Abstract

Computing the spectrum of the Hodge-Laplacian on homogeneous spaces is a challenging problem. In a joint project with E.,A. Lauret, we studied the spectrum on $1$-forms for general left-invariant metrics on $S^3 cong SU(2)$ and its quotient $SO(3)$. We successfully computed the explicit full spectrum for Berger spheres and formulated a highly plausible conjecture for the first eigenvalue of general homogeneous metrics. However, despite extensive numerical evidence, a rigorous abstract proof for this general case remained difficult to find for over a year.           Just before submitting our manuscript, we decided to run a final test using the newly released AI model ChatGPT 5.4 Pro (a premium model costing $200 per month, limited to 15 prompts). Provided with a single prompt, the model reasoned completely on its own for 100 minutes. Surprisingly, it came up with an elegant and flawless abstract proof for our conjecture. Crucially, the AI took a completely different path from our previous manual attempts: by defining a clever diagonal transformation to pass to the round metric and using the self-adjoint Curl operator, it successfully established the required lower bound. Our research group reviewed the generated proof over the following days and confirmed its validity.           This proposed flash talk will share a brief, anecdotal report of this remarkable AI-assisted discovery. It will show the mathematical setup of the problem and demonstrate how advanced LLMs can navigate intricate algebraic and geometric barriers, offering a glimpse into the potential of AI as a reasoning assistant in pure mathematics.