Beijing-based neurosurgeon Jin Shanmu has managed to prove Crouzeix’s conjecture, a two-decade-old problem in numerical linear algebra, with the help of OpenAI’s latest flagship AI model, GPT-5.6-Sol, after a 16-hour autonomous run on the ChatGPT Work platform. He was not looking to make mathematical history; he was simply trying to crack a problem related to brain ultrasounds.
How Crouzeix’s conjecture was tackled
Jin, a self-taught mathematics enthusiast who is a postdoctoral researcher and resident at the Peking Union Medical College Hospital, ran GPT-5.6-Sol autonomously for 16 hours on ChatGPT Work to arrive at the proof. The problem he solved had frustrated experts around the world since 2004.
About the conjecture
Proposed by French mathematician Michel Crouzeix, the conjecture posits that the norm of applying any function to a matrix is no larger than twice the function’s maximum value on that matrix’s numerical range. Although abstract, the conjecture has been an intriguing problem in matrix analysis — a field Jin says he stumbled into while undertaking research on transcranial ultrasounds.
