Quantum automorphisms of graphs

• Applicant: Professor Dr. Moritz Weber
• Subject Area: Mathematics
• Term: from 2017 to 2020
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 387699909

The theory of graphs is an indispensable tool in many domains of mathematics. Many famous problems have been formulated and solved using graphs such as the Four Color Theorem, travelling salesman problems or graph rewriting algorithms in computer science. One of the main invariants for distinguishing graphs is the aspect of symmetry, i.e. the investigation of their groups of automorphisms. Symmetry as a mathematical concept is as old as mathematics itself. However, in the past few decades modern mathematics required more general notions of symmetry, modelled by more general objects than groups: The use of quantum groups has gained growing interest over the past years as it is designed for the features of noncommutativity, one of the major challenges in nowadays mathematics and theoretical physics. The notion of a quantum automorphism group of a graph has been defined in the realm of topological quantum groups only a few years ago and it still awaits its systematic investigation. This is in the focus of this proposal.More precisely, we aim at- investigating new means to distinguish graphs, in particular for those having the same automorphism groups but different quantum automorphism groups; ideally culminating amongst others in quantum versions of classical graph theorems such as Erdös-Renyi's Theorem or Cameron's Theorem,- studying the C*-algebras associated to graphs using the knowledge about their quantum automorphism groups and to develop a quantum point of view on the notion of symmetries for graph C*-algebras,- and finally finding new examples of compact quantum groups; moreover, explaining the representation theory of certain quantum groups associated to graphs as recently defined by the applicant.

• DFG Programme: Research Grants