Disputas: Bas P. A. Jordans
M. Sc. Bas P. A. Jordans ved Matematisk institutt vil forsvare sin avhandling for graden ph.d.:
Random walks on discrete quantum groups and associated categories
Bas P. A. Jordans
Tid og sted for prøveforelesning
- Associate Professor Claudia Pinzari, Sapienza Università di Roma
Associate Professor Roland Vergnioux, Université de Caen Normandie
- Professor Paul Arne Østvær, Universitetet i Oslo
Leder av disputas
Professor Tom Louis Lindstrøm, Matematisk institutt, Universitetet i Oslo
- Professor Sergiy Neshveyev, Matematisk institutt, Universitetet i Oslo
- Professor Lars Tuset, Institutt for informasjonsteknologi, Høgskolen i Oslo og Akershus
In this thesis we study random walks on discrete quantum groups. We compute some concrete Martin boundaries and investigate convergence to the boundary. In addition we generalise this theory to monoidal categories.
Random walks describe the behaviour of a point moving randomly through a space. A lot of structure of the space can be captured by these random walks. It therefore provides very useful tools to study all kinds of spaces. Originally random walks are mainly applied to lattices, graphs and groups.
In the middle of the eighties quantum groups were introduced and in the subsequent years this theory was further developed. Izumi brought the fields of quantum groups and random walks together and defined random walks on discrete quantum groups.
One of the main topics we study in this thesis is probabilistic boundaries of random walks on discrete quantum groups. We compute the Martin boundary for classical random walks on the lattice of irreducible representations of deformed Lie groups and we study the full Martin boundary of SUq(N). Convergence to the boundary is an important theorem in classical random walk theory, but it is a conjecture in the quantum case. Here we give the first quantum example, by establishing convergence to the boundary for random walks on SUq(2).
The representation category of a compact quantum group forms a C*-tensor category. This close relation opens the way for random walks on such C*-tensor categories. We define the Martin boundary and convergence to the boundary for random walks on such categories and we establish compatibility with the quantum case. These C*-tensor categories are important for classifying compact quantum groups. In this thesis we give an intermediate step in this classification process and classify all C*-tensor categories with the same fusion rules as SU(N).
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