I plan to cover the following topics, in roughly this order. The references are to the course text.

1. Representation theory of $\mathrm{GL}(n, \mathbb{C})$ and Schur functors.

   This is a "warm-up" to motivate some of the things that will come later in the course: For a complex vector space $V$, all algebraic representations of $\mathrm{GL}(V)$ are direct sums of irreducible representations. Moreover, the irreducible representations are indexed by non-increasing integer sequences of length $\mathrm{dim}(V)$. The proofs will use some facts to be established later in the course, most importantly, the Weyl integration formula. A little bit about the relation with symmetric functions. (finished - Wednesday April 2)

2. Definition of Lie groups and Lie algebras, fundamental theorems.

   Definition of a Lie group. The exponential map and the Lie algebra, basic examples. (I.2, I.3). Lie's fundamental theorem: a Lie group is locally determined by its Lie algebra and the Baker-Campbell-Hausdorff formula. Sketch of proof. (finished at start of Friday April 11))

3. Manifolds

   A closed subgroup is a smooth submanifold (I.3). Quotients (I.4). Lie algebra via vector fields; proof of Lie's theorem via integrable subbundles of the tangent bundle. (finished Wednseady April 16, with the following changes: we haven't covered quotients, but we did discuss covering groups. We'll do quotients as they come up later.)

4. Integration and the Peter-Weyl theorem.

   Definition of Haar measure and its construction for a Lie group in terms of invariant differential forms (I.5). Direct construction for a compact Lie group via averaging. Statement and proof of Peter--Weyl theorem (III.2). (Probably will be finished on Friday April 25)

5. Basic representation theory.

   Recollection of the main theorems of character theory of finite groups, with proofs that also apply to compact groups. (II, especially 3, 4). Representation theory of SU(2), SL(2), and its Lie algebra, Weyl's unitary trick (II.5). (finished)

6. Maximal tori (IV)

   The basic examples: classical groups. Definition of maximal tori, Weyl group, and examples. In a compact, connected Lie group $G$, all maximal tori are conjugate and their union is the entire group. Weyl's integration formula. (Completed as of May 7, except for Weyl's integration formula)

7. Root systems (V)

   Definition of an abstract root system. (IV.3) Definition of the root system associated to a compact Lie group $G$ and proof that it verifies the axioms of a (reduced) root system. (V.1-V.3) Classification of root systems (V.5) and the classical groups (V.6). (Expect to be completed by May 14)

8. Highest weight theory (VI)

   Highest weight theory: irreducible $G$-representations are in bijection with Weyl-orbits in the character group of a maximal torus.

9. The existence and uniqueness theorems.

   Compact groups are classified by their root systems (plus a tiny bit more, to keep track of center); for each root system there is a corresponding compact group. We will not prove these, but, as time permits, we'll talk about some ideas that go into the proofs.