The Gauss-Bonnet Theorem
Bart van den Dries (Universiteit Utrecht)
The Gauss-Bonnet theorem gives a relation between the topology and the geometry of smooth surfaces: roughly, it states that the total curvature of a surface is fixed by its topology. This is remarkable: whereas geometry is all about distances, angles, curvature etc., topology is not about these things at all. Yet the topology of a surface puts constraints on the geometry (and vice versa) in a way that the Gauss-Bonnet theorem makes precise.
A large part of the talk will be devoted to the concept of (Gauss-)curvature of a surface; this is the main ingredient of the theorem. In the rest of the talk I will give some nice applications and, if time permits, say some words on the proof. Anyone with some basic knowledge of calculus and topology should be able to follow the talk.
