This talk is about select topics in Differential Topology, namely, oriented intersection theory and the Poincaré–Hopf Theorem. We will start by covering the preliminary topics about smooth manifolds, including derivatives/differentials, the preimage theorem, and transversality. Then we will talk about orientation of manifolds, oriented intersection numbers, and the Poincaré–Hopf Theorem.

The Poincaré–Hopf Theorem is about smooth vector fields on compact manifolds. It says that if such a vector field has only isolated zeros, then the sum of the indices of the zeros is equal to the Euler characteristic of the manifold. We will define all these terms in the talk, but one takeaway is that the types of zeros allowed on a vector field is a topological invariant.