Morse theory for two functions

Morse theory for analytic functions on surfaces

In this paper we deal with analytic functions f : S → R defined on a compact two dimensional Riemannian surface S whose critical points are semi degenerated (critical points having a non identically vanishing Hessian). To any element p of the set of semi degenerated critical points Q we assign an unique index which can take the values −1, 0 or 1, and prove that Q is made up of finitely many (cr...

Morse Functions of the Two Sphere

We count how many ”different” Morse functions exist on the 2-sphere. There are several ways of declaring that two Morse functions f and g are ”indistinguishable” but we concentrate only two natural equivalence relations: homological (when the regular sublevel sets f and g have identical Betti numbers), and geometric (when f is obtained from g via global, orientation preserving changes of coordi...

Two-orbit Kähler Manifolds and Morse Theory

We deal with compact Kähler manifolds M acted on by a compact Lie group K of isometries, whose complexification K has exactly one open and one closed orbit in M . If the K-action is Hamiltonian, we obtain results on the cohomology and the K-equivariant cohomology of M .

Morse Theory

Discrete Morse Theory for Filtrations

Acknowledgements Enumerating all the ways in which I am grateful to Konstantin would essentially double the length of this document, so I'll save all that stuff for my autobiography. But I will note here that he was simultaneously patient, engaged, proactive, and – best of all – ruthlessly determined to refine and sculpt all our vague big-picture ideas into digestible and implementable concrete...