REPRESENTATION THEORY AND HARMONIC ANALYSIS ON COMPACT
GROUPS

The goal of this course will be to give an introduction to the
representation theory of compact groups and its intimate connections to
harmonic analysis on these groups. A good understanding of this vast
subject is essential to many areas of mathematics, including: random matrix
theory, quantum mechanics, quantum algebra (e.g., Hopf algebras,
subfactors, tensor categories, quantum groups), and dynamics, ergodic
theory and operator algebras. This course will mainly focus on compact
groups in order to avoid some technicalities and to make the subject
approachable.  Topics covered will include: Group representations, Haar
measure, unitarizability, reducibility, Schur’s Lemma, Pontryagin duality,
non-commutative Fourier transform and the Plancharel theorem,
representations of SU (n), U (n), O(n), S_n, tensor categories and the
Tannaka-Krein reconstruction theorem.