CroCQG

The theory of locally compact quantum groups, dating back to the 1960's, is a rapidly evolving field with connections to operator algebras, representation theory, probability, algebraic topology, non-commutative geometry and quantum information theory. Beyond classical compact groups and duals of discrete groups, it encompasses Jimbo-Drinfel'd deformations of compact Lie groups, Wang's universal quantum groups, quantum symmetry groups of graphs etc. The aim of the project is to gather french specialists studying these very general objects from different perspectives and to foster the exchange of technical tools and research goals. We have identified a number of key problems within three main areas. The notion of Quantum Symmetry connects topological quantum groups with various geometric, analytical or combinatorial objects. We will investigate that connection in the settting of quantum channels in information theory, of graphs both finite and infinite and of non-formal deformation quantization. In Non-Commutative Probability, the study of quantum stochastic processes, both on compact and discrete groups, exhibit deep connections to their algebraic and analytic structure. We aim to explore the existence of quantum analogues of the RSK path transformation using quantum Schur-Weyl duality, and to study the structure of Gaussian processes on compact quantum groups as well as their asymptotic behavior. In Operator Algebras we will investigate the structure of von Neumann algebras associated to universal compact quantum groups using recent progress connecting boundary actions of discrete groups and maximal amenability. We will also investigate the structure of quantum automorphism groups locally finite trees using ideas from geometric group theory.