Morse theory


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Morse theory

[′mȯrs ‚thē·ə·rē]
(mathematics)
The study of differentiable mappings of differentiable manifolds, which by examining critical points shows how manifolds can be constructed from one another.
References in periodicals archive ?
Writing in the area of differential geometry at the graduate level, Moore gives foundations for a partial Morse theory of minimal surfaces in Riemannian manifolds.
On one hand, there have been many approaches to study periodic solutions of differential equations or difference equations, such as critical point theory (which includes the minimax theory, the Kaplan-Yorke method, and Morse theory), fixed point theory, and coincidence theory; see, for example, [4-20].
To this end, we need to mildly generalize some elements of Discrete Morse Theory in order to be able to work with nonregular CW-complexes or, correspondingly, face categories that are not posets (see Section 5.3).
The topological model built in this way is analogous to how Morse theory is used to construct a cell decomposition of a manifold using sublevel sets of a Morse function on the manifold [5-7].
Our approach combines variational techniques based on the critical point theory, together with Morse theory. We prove two multiplicity theorems.
The topics include some algebraic methods in semi-Markov chains, statistical topology through Morse theory persistence and non-parametric estimation, structural properties of the generalized Dirichlet distributions, projections on invariant subspaces, combining statistical models, regular fractions and indicator polynomials, and some hypothesis tests for Wishart models on symmetric cones.
Knudson bases his introduction to Morse theory on John Milnor's Morse Theory and Yukio Matsumoto's An Introduction to Morse Theory, both of which he considers elegant masterpieces.
The proof follows the same outline as the previous study except that we use discrete Morse theory to determine the homotopy type of the associated complexes of ordered set partitions.
The authors work with a new sequence of eigenvalues that employs the cohomological index to systematically develop alternative tools such as nonlinear linking and local splitting theories to apply Morse theory to quasilinear problems.
Lemma 3.2 is closely related to results of Jollenbeck-Welker, of Skoldberg, and of Kozlov (see (3), (8), and (4)) developing algebraic versions of discrete Morse theory.
They then move briskly through Toponogov's theorem, homogeneous spaces, Morse theory, closed deodesics and the cut locus, the sphere theorem and its generalizations, the differentiable space theorem, complete manifolds of non-negative curvature, and compact manifolds of non-positive curvature.
Other topics include automated reverse engineering of free form objects using Morse theory, dense depth and color acquisition of repetitive motions, examplar-based shape from shading, and generalized MPU implicits using belief propagation.