Representation and category theoretic aspects of logarithmic conformal field theories

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Representations of symmetry structures in physics are, in large classes of examples, completely reducible. Logarithmic conformal _eld theories are a class of quantum _eld theories with numerous applications in statistical mechanics and solid state physics; in these theories indecomposable but not irreducible representations arise.Some relevant representation categories are mathematically well-understood, e.g. via quantized universal envelopping algebras. We propose on the one hand side to make new classes of examples explicitly accessible. On the other hand, we propose to construct in general and in classes of examples physically relevant quantities using tools from representation theory and category theory.We have three detailed goals for dissertation projects:1. Construction of invariants of actions of mapping class groups which are candidates for physical correlators.2. Calculations of these invariants in concrete examples, e.g. representation categories of Nichols algebras or group algebras in \bad" characteristic.3. Description of physically relevant quantities like boundary conditions or defect types in terms of categorical quantities.