参与数学的方式给出有历史纵深的分析。导读提示：这是一篇观点性随笔而非成果发布，值得读者直接阅读原文。

这篇arXiv论文（2608.11195，2026年8月11日提交）以改进Grothendieck常数（Grothendieck constant）的界为例，展示如何用AI承担长程研究任务，而非仅仅“找答案”；作者之一Pravesh Kothari还接受了普林斯顿大学关于这项工作的采访。导读提示读者关注案例中AI参与研究流程的具体方式，以及Grothendieck常数这一界问题为何适合作为长程任务的试验场。

Tarik Aougab在arXiv上发表随笔《The grounded spirituality of mathematics》，主张对数学采取更'物理'的诠释，并讨论这一立场对使用AI工具意味着什么。对习惯以柏拉图主义视角看待数学的读者而言，这一观点可能颇为意外；文章的价值在于把数学本体论的立场与当下AI辅助证明的实践联系起来。

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。

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.

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.

The EWM has released a statement on ‘[AI and the profession](https://www.europeanwomeninmaths.org/ewm-statement-on-ai-and-the-profession/)‘. They are welcoming anyone who would like to help to join working groups, and urge the need for better representation in the current debates on AI and mathematics.

In [this essay,](https://vislumbres.github.io/#/post/raw-mathematics) Rivera advocates for ‘raw mathematics’, a culture that is more collaborative and is driven by genuine understanding and curiosity rather than prestige and productivity. Rivera draws inspiration from [Art Brut](https://artbrut.ch/en/what-is-raw-art), whose artists “forge a universe for their own use”. This is the first post on a new communal blog ‘[Vislumbres](https://vislumbres.github.io/#/)‘, which will not be exclusively focused only on AI and Mathematics, but will include accessible expositions across different fields about themes that usually don’t make it beyond academic circles.

[Tim Gehrunger](https://people.math.ethz.ch/~timge/) has created this [GitHub repository](https://github.com/GTimG/QuasiRiemannTracker) and [website](https://gtimg.github.io/QuasiRiemannTracker/) to track the best known zero-free region for the Riemann Zeta Function. The fact that there are no zeroes for $\operatorname{Im}(z) > 1$ is classical; OpenAI’s big breakthrough was reducing this to $\frac{7}{8}$, which has since been reduced slightly. We’re waiting for $\frac{1}{2}$, get to work everyone! Similarly, [OpenAI provided](https://github.com/openai/math/blob/main/preprints/Integer-multiplication-below-n-log-n-September-23-2026/paper.pdf) a deterministic algorithm to multiply two $n$\-bit integers in $O(n(\log n)^{1-\kappa})$ steps, with an absurdly small $\kappa$. [This website](https://beyond-n-log-n.netlify.app/) and [GitHub](https://github.com/CrocSwap/integer-mult-bounds) track the best improvements on $\kappa$ (at least in some instances).

[This website](https://curticapean.net/math-notes/) documents OpenAI’s claimed solutions, and related activities in the mathematical community at the time. Ir provides related preprints, comments by researches, and additional discussion; and anyone can comment contributions. A related initiative is the website [citedbyagi.com](https://citedbyagi.com/), which tracks every human author cited across [OpenAI’s Maths GitHub](https://openai.com/index/sharing-ai-progress-in-mathematics/) manuscripts.

This research was carried out by Antrhopic’s Red Team and is titled ‘[AI models may be capable of self-replicating without access to a datacenter](https://www.anthropic.com/research/self-replicating-ai)‘. As the discourse in the wider tech spheres about potential catastrophic loss-of-control scenarios escalates, expect to see more research of this sort.