# Machine learning the fine interior

### Prof Alexander Kasprzyk

#### Abstract

First described in 1983, the Fine interior is a key combinatorial tool in Mirror Symmetry. Broadly speaking, a convex lattice polytope *P* corresponds to a hypersurface *Z* in a toric variety. Associate to *P* is the Fine interior *F(P):* the rational polytope given by moving all supporting hyperplanes of *P* in by lattice distance 1. Many geometric properties of *Z* can be deduced from combinatorial properties of *F(P)*. For example, there exists a unique canonical model of Z if *F(P)* is non-empty, and the Kodaira dimension is determined by the dimension of *F(P)*. Computing the Fine interior *F(P)* is computationally challenging and, despite being so important, almost nothing is known about how the combinatorics of *P* determines the dimension of *F(P)*. This is an area perfect for investigation via Machine Learning.

In this project we will explore the classification of certain four-dimension lattice simplices -- those containing a single interior lattice point. Each of these 338,752 simplices can be described uniquely by an integer-valued vector *(a_0,...,a_4)*, and in nearly every case we know the Fine interior as a result of brute-force computations totalling many decades of CPU time. We will ask whether Machine Learning can predict the dimension of *F(P)* directly from the vector *(a_0,...,a_4)* and, if successful, attempt to understand how the machine is performing this calculation. This should present us with unique insights into the combinatorics of the Fine interior, which in turn will generate a richer understanding of the underlying geometry.