Number Theory Seminar

We argue how AI can assist mathematics in three ways: theorem-proving, conjecture formulation, and language processing.

 

Inspired by initial experiments in geometry and string theory in 2017, we summarize how this emerging field has grown over the past years, and show how various machine-learning algorithms can help with pattern detection across disciplines ranging from algebraic geometry to representation theory, to combinatorics, and to number theory. 

 

At the heart of the programme is the question how does AI help with theoretical discovery, and the implications for the future of mathematics.

In 2004, Kisin proved modularity lifting theorems for two-dimensional Barsotti-Tate representations of totally real fields. A key ingredient in his proof was the construction of resolutions of moduli spaces of crystalline representations of finite extensions of $\mathbb{Q}_p$ using p-adic Hodge-theoretic data.

 

In this talk I will discuss recent joint work with Bao Le Hung and Brandon Levin which extends these results to three-dimensional Galois representations of minimal regular weight. I will begin by recalling some of Kisin's main ideas, before focusing on the role played in our work by certain affine Springer loci inside the affine Grassmannian. In particular, I will indicate how sufficient control of the singularities of these loci, which we obtain for the quasi-minuscule coweight (2,1,0), largely reduces the problem to a dimension estimate.