Symmetry - mathematical structures and applications

Instruction is in the form of lectures, calculation exercises, and individual projects. The lectures cover the following themes: - definition, examples, and aspects of structure theory for various algebraic structures: finite groups, associative algebra, Hopf algebra, Frobenius algebra, finite-dimensional Lie groups and Lie algebra, quantum groups, and supersymmetries - the basics of representation theory for groups and Lie algebras, including construction of character table for finite groups, factorisation of the tensor product of representations, basic functions of irreducible representations, PBW theorem and Weyl's character formula, - classification of finite-dimensional complex simple Lie algebras and introduction to Kac-Moody and affine Lie algebras, - quantum mechanics applications, including Bloch's theorem, symmetry-adapted wave functions of molecular orbitals and crystalline fields splitting of nuclear orbitals, - spectroscopy applications, including Unsold's theorem and electrical/magnetic dipole transfers, - applications in molecular vibrations and crystalline symmetries, - tensor categories, graphical tensor calculus, application of the graphical tensor calculus to visualise algebraic relations, and graphical proof, - applications of the graphical calculus in quantum mechanics and quantum information.