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Machine learning the dimension of a Fano variety

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Listed:
  • Tom Coates

    (Imperial College London)

  • Alexander M. Kasprzyk

    (University of Nottingham)

  • Sara Veneziale

    (Imperial College London)

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.

Suggested Citation

  • Tom Coates & Alexander M. Kasprzyk & Sara Veneziale, 2023. "Machine learning the dimension of a Fano variety," Nature Communications, Nature, vol. 14(1), pages 1-11, December.
  • Handle: RePEc:nat:natcom:v:14:y:2023:i:1:d:10.1038_s41467-023-41157-1
    DOI: 10.1038/s41467-023-41157-1
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    References listed on IDEAS

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    1. Alex Davies & Petar Veličković & Lars Buesing & Sam Blackwell & Daniel Zheng & Nenad Tomašev & Richard Tanburn & Peter Battaglia & Charles Blundell & András Juhász & Marc Lackenby & Geordie Williamson, 2021. "Advancing mathematics by guiding human intuition with AI," Nature, Nature, vol. 600(7887), pages 70-74, December.
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