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podcast by Google NotebookLM(20260115목) Subjects: Algebraic Geometry (math.AG); Algebraic Topology (math.AT) Authors: JIM BRYAN, BALA´ ZS ELEK, FREDDIE MANNERS, GEORGE SALAFATINOS, AND RAVI VAKIL How Mathematicians and AI Teamed Up to Prove a New Theorem Introduction: A New Kind of Research Partner The traditional image of mathematical research often involves a lone genius, chalk in hand, wrestling with abstract concepts on a blackboard. But a recent academic paper offers a fascinating glimpse into a different future, one where mathematicians collaborate with a new kind of research partner: artificial intelligence. In a landmark paper, a team of researchers revealed that their proof of a new theorem was achieved "in conjunction with Google Gemini and related tools." They tackled a theorem in algebraic geometry, a deeply abstract field, exploring how it connects to fundamental questions in topology. This collaboration wasn't about using a powerful calculator; it was a dynamic partnership that signals a profound shift in how scientific discovery can happen. AI Is More Than a Calculator—It's a Creative Collaborator Crucially, the AI was not merely a high-powered calculator; it was a source of creative insight. Its role extended far beyond simple computation or data retrieval. The authors found that the AI’s contribution involved a genuine combination of synthesis, retrieval, generalization, and innovation, assembling existing techniques into novel lines of argument. This suggests that AI can function as a creative collaborator in the discovery process, generating new ideas and strategies. The authors were so struck by this that they concluded: So, absent some future discovery to the contrary, the model’s contribution appears to involve a genuine combination of synthesis, retrieval, generalization and innovation of these existing techniques. Discovery Wasn't a "Eureka!" Moment, but a Human-AI Dialogue Rather than a single "Eureka!" moment, the proof emerged from a dynamic conversation between the researchers and the AI. The paper describes the process as an "iterative human/AI interaction"—a collaborative loop where human intuition guided machine intelligence. The process unfolded in several stages: 1. First, the human mathematicians framed the complex problem as a "scaffolding of sub-problems" of increasing difficulty. 2. The AI successfully solved some initial parts but couldn't solve the whole problem at once. Its initial success was on a simplified, numerical version of the problem. 3. The mathematicians then analyzed the AI's partial solutions, isolating key intermediate statements that their own expertise allowed them to generalize into a broader proof strategy. 4. Guided by this human insight, the AI generated a complete proof for the numerical problem and was subsequently prompted to extend the argument to the full, more abstract theorem. This back-and-forth highlights the indispensable role of human expertise. The mathematicians' ability to see the bigger picture, recognize patterns, and strategically guide the AI was essential to the breakthrough. The AI's Output Was "Very Accomplished" The quality of the proofs generated by the AI left a strong impression on the authors. They noted that the output, particularly from a specialized, unpublished internal tool provisionally named "FullProof," was "very accomplished." To showcase the AI's capability, the researchers included a complete, unedited AI-generated solution to a complex sub-problem as an appendix in their final paper. This provides a concrete and remarkable example of the AI's ability to produce rigorous, human-readable mathematical arguments. Humans Remain Firmly in the Driver's Seat Despite the AI's substantial contributions during the discovery phase, the authors make a crucial point about responsibility and authorship. The final academic paper was written entirely by the human researchers, who took full ownership of the work. The AI was a partner in discovery, but the humans were the final arbiters of truth, verifying the logic and ensuring the correctness of every argument. The authors are explicit on this point: We should first emphasize that the text of this paper was written from scratch by the authors... In particular, the authors are responsible for the correctness of the arguments presented here in the usual way. The Dawn of a New Scientific Age? This paper stands as a powerful demonstration of a new partnership between human intellect and artificial intelligence. It shows that AI can be more than a passive tool; it can be an active, creative collaborator in the quest for knowledge—a synergetic dialogue where human intuition guided machine intelligence to a solution neither could have reached alone. This partnership even extended to exploring the boundaries of the theorem, with the AI helping to generate a counterexample showing where the conclusions did not apply—a hallmark of true scientific exploration.