Curso - 30 aulas.

Part I: In the first part, we discuss geometric properties of Lie groups and Lie algebras including the classification of semisimple Lie algebras (root space decomposition and Dynkin diagrams). Part II: In the second part we cover the basics of representation theory of semisimple Lie algebras and of compact Lie groups. In both cases, a geometric view point will be emphasized, applying algebraic properties of the Lie algebra to geometric properties of the underlying Lie group. In part II we will e.g. discuss both real and complex representations of compact Lie groups as an application of the usual theory of complex representations of complex semisimple Lie algebras. Part III: If time permits, we will also discuss the theory of symmetric spaces, which are basic examples not only in geometry, but in many areas of mathematics. Unlike Helgason, the basic reference for the subject, the emphasis will be on the geometric aspects as opposed to algebraic ones.