Divisibility questions on the partition function and their connection to modular forms

Abstract: The partition function records the number of ways an integer can be written as a sum of positive integers. Already studied by Euler, it has turned out to be a great source of inspiration in the theory of modular forms over the course of the past century. This development was ignited by Ramanujan. At at a time when it was a challenge to merely calculate values of the partition function, he anticipated divisibility properties of astonishing regularity. We will explain some of the ideas that emerged from Ramanujan's conjectures and some of their modern manifestations. Many of these are connected to modular forms and via these to Galois representations. They help us to understand in an increasingly precise sense how frequently Ramanujan's divisibility patterns and their generalizations occur.

other condensed matterstatistical mechanicsstrongly correlated electronsgeneral relativity and quantum cosmologyHEP - theorymathematical physicsalgebraic geometrydifferential geometrydynamical systemsgroup theorynumber theoryquantum algebrarepresentation theory

Audience: researchers in the discipline

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Number theory, Arithmetic and Algebraic Geometry, and Physics

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