Morse Theory for Continuous Functionals

A New Cohomology for the Morse Theory of Strongly Indefinite Functionals on Hilbert Spaces

Take a C function f :M → R on a complete Hilbert manifold which satisfies the Palais–Smale condition. Assume that it is a Morse function, meaning that the second order differential df(x) is non-degenerate at every critical point x. Recall that the Morse index m(x, f) of a critical point x is the dimension of the maximal subspace on which df(x) is negative definite. Then the basic result of Mors...

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...

Morse Theory on Graphs

Let Γ be a finite d-valent graph and G an n-dimensional torus. An " action " of G on Γ is defined by a map which assigns to each oriented edge, e, of Γ, a one-dimensional representation of G (or, alternatively, a weight, αe, in the weight lattice of G. For the assignment, e → αe, to be a schematic description of a " G-action " , these weights have to satisfy certain compatibility conditions: th...

Morse Theory on Meshes

In this report, we discuss two papers that deal with computing Morse function on triangulated manifolds. Axen [1] gives an algorithm for computing Morse function on a triangulated manifold of arbitrary dimension but it not practical because of its space requirement. Hence, he describes an algorithm for computing critical points and their Morse indices for a 2-manifold. Edelsbrunner et al. [2] d...