Machine learning the dimension of a Fano variety
Tom Coates Alexander M. Kasprzyk Sara Veneziale
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.