Machine Learning Detects Terminal Singularities

Abstract: I will describe an example of AI-assisted mathematical discovery, which is joint work with Al Kasprzyk and Sara Veneziale. We consider the problem of determining whether a toric variety is a $\mathbb{Q}$-Fano variety. $\mathbb{Q}$-Fano varieties are Fano varieties that have mild singularities called terminal singularities; they play a key role in the Minimal Model Programme. Except for the special case of weighted projective spaces, no efficient global algorithm for checking terminality of toric varieties was known.

We show that, for eight-dimensional Fano toric varieties $X$ of Picard rank two, a simple feedforward neural network can predict with 95% accuracy whether or not $X$ has terminal singularities. The input data to the neural network is the weights of the toric variety $X$; this is a matrix of integers that determines $X$. We use the neural network to give the first sketch of the landscape of $\mathbb{Q}$-Fano varieties in eight dimensions.

Inspired by the ML analysis, we formulate and prove a new global, combinatorial criterion for a toric variety of Picard rank two to have terminal singularities. This gives new evidence that machine learning can be a powerful tool in developing mathematical conjectures and accelerating theoretical discovery.

machine learningmathematical physicsalgebraic geometryalgebraic topology

Audience: researchers in the topic

DANGER3: Data, Numbers, and Geometry