Event Type:

Geometry-Topology Seminar

Date/Time:

Monday, October 23, 2017 - 12:00 to 13:00

Location:

Stag 162

Local Speaker:

Abstract:

Given a real polynomial mapping $F:\mathbb{R}^n\to\mathbb{R}$, a natural question is to determine the topology of the fiber $F^{-1}(0)$. It turns out the Euler characteristic of such fibers can be formulated in terms of the local degree of a certain gradient field. This fact is ultimately a consequence of the Lefschetz-Hopf theorem, from which one deduces the Poincar\'e-Hopf theorem. I'll cover these classic theorems for context, and discuss how this machinery allows for concrete computations.