Abstract: For any compact surface without boundary, the classical Gauss-Bonnet theorem
asserts that the integral of the curvature equals a universal constant times the Euler
characteristic. I will sketch several proofs of this theorem, in varying degrees of generality,
and will also discuss a generalization that applies to surfaces-with-boundary. After
indicating some analogs of this result for higher-dimensional
manifolds, I will go on to briefly discuss other relations between curvature and topology
in higher dimensions. I will then close by mentioning a few of the ways
that Stony Brook has contributed to the development of the subject.
