Cohomology theories are a fundamental tools in topology and geometry which measure deep global invariants of a space, or of a geometric object on a space. Sometimes, these objects have certain points which are particularly interesting: singular points on varieties, equilibria of vector fields, fixed points of self-maps. The study of these is, by definition, a local study. Local-to-global theorems, which relate the local behavior around these special points to global information measured in the cohomology spaces, are both powerful and fascinating.

I want to talk about local-to-global theorems in the case of self-maps on compact [insert your favorite adjective] manifolds - i.e. fixed point theorems. Time permitting, I will survey these results from Brouwer's fixed point theorem to the Woods-Hole formula for coherent cohomology in smooth complex projective varieties. I will only assume some basic knowledge of manifold theory and algebraic topology.