  The following two loosely connected sets of topics are reviewed in these
lecture notes: 1) Gauge invariance, its treatment in field theories and its
implications for internal symmetries and edge states such as those in the
quantum Hall effect. 2) Quantisation on multiply connected spaces and a
topological proof the spin-statistics theorem which avoids quantum field theory
and relativity. Under 1), after explaining the meaning of gauge invariance and
the theory of constraints, we discuss boundary conditions on gauge
transformations and the definition of internal symmetries in gauge field
theories. We then show how the edge states in the quantum Hall effect can be
derived from the Chern-Simons action using the preceding ideas. Under 2), after
explaining the significance of fibre bundles for quantum physics, we review
quantisation on multiply connected spaces in detail, explaining also
mathematical ideas such as those of the universal covering space and the
fundamental group. These ideas are then used to prove the aforementioned
topological spin-statistics theorem.e of the universal covering space and the
fundamental group.
