In this talk, I will discuss some ways in which AI can be used in mathematical research. Through a series of examples, I will illustrate how AI can help explore problems, suggest ideas, test conjectures, and sometimes obtain partial results. I will also describe several instances where these methods contributed to ongoing work, including partial progress that eventually appeared in a paper.

Part of the talk will be conducted in a live, experimental format. I will demonstrate how one can use AI in real tim by selecting a mathematical paper and attempting to engage with it on the spot. By this, I mean using AI to investigate the paper, explore related questions, and try to prove partial results inspired by it. The goal is not to present AI as a replacement for mathematical thinking, but rather as a new kind of experimental tool that may assist with exploration and discovery.

Nonstandard analysis is a set of techniques whose central idea is to enlarge a given mathematical structure by adding certain “ideal” elements in such a way that the enlarged structure maintains the same “logical” properties as the original structure.  Nonstandard analysis has found applications in nearly every area of mathematics.  In this talk, we will explain how this technique works by giving some simple proofs of important theorems from Ramsey theory, which is a branch of combinatorics.  These applications involve a relatively recent idea, namely the notion of an iterated nonstandard extension.