I- LIE GROUPS

• Geometry of the exponential map
• Fundamental group and covering groups
• Group actions, invariant metrics, homogeneous spaces
• The isometry groups of Riemannian space forms
• Invariant integration on compact Lie groups

• Low-dimensional Lie algebras
• Nilpotent and solvable Lie algebras/Lie groups
• Structure of complex semisimple Lie algebras
• Representations of  sl(2,C), su(2), SO(3,R) and SU(2)
• Representations of sl(n,C) and so(n,C)

• Characters and irreducible representations; Peter-Weyl theorem
• Tensor products, Symmetric groups and Weyl's construction
• Representations of SU(n) and SO(n)
• Weyl's character formula; dimension and multiplicity formulas
• Complex homogeneous spaces;  Borel-Weil-Bott description of representations

• Structure of real semisimple Lie groups
• Riemannian symmetric spaces
• Unitary representations of  SL(2,R) and of the Poincare' group (outline)