Mapping class groups and algebraic cycles

Richard Hain, Duke University The topology, geometry and arithmetic of moduli spaces of curves are well known to be intertwined. Many geometric properties of algebraic curves are reflected in the topology and geometry of the corresponding moduli space. In this talk I will give an overview of one aspect of this connection --- namely results that relate algebraic cycles on powers C^n of a genus g curve C to the structure of the mapping class group of a closed surface of genus g. This is a subject with a rich history with roots in the work of Riemann and Abel in the 19th century. I will review relevant background material, including the definition of and basic results about mapping class groups and algebraic cycles. I will explain why cycles that are defined for all curves, such as those defined by Ceresa and Gross--Schoen, are related to the Torelli subgroups of mapping class groups. If time permits, I will describe a new (higher) algebraic cycle defined for hyperelliptic curves that Wanlin Li and I have constructed. Thursday 12th March, 2026