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The present volume is a collection of seven papers that are either based on the talks presented at the workshop "Conformal field theories and tensor categories" held June 13 to June 17, 2011 at the Beijing International Center for Mathematical Research, Peking University,  or are extensions of the material presented in the talks at the workshop. These papers present new developments beyond rational conformal field theories and modular tensor categories and new applications in mathematics and physics. The topics covered include tensor categories from representation categories of Hopf algebras, applications of conformal field theories and tensor categories to topological phases and gapped systems, logarithmic conformal field theories and the corresponding non-semisimple tensor categories, and new developments in the representation theory of vertex operator algebras. Some of the papers contain detailed introductory material that is helpful for graduate students and researchers looking for an introduction to these research directions. The papers also discuss exciting recent developments in the area of conformal field theories, tensor categories and their applications and will be extremely useful for researchers working in these areas.

Quantum field theory --- Conformal invariants --- Categories (Mathematics) --- Tensor products --- Tensor algebra --- Algebra, Tensor --- Products, Tensor --- Conformal invariance --- Invariants, Conformal --- Mathematics. --- Associative rings. --- Rings (Algebra). --- Category theory (Mathematics). --- Homological algebra. --- Nonassociative rings. --- Mathematical physics. --- Mathematical Applications in the Physical Sciences. --- Associative Rings and Algebras. --- Category Theory, Homological Algebra. --- Non-associative Rings and Algebras. --- Mathematical Physics. --- Algebras, Linear --- Calculus of tensors --- Conformal mapping --- Functions of complex variables --- Algebra. --- Mathematics --- Mathematical analysis --- Physical mathematics --- Physics --- Rings (Algebra) --- Homological algebra --- Algebra, Abstract --- Homology theory --- Category theory (Mathematics) --- Algebra, Homological --- Algebra, Universal --- Group theory --- Logic, Symbolic and mathematical --- Topology --- Functor theory --- Algebraic rings --- Ring theory --- Algebraic fields --- Calculus of tensors. --- Quantum groups. --- Conformal geometry.