Machine learning the dimension of a Fano variety

Kasprzyk, Alexander M.; Coates, Tom; Veneziale, Sara

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.

Citation

Kasprzyk, A. M., Coates, T., & Veneziale, S. (2023). Machine learning the dimension of a Fano variety. Nature Communications, 14, Article 5526. https://doi.org/10.1038/s41467-023-41157-1

• Journal Article Type: Article
• Acceptance Date: Aug 23, 2023
• Online Publication Date: Sep 8, 2023
• Publication Date: 2023
• Deposit Date: Aug 31, 2023
• Publicly Available Date: Sep 12, 2023
• Journal: Nature Communications
• Electronic ISSN: 2041-1723
• Publisher: Nature Publishing Group
• Peer Reviewed: Peer Reviewed
• Volume: 14
• Article Number: 5526
• DOI: https://doi.org/10.1038/s41467-023-41157-1
• Keywords: General Physics and Astronomy; General Biochemistry, Genetics and Molecular Biology; General Chemistry; Multidisciplinary
• Public URL: https://nottingham-repository.worktribe.com/output/24852502
• Additional Information: This is the peer reviewed version of the following article: Kasprzyk, A. M., Coates, T., & Veneziale, S. (2023). Machine learning the dimension of a Fano variety. Nature Communications, 14, Article 5526, which has been published in final form at https://doi.org/10.1038/s41467-023-41157-1