Murmurations in Arithmetic Geometry and Related Topics: November 11-15, 2024

Organized by:

  • Yang-Hui He (London Institute for Mathematical Sciences)
  • Abhiram Kidambi (Max Planck Institute for Mathematics, Leipzig)
  • Kyu-Hwan Lee (University of Connecticut)
  • Thomas Oliver (University of Westminster)

Mathematicians have studied elliptic curves for many decades, owing to their beautiful abstract structure, powerful applications in number theory and algebraic geometry, and practical relevance in cryptography. It is surprising, therefore, that the socalled murmuration phenomenon was first observed only in 2022.

Murmurations of elliptic curves were first discovered in 2022 in the context of applying machine learning to datasets of elliptic curves. This murmuration phenomenon can be thought of as striking, unexpected oscillatory patterns in the statistics of a large families of elliptic curves. Murmurations have turned out to be ubiquitous for arithmetic and automorphic objects such as higher genus curves, modular forms, and Maass forms. Though some cases of murmuration have been rigorously proven, and the relationship between murmurations and $1$-level densities of zeroes of $L$-functions have been made clear, the current status of understanding of murmurations is still in its nascent stage, and many questions remain open. This workshop will provide the opportunity to report on recent developments in the study of murmurations and to gather together a wide array of perspectives with a view towards answering open questions and developing structural theorems and applications. Furthermore, possible extensions of the murmuration phenomenon to K3 surfaces and Calabi–Yau 3-folds will be investigated by bringing together researchers in arithmetic geometry and string theory, who have been establishing databases for these objects and applying machine-learning tools to understand their arithmetic structures.

Photo by Walter Baxter / A murmuration of starlings at Gretna / CC BY-SA 2.0.