Group Theory and Symmetries

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add_post_meta($ID, 'Author', 'G.Mussardo', True);
add_post_meta($ID, 'Credits', '2', True);

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$CREDITS = get_post_meta(get_the_ID(),'Credits',True);
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by $AUTHOR ($CREDITS credits)


Starting and using several examples coming from Physics, the course aims to present the basic properties of one of the most useful and beautiful branches of Mathematics, Group Theory.

Main topics

  • The Great Game, an excursus across themes and history of group theory
    • Finite Groups
      • Properties
      • Representations
      • Examples: Platonic Solid, Crystals, Space and Plane Tessellation
      • Energy spectrum in periodic and a-periodic systems
      • Point Groups
      • Permutations: Cayley’s theorem and Young Tableaux
      • Potts model and random systems
      • Echoes from number theory
    • Lie Groups
      • Generators and Commutators
      • Baker-Haussdorf formula
      • Lie Algebras
      • Simple and semi-simple algebras
      • The adjoint representation
    • The building block of all: SU(2)
      • Raising and lowering operators
      • Irreducible representations
      • Coherent states
      • Tensor products and 6j-symbols
      • Quantum spin chains
      • Cold atoms in a two-well trap
    • Wigner-Eckart theorem
      • Structure theory
        • Cartan algebra
        • Roots and weights
        • Raising and lowering operators
        • Structure constants and normalizations
        • Constructing the Lie algebras
          • Positive weights
          • Simple roots
          • A lot of SU(2)’s groups
          • The kaleidoscope of the Weyl group
          • The Dynkin diagrams
          • Examples from rank 2 algebras:
            • 1) SU(3) and eightfold way
            • 2) Being exceptional, G2
          • Unitary Groups
            • Irreducible representations and Young tableaux
            • The classical and exceptional groups
              • Ising model in a magnetic field, glimpse on E8 structure of nature
              • The Classification Theorem of semi-simple Lie algebras
                • Global and gauge symmetries in field theories
                  • Two famous bosons: Goldstone and Higgs

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