Algebraic geometry is the study of geometrical shapes defined as solutions to polynomial equations -- algebraic varieties. Amongst varieties, the positively curved ones -- Fano varieties -- distinguish themselves for their importance: they can be considered the building blocks of algebraic geometry. A new approach to the classification of Fano varieties comes from mirror symmetry and involves an invariant called the quantum period. This is a sequence of integers giving a numerical fingerprint for a Fano variety. In this thesis, we approach questions in the classification of Fano varieties using both traditional mathematics and a novel methodology informed by machine learning.

In the first part, we build accurate machine learning models that predict the dimension of certain Fano varieties from their quantum period. Guided by the models, we can establish rigorous asymptotic formulae for the quantum period making the dependence on the dimension clear. This result is proof-of-concept of two main ideas: firstly how the machine learning models can guide intuition in formulating rigorous mathematical statements. Moreover, it gives positive evidence for the conjecture that the quantum period carries geometric information about the variety.

Secondly, we use machine learning differently. When classifying Fano varieties, we have to admit objects with some bad points, called terminal singularities. Verifying such a condition is fundamental, but challenging. We develop an accurate neural network that predicts whether certain (nicely behaved) Fano varieties have terminal singularities. We use this classifier to accelerate computer algebra routines, to generate the first sketch of the landscape of this class of Fanos. Inspired by this, we formulate and prove a new combinatorial criterion for checking if this class of Fanos has terminal singularities. Together with the first sketch of their landscape, this gives strong evidence that machine learning can be essential in accelerating theoretical discovery.