thomas.lanard@gmail.comMachine learning is not only a tool for technology; it can also serve as a powerful compass for mathematical discovery. By identifying hidden patterns in large datasets, it can surface relationships that might otherwise go unnoticed, guiding mathematicians toward new conjectures that can then be rigorously proved.
In this talk, I will present how, in collaboration with Alberto Mínguez, we applied this approach within the Langlands Program. Our focus is the Aubert–Zelevinsky duality, an involution on the irreducible representations of a p-adic group, playing a central role in representation theory. A result of Mœglin and Waldspurger gives an explicit combinatorial recipe to compute this duality for the group GL_n. For classical groups, such as Sp_{2n} or SO_{2n+1}, no such explicit recipe was known. By training machine learning models using an algorithm of Atobe and Mínguez, we discovered structural patterns that led us to a new combinatorial algorithm, which we then proved rigorously.
I will also present langlandsprograms.com, a web application developed with Petar Bakić and Elad Zelingher, where this algorithm is implemented and can be explored interactively.