Function taxonomy based on contour topology

Question

I only have a basic knowledge of topology so please correct any false assumptions. It seems that certrain functions have some similar (homotopic) contour curves, eg the gaussian and $-x^2$ have some homotopic contours around the global max. Has there been any research regarding such a taxonomy of n-ary functions? Is it even interesting?

Answer 1

Nothing terribly sophisticated. Any $C^2$ function near a nondegenerate critical point will be locally like a quadric. See Second partial derivative test. In the degenerate case, add higher order terms.