Modular forms, Hecke algebras and noncommutative geometry

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In recent work of Mesland-Sengun, a natural action of Hecke operators on the K-theory of arithmetic groups has been constructed using bivariant K-theory. The aim of this project is to investigate and understand categorical actions of Hecke pairs on C^*-categories much more broadly. To this end the student will develop the foundations of categorical representation theory of groups on C^*-categories, in particular define and study induction and restriction functors and their biadjointness. This will be accompanied by the analysis of concrete examples, in particular those coming from finite groups, classical Hecke symmetries, and the integral Bost-Connes system studied by Connes-Consani-Marcolli. This research will enhance the understanding of symmetry in the realm of operator algebras. A key novelty of the project is the use of higher categorical techniques.