Jin Shanmu is a postdoctoral researcher at the Peking Union Medical College in Beijing, China, and also the cynosure of mathematicians from around the globe after he proved the famous Crouzeix's conjecture with a little bit of help from ChatGPT. The math problem had troubled mathematicians for over two decades but was solved by Jin after the release of OpenAI's latest reasoning model.
While most people use tools like ChatGPT to handle everyday tasks, for some like Jin Shanmu, it has opened up new opportunities for learning and discovery. Jin, who has limited math education, was using the tool to research transcranial ultrasounds. If the sound of the technical term is making you look for something more palatable to read, know this first: Jin is a doctor, but he graduated with geology as his undergraduate major. So before stumbling across a famous math problem, Jin put himself through medical school and then began research using ultrasound scans of the brain, where he eventually did something incredible.
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. These matrices can use mathematical operations like addition, subtraction, and multiplication among themselves and lead to the formation of new matrices that store the result of the operation in them. 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. While abstract, it has been a hurdle for mathematicians ever since.
With the advent of large language models (LLMs) and tools like ChatGPT, mathematicians too began using them to solve such problems that have remained unsolved for years. Alex Townsend, a mathematician at Cornell University, and Anne Greenbaum, a professor at the University of Washington, too were using GPT models to solve Crouzeix's Conjecture for the past year but late last month, came across a surprise.
When AI Solves Your Problems
Townsend had been prompting AI to solve the Crouzeix's Conjecture, but for this attempt on July 30, ChatGPT informed him that the problem had been solved three days ago. 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 Although the paper is not peer-reviewed, Townsend reviewed the manuscript and even shared it with Greenbaum and Crouzeix, who confirmed that Jin's proof was correct.
While this manuscript shows the thirst for knowledge that Jin has, it also showcases the capability of models like ChatGPT, which have helped him crack the math, even without highly specialized training in the subject. Earlier in May, OpenAI 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. Another AI developer 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.