Algebraic structures in mathematical physics and category theory

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Symmetries in Nature have fascinated the mankind ever since the beginning of civilization, since they help us understanding its most fundamental laws. An easy instance is rotational symmetry, invariance of a phenomenon if we observe it independent of the viewing angle. The more uncommon a symmetry is, the more interesting it is from a physics point of view and more exotic and original mathematical structures describe them. In the last decades there has been a substantial effort to understand those related to conformal symmetry, a symmetry particularly rare which preserves angles (but not e.g. sizes). These advances have constituted an active, rich and cutting-edge field of world-wide research. This PhD project will study systematic ways of detecting and classifying certain algebraic structures arising in this setting and describing certain physical entities, in particular algebra objects in modular tensor categories obtained from representations of vertex operator algebras. These objects have a beautiful physical description and are connected to other mathematical formalizations of physical theories like e.g. r-spin topological field theories.