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Machine learning is transforming mathematical discovery, enabling advances on longstanding open problems. In this talk, I will discuss AlphaEvolve, a general-purpose evolutionary coding agent that uses large language models to autonomously discover old and new mathematical constructions and potentially go beyond them. AlphaEvolve tackles a wide variety of problems across analysis, geometry, combinatorics, and number theory. In some instances, AlphaEvolve is also able to generalize results for a finite number of input values into a formula valid for all input values. Furthermore, we are able to combine this methodology with Deep Think and AlphaProof in a broader framework where the additional proof-assistants and reasoning systems provide automated proof generation and further mathematical insights. This illustrates how general-purpose AI systems can systematically successfully explore broad mathematical landscapes at an unprecedented speed, leading us to do mathematics at scale.

Skein modules are algebraic objects that somehow “encode” links in a 3-dimensional manifold, in a way reminiscent of homology; they have many interesting connections to physics, representation theory, knot theory (via the Jones polynomial) and non-commutative algebra. In this talk I will give a broad introduction and overview of the topic and then discuss a new, elementary proof (due to myself and Renaud Detcherry) of finite dimensionality of the Kauffman bracket skein modules, originally done by Gunningham-Jordan-Safronov.