(1) Representations of a finite group G
(a) Groups, actions of groups on sets, natural constructions (depending on the audience).
(b) Basic definitions: representation of a group G, morphisms of representations.
(c) Irreducible representations. Schurs lemmas.
(d) Natural constructions with representations.
(e) Complete reducibility.
(f) Intertwining numbers and their properties.
(g) Decomposition of the regular representation.
(h) Group algebra and its structure.
(i) Burnside theorem and its corollaries.
(j) Characters, Orthogonality relations. Character rings.
(n) Representations of finite abelian groups. Fourier transform.
(k) Brauers theorem (optional)
(l) Restriction and induction
(m) Mackey theory
(2) Some results about representations of topological groups.
(a) Representations of commutative groups and Fourier transform.
(b) Basic results about representations of the compact group G = SO(3).
(3) Representations of general compact groups
(a) Basic definitions and properties
(b) Peter-Weyl theorem
(4) Lie groups and Lie algebras
(5) Representations of compact Lie groups via representations of Lie algebras (if time permits)
(6) Representations of general Lie groups (if time permits)
(a) Lie groups and Lie algebras
(b) The space of smooth vectors, Garding theorem on density, Dixmier-Malliavin theorem, the
action of the Lie algebra
(c) Cocompact subgroups, smooth induction
(d) Representations of SL(2,C)