Mathematical Foundations of Topological Quantum Field Theories

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The field of Topological Quantum Field Theories (TQFT) has always been a source of rich interaction between Mathematics and Physics. This interplay highly relies on the ability to 'translate' the ideas fromPhysics into a rigorous mathematical setting allowing the application of powerful theorems such as the Baez-Dolan Cobordism Hypothesis, which were discovered as purely mathematical statements. The most important tools for studying TQFT in mathematics are highercategory theory and, in particular, the fully-extended bordism category. Realising new 'physical' ideas as mathematical structures in these areas is not only a necessary step in the process of understanding them, but it also is an ample source of examples and motivation of the study of higher categorytheory as a subject, which is already interesting by itself.In the topic of theoretical condensed matter physics TQFTs arise as lowtemperature limits. They have been of particular interest for the purpose of using topological protection of quantum states in quantum computing.Such TQFTs have been grouped together into different classes, so-called topological phases. Which topological phases can occur in a specific physical situation depends on the parameters of that situation, mainly the dimensiond and the symmetry group G.There has since been a great interest in classifying topological phasesgiven the parameters (G, d) as such a classification would predict underwhich requirements particular phases, such as topological insulators canbe expected.Reflection positivity (rp) is a phenomenon that is observed in most physical TQFTs, but so far had no counter-part in the mathematical world. Recently Freed and Hopkins [2] proposed a definition of reflection positivityfor invertible topological phases and argued why one should expect it to be implemented for all TQFTs relevant in Physics. This improvement from the standard definition of a TQFT to a rpTQFT seems to close the gap between the results obtained from the mathematical model and those appearing in the theoretical physics literature. Using tools from stable equivariant homotopy theory they classified invertible rpTQFTs in low dimensions for most interesting symmetry groups reproducing the results known from the theoretical physics literature and generating new results in many interesting cases. From a mathematical perspective it is quite unsatisfying that the FreedHopkins definition of rp only works out for invertible TQFTs. It is therefore a natural question to ask for this definition to be extended to not necessarily invertible TQFTs and to generalize their classification results to this case.This project mainly falls into the research area of Geometry and Topology, but it also has strong links to several other areas in mathematics and physics. The novelty of the research method lies in combining the physical aspects of TQFT with the theory of equivariant higher categories, and in the invertible case with stable equivariant homotopy theory. This interconnectedness will be useful in either direction.