Morse theory (by Marston Morse, around 1935) analyzes any smooth manifold by
looking at an arbitrary smooth function defined on it. Discrete Morse Theory (by Robin Forman,
around 2000) is more elementary: It works on simplicial complexes, and it allows to "simplify"
them without changing the homotopy. Both theories are used in a variety of applications, for
example in shape recognition. Since every smooth manifold can be triangulated (in the sense that one can find a simplicial complex homeomorphic to it), we may compare the two theories on the common battleground of
"triangulations of manifolds". For example, both theories yield upper bounds for the Betti
numbers (via the so-called Morse inequalities). Which of the two theories gives sharper bounds?
This is joint work with Karim Adiprasito.