Machine learning the dimension of a Fano variety

TL;DR

This paper demonstrates that machine learning can accurately determine the dimension of Fano varieties from their quantum periods, providing evidence for the conjecture that quantum periods uniquely identify Fano varieties.

Contribution

It applies machine learning to extract geometric information from quantum periods and establishes asymptotic results linking quantum periods to the dimension of Fano varieties.

Findings

Neural network achieves 98% accuracy in predicting dimension

Asymptotic formulas for quantum periods are established

Results support the conjecture that quantum periods determine Fano varieties

Abstract

Fano varieties are basic building blocks in geometry - they are `atomic pieces' of mathematical shapes. Recent progress in the classification of Fano varieties involves analysing an invariant called the quantum period. This is a sequence of integers which gives a numerical fingerprint for a Fano variety. It is conjectured that a Fano variety is uniquely determined by its quantum period. If this is true, one should be able to recover geometric properties of a Fano variety directly from its quantum period. We apply machine learning to the question: does the quantum period of X know the dimension of X? Note that there is as yet no theoretical understanding of this. We show that a simple feed-forward neural network can determine the dimension of X with 98% accuracy. Building on this, we establish rigorous asymptotics for the quantum periods of a class of Fano varieties. These asymptotics determine the dimension of X from its quantum period. Our results demonstrate that machine learning can pick out structure from complex mathematical data in situations where we lack theoretical understanding. They also give positive evidence for the conjecture that the quantum period of a Fano variety determines that variety.