This article grew out of several talks that the author presented at the Banach Institute and at the University of Bialystok in Poland during November of 2001. It describes six problems from the geometry of submanifolds. Some of the problems come from the theory of constant curvature submanifolds in Euclidean space, as well as applications of Morse theory of the height function to the problem of...

We present an algorithm for computing the main topological characteristics of three-dimensional bodies. The algorithm is based on a dis-cretization of Morse theory and uses discrete analogs of smooth functions with only nondegenerate (Morse) and the simplest degenerate critical points.

Keywords: Discrete Homological nontrivial critical point Morse theory Local linking a b s t r a c t In this paper, the existence and multiplicity results of solutions are obtained for the discrete nonlinear two point boundary value problem (BVP) ÀD 2 uðk À 1Þ ¼ kf ðk; uðkÞÞ k 2 þ is a parameter. By using the critical point theory and Morse theory, we obtain that the above (BVP) has solutions fo...

I develop algebraic-topological theories, algorithms and software for the analysis of nonlinear data and complex systems arising in various scientific contexts. In particular, I employ discrete Morse-theoretic techniques to substantially compress cell complexes built around the input data without modifying their core topological properties. Recently, I have generalized discrete Morse theory its...

Let M be a compact oriented simply-connected manifold of dimension at least 8. Assume M is equipped with a torsion-free semi-free circle action with isolated fixed points. We prove M has a perfect invariant Morse-Smale function. The major ingredient in the proof is a new cancellation theorem for the invariant Morse theory.

A gentle introduction to stratified Morse theory and Kashiwara’s conormal cycle.

The travel time brachistochrone curves in a general relativistic framework are timelike curves, satisfying a suitable conservation law with respect to a an observer field, that are stationary points of the travel time functional. In this paper we develop a global variational theory for brachistochrones joining an event p and the worldline of an observer γ in a stationary spacetime M. More speci...

Several algorithms for generating two-dimensional unstructured meshes use triangle insertion and removal as their basic operations. This work presents a robust mathematical framework, based on Morse Theory, which allows full control of the topological changes caused by the insertion or removal of triangles into 2D meshes. Topological control is ensured by a set of Morse operators and simplifies...