陶哲轩用GPT5-Pro跨界挑战！3年无解的难题，11分钟出完整证明

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][4][22]. Group 1: Problem Solving Process - The original problem involved determining if a smooth topological sphere in three-dimensional space, with a principal curvature absolute value not exceeding 1, encloses a volume at least equal to that of a unit sphere [8]. - 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 [9]. - GPT-5 Pro completed the calculations in 11 minutes and 18 seconds, providing a complete proof for the star-shaped case using various inequalities and identities, some of which Tao was familiar with [10]. Group 2: AI's Role and Limitations - Although AI made minor errors in estimating a perturbation nonlinear term, it also identified a special case that reverted to the star-shaped result [17]. - The AI's performance was effective for small-scale problems, contributing useful ideas from literature that Tao was previously unaware of [23]. - However, for medium-scale strategies, AI reinforced Tao's incorrect intuition without questioning it, indicating a limitation in critical analysis [26][27]. Group 3: Insights on AI in Research - Tao reflected on the multi-scale value of AI tools, emphasizing the need for human oversight to maintain awareness of task structures across different scales [36]. - He proposed that the optimal level of automation lies between 0% and 100%, allowing for sufficient human involvement to address local issues while reducing repetitive tasks [36]. - The experience reinforced Tao's earlier assertion that the effectiveness of a tool must be evaluated across multiple scales [33]. Group 4: Historical Context of AI Collaboration - Tao's exploration of AI's potential in mathematics began three years ago with the release of ChatGPT, where initial interactions yielded disappointing results [41]. - A turning point occurred with GPT-4, which demonstrated significant efficiency in handling statistical data and familiar mathematical problems, leading to increased expectations for AI integration in research tools [43]. - By July, following OpenAI's achievements, Tao began tackling more complex mathematical problems with AI, finding it particularly useful for numerical searches, which saved considerable time [52]. Group 5: Future Implications - Tao concluded that AI is reshaping the scientific paradigm, serving as a "co-pilot" for mathematicians rather than replacing human creativity and intuition [54]. - The collaboration with AI is expected to lead to more experimental approaches in mathematics, moving beyond purely theoretical work [55].