Chinese Neurosurgeon Cracks Decades-Old Crouzeix's Conjecture Using ChatGPT
核心洞察
Beijing-based neurosurgeon Jin Shanmu (搜索), a postdoctoral researcher at Peking Union Medical College Hospital, proved Crouzeix's conjecture, a decades-old problem in numerical linear algebra.
Jin used a 16-hour autonomous run on GPT-5.6-Sol operating on the ChatGPT Work platform while researching transcranial ultrasounds.
Mathematicians Alex Townsend, Anne Greenbaum, and Michel Crouzeix reviewed the manuscript and confirmed the proof was correct, though it is not yet peer-reviewed.
When Beijing-based neurosurgeon Jin Shanmu (搜索) pulled up a chair at his computer, he was not looking to make mathematical history. He was simply trying to crack a problem related to brain ultrasounds. Jin, a postdoctoral researcher and resident at the Peking Union Medical College Hospital, managed to prove Crouzeix's conjecture, a problem in numerical linear algebra that had remained unsolved for roughly two decades.
Jin's tool was a 16-hour autonomous run on GPT-5.6-Sol, operating on the ChatGPT Work platform. His path to the discovery was unconventional: Jin is a doctor who graduated with geology as his undergraduate major, then put himself through medical school before beginning research using ultrasound scans of the brain.
What Is Crouzeix's Conjecture?
Crouzeix's conjecture is a problem about matrices, which are typically used to solve linear equations or handle large amounts of data in computer science or physics. Matrices store values in a rectangular grid of numbers or symbols, and can use mathematical operations such as addition, subtraction, and multiplication among themselves to form new matrices that store the result of the operation.
About two decades ago, French mathematician Michel Crouzeix posited that any function applied to a matrix is no larger than twice the function's maximum value. More precisely, 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. While abstract, it has long been an intriguing problem in matrix analysis — a field Jin stumbled into while undertaking research on transcranial ultrasounds.
How the Discovery Unfolded
With the advent of large language models (LLMs) and tools like ChatGPT, mathematicians began using them to solve problems that had remained unsolved for years. Alex Townsend, a mathematician at Cornell University, and Anne Greenbaum, a professor at the University of Washington, had also been using GPT models to solve Crouzeix's conjecture for the past year.
Late last month, however, they came across a surprise. Townsend had been prompting AI to solve the conjecture, but on July 30, ChatGPT informed him that the problem had been solved three days earlier. When Townsend dug deeper, he found a paper submitted by Jin in which he had given a definitive answer for the math problem.
Although the paper is not peer-reviewed, Townsend reviewed the manuscript and shared it with Greenbaum and Crouzeix, who confirmed that Jin's proof was correct.
Broader Implications for AI in Mathematics
While the manuscript demonstrates Jin's thirst for knowledge, it also showcases the capability of models like ChatGPT to help crack mathematical problems even without highly specialized training in the subject.
This is not an isolated development. Earlier in May, OpenAI (搜索) confirmed that a general-purpose reasoning model it had tested internally was able to solve the planar unit distance problem, posed by Hungarian mathematician Paul Erdos way back in 1945. Earlier this month, the company also listed 10 other mathematical problems that its upcoming model, Astra, has nearly solved or made major progress on.
Competing AI company Anthropic (搜索), whose Claude model is widely used by businesses, also said the latest version of its model is attempting to solve the famous Riemann hypothesis — showing once again that AI could answer major math puzzles that have been pending for decades.