Coates Tom, Kasprzyk Alexander M, Veneziale Sara
Department of Mathematics, Imperial College London, London, UK.
School of Mathematical Sciences, University of Nottingham, Nottingham, UK.
Nat Commun. 2023 Sep 8;14(1):5526. doi: 10.1038/s41467-023-41157-1.
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.
法诺簇是几何学中的基本构建块——它们是数学形状的“原子碎片”。法诺簇分类的最新进展涉及分析一个名为量子周期的不变量。这是一个整数序列，它为法诺簇提供了一个数值指纹。据推测，一个法诺簇由其量子周期唯一确定。如果这是真的，那么人们应该能够直接从其量子周期恢复法诺簇的几何性质。我们将机器学习应用于这个问题：X的量子周期能确定X的维数吗？请注意，目前对此尚无理论上的理解。我们表明，一个简单的前馈神经网络可以以98%的准确率确定X的维数。在此基础上，我们为一类法诺簇的量子周期建立了严格的渐近性。这些渐近性从其量子周期确定X的维数。我们的结果表明，在我们缺乏理论理解的情况下，机器学习可以从复杂的数学数据中提取结构。它们也为法诺簇的量子周期决定该簇这一猜想提供了积极的证据。
