mathematics

Darij Grinberg's 454-page graduate-level textbook 'Introduction to Graph Theory' (Spring 2025 edition, Drexel University Math 530) covers simple graphs, trees, Eulerian/Hamiltonian cycles, colorings, matchings, and network flows, with around a hundred exercises.

A guest post on Terence Tao's blog by Karthik C. S., Amit Sahai, and Jie Xue reflecting on how AI forces mathematicians to reconsider the purpose and value of their work, arguing for a Platonic, 'egoless' view of mathematical discovery over one tied to individual achievement and result production.

A math blog post arguing that the projective plane and homogeneous coordinates should be taught bottom-up, motivated by eliminating exceptional cases (like vertical lines and points at infinity) rather than presented as arbitrary abstractions, with examples from graphics programming and elliptic curve cryptography.

A claim surfaced that the upcoming next-generation GPT model used 10,000 agents working for about four days to prove the first Millennium Prize problem, with the model expected to be released soon.

OEIS announces the winning sequences of its 400k celebration, with Eric Fox's City-Building sequence A400000 taking the top prize and seven runners-up recognized by the selection committee.

Terence Tao criticizes how AI is being applied to mathematics, arguing that automated "strip mining" of problems like Navier-Stokes damages the long-term development of the field even when the proofs are correct.

A new MIT paper surveys 268 early-career mathematicians and finds nearly 70% expect AI to make tenure-track jobs harder to get, with fear of being scooped leading some to withhold unfinished work. Among those using AI in research, 54.8% report disliking it.

Sam Altman acknowledged OpenAI's disruptive AI-generated math releases but said the company will continue publishing, arguing humanity deserves fundamental knowledge despite criticism from mathematicians about poor communication and gatekeeping of breakthroughs.

A reflection on a two-year-old r/math post questioning whether AI could ever solve unsolved mathematical problems, noting that AI has since solved a Millennium Prize problem and raising questions about how close we are to AGI.

The video explains outer billiards systems: from periodic cycles, singularities, and stable islands on parallelogram tables to unbounded orbits on the Penrose kite, finally introducing the hyperbolic plane tiled by regular polygons to show how simple rules can generate unpredictable structures.

Axiom Math released the first complete computer-checked proof of the Classification of Finite Simple Groups in Lean, using swarms of dozens of AI agents over 24 days to produce ~43.1 million lines of code, while also surfacing 34 deviations from the published record and a new result in group theory.

OpenAI announced AI-generated breakthroughs resolving the Kakeya conjecture in mathematics, challenging the exponential time hypothesis in computer science, and advancing quantum chromodynamics research in physics, but is facing growing frustration from researchers over its inconsistent and opaque release practices. The company says it is now working with external expert 'shepherds' to guide how findings are shared, with the physics result withheld entirely pending expert guidance.

Postdocs at MIT, Harvard, and Oxford claim substantial progress toward the Yang-Mills existence problem using AI-assisted (Claude and ChatGPT) human-guided research on a ~$500 budget, building on Bałaban's work, and are publicizing it via GitHub as an experiment in community-generated mathematics.

A complete formalization of the Classification of Finite Simple Groups in Lean 4, with every logical step checked by Lean's kernel, spanning 43.1M lines of Lean and following the first-generation proof. Along the way, the team discovered an issue in a published argument for the Unbalanced Group Theorem and devised an alternative route.

OpenAI 宣称发布了一篇证明指数时间假设（ETH）为假的论文，主定理已通过 Lean 验证，并由 Impagliazzo、Paturi 和 Williams 领导的独立工作组进行审查；文章通俗解释了 3-SAT、NP 完全性以及 ETH 被证伪可能带来的深远影响。

OpenAI announces three major scientific discoveries in mathematics, computer science, and physics—including a refutation of the Exponential Time Hypothesis—while unveiling a new collaboration model using external academic 'shepherds' to guide and verify the results.

OpenAI发布了openai/math仓库，包含其未公开内部模型产生的721篇数学手稿（373个结果族），涵盖多个数十年未解问题的突破，并附带约42%结果的Lean形式化证明与模型推理摘要。

OpenAI 发布关于与学术界合作构建科学理解的文章，探讨其在 Yang-Mills 千禧年大奖难题等基础科学问题上的研究与社区协作。

OpenAI announces three new AI-produced discoveries in mathematics (local smoothing/Kakeya conjecture), complexity theory, and quantum field theory, along with a new release process using external expert 'shepherds' and collaborative groups to validate and contextualize the results.

《Start Doing Graph Theory》是一本关于用图论解谜、证明定理的入门教材，采用 Creative Commons 许可免费在线阅读或下载 PDF，现公开发布。