Type: Seminar

Distribution: World

Expiry: 12 Aug 2010

Auth: yzhang(.pmgrad;1504.2007)@p494.pc.maths.usyd.edu.au

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.