Modular forms, Hecke algebras and noncommutative geometry

在Mesland-Sengun最近的工作中，Hecke算子在算术群的K-理论上的自然作用被构造成了双变K-理论。这个项目的目的是更广泛地研究和理解Hecke对在C^*-范畴上的范畴作用。为此，学生将发展C^* 范畴上群的范畴表示理论的基础，特别是定义和研究归纳和限制函子及其双伴随性。这将伴随着具体的例子的分析，特别是那些来自有限群，经典Hecke对称性，并研究康纳斯-康萨尼-马科利积分Bost-Connes系统。这一研究将加深对算子代数领域中对称性的理解。该项目的一个关键新奇是使用了更高的分类技术。

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.