Core Insights - The article discusses the transformation of AI from a "mathematical problem-solving tool" to a "research collaboration partner," exemplified by Tsinghua University's AI mathematician system (AIM) successfully solving a complex mathematical proof [1][2][3] Group 1: AI's Role in Mathematical Research - The research demonstrates the feasibility of AI as a collaborative partner in tackling complex mathematical problems, marking a significant shift in how mathematical discoveries can be approached [2][3] - The study addresses the limitations of current AI systems in mathematics, which often excel in standardized tasks but struggle with real-world research needs [4][5] - The AIM system's collaboration with human researchers led to a comprehensive 17-page mathematical proof, showcasing the potential of human-AI synergy in advanced mathematical research [8][29] Group 2: Methodological Framework - The research outlines five effective human-AI interaction modes that serve as operational guidelines for AI-assisted mathematical research [13][30] - These modes include Direct Prompting, Theory-Coordinated Application, Interactive Iterative Refinement, Applicability Boundary and Exclusive Domain, and Auxiliary Optimization, each designed to enhance the collaborative process [14][17][19][21][22] - The systematic approach to human-AI collaboration not only improves the efficiency of mathematical proofs but also provides a reusable framework for future research [30] Group 3: Future Directions - The study emphasizes the need for further development of human-AI interaction models to enhance mathematical research capabilities and explore their applicability across different mathematical fields [32][34] - Future research will focus on optimizing the AIM system's architecture to improve its reasoning capabilities and overall performance in mathematical theory research [36]

陶哲轩与GPT-5 Pro这对搭档再大发神威，解决了一个3年无人解决的难题。 而且是"不太在自己专业范围内"的问题：微分几何领域的开放问题。 有了这次跨界解题的经验，陶哲轩对AI在学术研究上的作用也有了新的思考，他总结到： AI在小尺度上很有用，中尺度上有些无益，大尺度上又有帮助。 AI从计算到证明一气呵成 要知道，陶哲轩擅长的分析、数论、组合学等研究的往往是整数、函数、算子的性质。而微分几何更侧重于流形的性质，常用的工具也很不一样。 陶哲轩只是按自己直觉给了一个大致方向，GPT-5 Pro就从复杂计算到严谨证明一气呵成，帮助陶哲轩捕捉关键逻辑，突破传统思维局限。 甚至在AI帮助下最终发现自己的直觉有误，但通过这个过程更好理解了问题。 $$\forall0|\,V\geq\forall0|\,B,$$ 陶哲轩最初的思路是将问题限制在星形区域上，借助积分不等式推进。但他的微分几何有些生疏，所以请AI帮他进行这些计算。 先来看看原始问题，是3年前就在mathoverflow上提出的： 设一个三维空间中的光滑拓扑球面围成的区域，且曲面的主曲率绝对值不超过1，那么它包围的体积是否至少等于单位球的体积？ 结果GPT-5 ...