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]

Core Insights - The collaboration between Terence Tao and GPT-5 Pro successfully addressed a three-year-old unsolved problem in differential geometry, showcasing the potential of AI in academic research [1][10]. Group 1: Problem Solving Process - The original problem involved determining if a smooth topological sphere in three-dimensional space, with principal curvature absolute values not exceeding 1, encloses a volume at least equal to that of a unit sphere [3]. - Tao's initial approach was to restrict the problem to star-shaped regions and utilize integral inequalities, but he sought AI assistance for complex calculations [4]. - GPT-5 Pro completed all calculations in 11 minutes and 18 seconds, providing a complete proof for the star-shaped case using various inequalities, some of which Tao was familiar with, while others were new to him [5]. Group 2: AI's Performance Evaluation - AI demonstrated effectiveness in small-scale problems, contributing useful ideas and only minor errors, but it reinforced Tao's incorrect intuition on medium-scale strategies [11][12]. - In large-scale understanding, AI was beneficial in accelerating research and helping Tao abandon unsuitable methods [14]. - Tao's experience highlighted the necessity of human expertise for further advancements in complex problems, indicating that AI's role is more supportive than substitutive [11][16]. Group 3: Historical Context and Evolution of AI Tools - Tao's exploration of AI's potential in mathematics began with the release of ChatGPT, where initial interactions yielded disappointing results due to a lack of depth in understanding mathematical problems [21][22]. - The introduction of GPT-4 marked a turning point, as it significantly improved efficiency in handling statistical data and mathematical tasks, leading to a more optimistic view of AI's integration into research [22][29]. - Tao's ongoing experiments with AI tools have shown that while AI can assist in numerical searches and problem-solving, it still requires careful oversight to mitigate issues like hallucinations or irrelevant outputs [29][31].