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.

翻译：Fano簇是几何学中的基本构造模块——它们是数学形状的"原子单元"。近期Fano簇分类研究的进展涉及分析称为量子周期的不变量。这是一个整数序列，为Fano簇提供数值指纹。据推测，Fano簇由其量子周期唯一确定。若此猜想成立，则应能直接从其量子周期恢复Fano簇的几何性质。我们应用机器学习探讨以下问题：X的量子周期能否揭示X的维数？注意，目前对此尚无理论理解。我们证明，简单的前馈神经网络能以98%的准确率确定X的维数。在此基础上，我们建立了一类Fano簇量子周期的严格渐近性质。这些渐近性质可通过量子周期确定X的维数。我们的结果表明，在缺乏理论理解的情况下，机器学习能从复杂数学数据中提取结构。这些结果也为"Fano簇的量子周期决定该簇"这一猜想提供了正面证据。