Neurosurgeon in Beijing has astounded world's mathematicians with ChatGPT. He used artificial intelligence program to generate proof for Crouzeix's conjecture which had puzzled mathematicians for over 20 years.Dr. Shanmu Jin (who is a neurosurgery resident and a postdoctoral fellow) began turning to math problem after doing transcranial ultrasound research.About The AuthorHey there, i am a technology enthusiast with a deep passion for gadgets, consumer electronics, emerging technologies, and the fast-paced world of digital innovation. Constantly exploring the latest tech trends, product launches, and industry developments, I enjoy translating complex technological advancements into engaging and accessible stories for readers. My interests span smartphones, wearables, artificial intelligence, smart devices, and the broader technology ecosystem. As I begin my journey as a Tech Journalist at Gadgets Now, I am excited to contribute to a platform that informs millions of readers, combining my passion for technology with storytelling to deliver insightful, accurate, and timely tech coverage.On July 30th, he prompted ChatGPT 5.6 to tackle the conjecture independently.According to mathematicians Alex Townsend and Anne Greenbaum, the system generated a decisive argument during an approximately 16-hour run using GPT-5.6 Sol in ChatGPT Work. Jin then developed the resulting material into a mathematical preprint, which experts subsequently examined and found convincing.Crouzeix’s Conjecture: The Decades-Old Mathematical ChallengeCrouzeix’s conjecture was proposed in 2004 by French mathematician Michel Crouzeix and concerns numerical linear algebra, particularly how functions behave when applied to matrices. In simplified terms, it predicts that the spectral norm of a function of a matrix can be bounded by twice the maximum value of that function over the matrix’s numerical range.The problem became important because numerical range provides information about a matrix that eigenvalues alone cannot capture, particularly for nonnormal matrices.Crouzeix proved progressively stronger partial results, while a 2017 result with Clément Palencia reduced the best known universal bound to (1+\sqrt2). Townsend and Greenbaum said Jin’s work now establishes the conjectured constant of 2.How Shanmu Jin Used GPT-5.6 Sol to Attack the ProblemJin did not simply ask ChatGPT for a proof and copy its response. Townsend and Greenbaum reported that he used a specially designed prompt adapted from an approach previously used by OpenAI for another mathematical problem.More articles by AuthorTrending StoriesThe prompt instructed the AI to work autonomously, explore substantially different proof strategies and use multiple subagents rather than prematurely settling on one attractive idea. Candidate arguments were subjected to adversarial checking, with approaches discarded when counterexamples emerged.Jin also instructed the system to continue searching until it obtained a complete proof that survived internal scrutiny. The resulting GPT-5.6 Sol run lasted about 16 hours and produced the key theorem on which Jin’s manuscript rests.The Key Breakthrough Emerged From an Autonomous AI RunThe decisive development came when GPT-5.6 Sol found an unexpectedly simple route through the hardest part of the conjecture. According to Townsend and Greenbaum, the AI’s key theorem avoided the stronger estimates that mathematicians had previously expected to be necessary. Instead, its argument reduced the central difficulty through a careful sampling strategy to a comparatively straightforward positivity condition. Jin subsequently incorporated and developed the result in his preprint, titled “The Numerical Range Is a 2-Spectral Set.”Townsend and Greenbaum said that when they first encountered the manuscript, they approached it cautiously because the conjecture had resisted mathematical efforts for years. After several hours of examination, however, they concluded that the central argument appeared to be genuine.Townsend and Greenbaum Confirmed the Proof’s Mathematical StrengthThe significance of Jin’s result was not accepted merely because an artificial intelligence system had generated it. Alex Townsend and Anne Greenbaum independently scrutinised the manuscript and reported that both of them, together with Michel Crouzeix, had checked the proof and believed it to be correct. Their assessment is important because AI-generated mathematical arguments can contain subtle errors, unsupported assumptions or missing lemmas.Townsend and Greenbaum noted that ChatGPT had previously produced bogus proofs or arguments that stalled when asked about the conjecture. This time, however, the generated argument survived their examination. Their assessment therefore shifts the story from an AI-generated claim to a mathematical result that has undergone substantial expert verification.A Neurosurgeon’s Unusual Route Into Advanced MathematicsJin’s background makes the breakthrough particularly striking. He is a medical doctor and neurosurgery resident whose mathematical education beyond standard science courses was largely self-directed. Townsend and Greenbaum reported that his interest in matrix analysis grew from his research involving transcranial ultrasound. While teaching himself the subject, Jin encountered Crouzeix’s conjecture and became attracted to its elegant formulation and geometric interpretation of a matrix’s numerical range.His work can serve as an illustration for AI, potentially reducing barriers among these disparate academic fields and encouraging others outside of traditional mathematical careers to tackle daunting theoretical problems.Jin, however, has modestly downplayed his accomplishment and suggests that serendipity played no small part in finding the key that unlocked it.Why the Result Matters Beyond a Single Mathematical ConjectureCrouzeix’s conjecture is not merely an abstract puzzle. Townsend and Greenbaum explained that the inequality has practical implications for approximation theory, matrix functions and computational methods such as GMRES and polynomial and rational Krylov techniques. If the conjectured bound holds, approximations of functions over a matrix’s numerical range can be translated into useful bounds for corresponding matrix functions. That makes the result relevant to numerical analysis as well as pure mathematics.This breakthrough also opens up a broader debate about the future of mathematics as smarter AIs become available. It signals that AI systems may not replace mathematicians all together, but rather produce initial hypotheses which human researchers can then review, express in concrete terms and check for proof.AI Opens a New Model for Mathematical DiscoveryThe development also highlights a changing relationship between human researchers and artificial intelligence. Jin supplied the research direction, constructed the prompt and later worked with the mathematical material, while GPT-5.6 Sol generated the crucial proof strategy during its autonomous run.Townsend and Greenbaum described the episode as evidence that major mathematical contributions could increasingly come from researchers working outside conventional mathematical specialisations with assistance from large language models.Another independent proof appeared only days after Jin’s preprint, from Emiel Lorist and Felix Schwenninger; their approach was different and they also disclosed using ChatGPT 5.6 to explore proof strategies. The episode therefore points toward a future in which AI-assisted discovery becomes another tool in the mathematician’s research process. FAQsWho is Dr. Shanmu Jin and what breakthrough did he achieve in mathematics?Dr. Shanmu Jin is a Beijing-based neurosurgeon and postdoctoral researcher who made a remarkable breakthrough by using ChatGPT 5.6 to produce a proof of Crouzeix's conjecture, a problem that had been unresolved for over two decades.How did Dr. Jin utilize AI in his research on Crouzeix's conjecture?Dr. Jin used a specially designed prompt in ChatGPT 5.6 that allowed the AI to work autonomously and explore various proof strategies. The AI was instructed to seek out a complete proof, which it developed during a 16-hour run.What are the implications of Crouzeix's conjecture in practical terms?Crouzeix's conjecture has significant implications for approximation theory, matrix functions, and computational methods. If validated, it can translate approximations of functions over a matrix's numerical range into useful bounds for corresponding matrix functions, affecting both numerical analysis and pure mathematics.end of article