Starting with very concrete examples of symmetry groups, the course will develop, in an interactive manner based on examples, the theory of group representations and its relations to different areas of mathematics and physics.
Representation theory studies the ways in which a group acts on vector spaces.
The course will emphasize the way in which the set of such representations has itself a group-like structure. There is a deep connection between representation theory and the structure of 3-dimensional space. Structure coefficients coming from representation theory can be used to define topological invariants of knots and 3-dimensional manifolds. These invariants are related to current attempts to model quantum gravity in mathematical physics. The course will start from the representation theory of the dihedral group, which is the symmetry group of a regular n-gon.