Symmetry and Equivariance

Catalog description

A self-contained pedagogical introduction to Groups acting-on things like sets, modules, vector spaces, manifolds, and other groups, and to the Morphisms and Equivariant Functions between them. The course will intertwine ideas on the Structure of (mostly finite-order) Groups with those on their Representation, making judicious use of Equivalence Relations throughout. In contrast to the usual pedagogy, we will organize the course around the question of where do groups live in physics. Specific examples will vary from year to year but include: Translational and Rotational Symmetry - Isometry, orbits & stabilizers; gauges, wavefunctions, & boundary conditions - U(1) and its finite-order Cyclic subgroups, Character Theory & the Group Algebra; Bloch & Wannier Functions - Morphisms b/w group actions, Pontryagin Duality; Molecules & Bonding - Point Groups, Permutation& Induced Reps; Selection Rules & Tensors - Irreps & Schur's Lemma; Spin-1/2 and Angular Momentum - Complex Numbers, Quaternions, Projective Spaces, groups acting on n-Spheres & Covers.