Extrema without convexity and stability without Lyapunov

The great majority of optimization problems where there is a global minimum are convex, and a great variety of demonstrations of equilibrium point stability of nonlinear systems involve Lyapunov functions. This work illustrates alternative techniques which may allow dispensing with a convexity assumption, or dispensing with use of a Lyapunov function. The techniques are grounded in topology, in particular Morse Theory, and results of Lefschetz-Hopf and Poincare-Hopf. Illustrations are provided from the literature.