Murmurations: a case study in AI-assisted mathematics
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Abstract (arXiv)
We report the emergence of a striking new phenomenon in arithmetic, which we call murmurations. First observed experimentally through averages over large arithmetic datasets, murmurations can be detected and analyzed using standard interpretability tools from machine learning, including principal component weightings, saliency curves, and convolutional filters. Although discovered computationally, they constitute a genuinely new and intriguing phenomenon in arithmetic that can be formulated and investigated using established tools of number theory. In particular, murmurations encode subtle information about Frobenius traces and naturally belong to the framework of arithmetic statistics. More precisely, murmurations connect to central themes surrounding the conjecture of Birch and Swinnerton-Dyer and perspectives from random matrix theory. In this paper, we present an overview of murmurations, contextualizing them within number theory and AI.
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References · 19
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7 more
- 2020Machine-Learning Number Fieldscited by 9
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- 2019Machine Learning meets Number Theory: The Data Science of Birch-Swinnerton-Dyercited by 8
- 2023Mathematical conjecture generation using machine intelligencecited by 8
- 2023Machine learning the dimension of a Fano varietycited by 7
- 2022Machine Learning Class Numbers of Real Quadratic Fieldscited by 6
- 2023Machine-Learning Kronecker Coefficientscited by 1
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