Course Description:   Metric spaces: continuity, connectedness, compactness.  Paths and homotopies.  The winding number and covering spaces.

Interpretations of the winding number in algebraic, analytic, topological and piecewise linear contexts.

Applications of the winding number, selected from: the fundamental theorem of algebra, the Brouwer and Borsuk-Ulam theorems the ham sandwich theorem, complex integration, the Riemann-Hurwitz formula, the Toeplitz index theorem, finding potentials for vector fields, Berry’s phase, winding and rotation numbers and applications to polygons, Jordan curve theorem, the Big Picard theorem (via Ahlfors’ theory of Uberlagerungsflachen).

Generalizations of the winding number, as time permits: Degree theory. The Bott Periodicity Theorem.

Prerequisites:  The main prerequisite for this class is a knowledge of Analysis (basic properties of real numbers, the \(\epsilon – \delta\) definition of continuity, the Heine-Borel theorem and so on.  At some points we’ll also need ideas from Calculus III (Gauss-Green-Stokes), linear algebra (matrices and determinants), and abstract algebra (group theory).