SUMS: Menzies -- More Pretty Pictures and Winding Numbers

The complex numbers are truly the Jon Stewart of mathematics.  Equipped with an
algebraic structure as an algebraically closed field, an analytic structure, and a
topological structure, all three branches of pure maths meet in the complex plane.  I
will talk about a very geometrically and analytically elegant tool of complex analysis:
winding numbers, which describe an important feature of smooth curves in the plane.
Winding numbers turn up in algebraic topology, differential geometry, and the
generalised Cauchy theorem, which I will prove.  This asserts that integrals of smooth
functions over smooth curves in the plane have very simple values that depend on the
topology of the surrounding domain.  This includes a rigorous proof of the residue
theorem.