Reeb graphs provide a method for studying the shape of a manifold by encoding the evolution and arrangement of level sets of a simple Morse function defined on the manifold. Since their introduction in computer graphics they have been gaining popularity as an effective tool for shape analysis and matching. In this context one question deserving attention is whether Reeb graphs are robust agains...

In this paper, we develop Morse theory, which allows us to determine topological information about manifolds using certain real-valued functions defined on the manifolds. We first prove the Morse lemma, which says that, near critical points, such functions can be written in a useful way that gives us topological information. We then show how the homotopy type of the manifold is related to the i...

Let F : (R,0)→(R,0) and g : (R,0)→R be analytic germs, let Br ⊂ R be a closed ball of small radius centred at the origin, and let Wz = F −1(z) ∩ Br. Let λ(g;Wz) = ∑ p(−1), where λ(p) is the Morse index at a critical point p of a Morse function which approximates g|Wz. For F and g satisfying some natural assumptions, we shall show that there is a finite family of analytic germs θi(z) such that λ...

We consider operators of Kramers-Fokker-Planck type in the semi-classical limit such that the exponent of the associated Maxwellian is a Morse function with two local minima and a saddle point. Under suitable additional assumptions we establish the complete asymptotics of the exponentially small splitting between the first two eigenvalues. Résumé On considère des opérateurs du type de Kramers-F...

Concrete fracture behavior is predicted by one of multi-scaling methods, called the virtual internal bond (VIB) model. The VIB model describes the microscopic interactions between the cement pastes and aggregates using the concept of homogenization. The microscopic behavior is connected to macroscopic behavior by the Cauchy-Born rule, which results in the strain energy function. From the macros...

In this paper we use the nudged elastic band technique from computational chemistry to investigate sampled highdimensional data from a natural image database. We randomly sample 8×8 and 9×9 high-contrast patches of natural images and create a density estimator believed as a Morse function. By the Morse function we build one-dimensional cell complexes from the sampled data. Using one-dimensional...

In this paper we study topology of the variety of closed planar n-gons with given side lengths l1, . . . , ln. The moduli space M` where ` = (l1, . . . , ln), encodes the shapes of all such n-gons. We describe the Betti numbers of the moduli spaces M` as functions of the length vector ` = (l1, . . . , ln). We also find sharp upper bounds on the sum of Betti numbers of M` depending only on the n...

• Topology of G/T : Flag manifolds G/T have cohomology only in even degrees, with Euler characteristic the degree of the Weyl group. The Euler characteristic can be calculated by a Lefschetz fixed point argument (see Adams [1], pgs. 90-92 for this). Identifying the flag manifold with a co-adjoint orbit, there is a Morse theory calulation of the cohomology that goes back to Bott, for an outline ...

In [13] a homology theory –Morse-Conley-Floer homology– for isolated invariant sets of arbitrary flows on finite dimensional manifolds is developed. In this paper we investigate functoriality and duality of this homology theory. As a preliminary we investigate functoriality in Morse homology. Functoriality for Morse homology of closed manifolds is known [1, 2, 3, 8, 14], but the proofs use isom...