In mathematics, stratified Morse theory is an analogue to
Morse theory
In mathematics, specifically in differential topology, Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a typical differentiab ...
for general
stratified space
In mathematics, especially in topology, a stratified space is a topological space that admits or is equipped with a stratification, a decomposition into subspaces, which are nice in some sense (e.g., smooth or flat).
A basic example is a subset o ...
s, originally developed by
Mark Goresky
Robert Mark Goresky is a Canadian mathematician who invented intersection homology with his advisor and life partner Robert MacPherson.
Career
Goresky received his Ph.D. from Brown University in 1976. His thesis, titled ''Geometric Cohomology a ...
and
Robert MacPherson. The main point of the theory is to consider functions
and consider how the stratified space
changes as the real number
changes. Morse theory of stratified spaces has uses everywhere from pure mathematics topics such as braid groups and
Lawrence–Krammer representation, representations to robot motion planning and potential theory. A popular application in pure mathematics is Morse theory on manifolds with boundary, and manifolds with corners.
See also
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Digital Morse theory
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Discrete Morse theory Discrete Morse theory is a combinatorial adaptation of Morse theory developed by Robin Forman. The theory has various practical applications in diverse fields of applied mathematics and computer science, such as configuration spaces, homology com ...
*
Level-set method
Level-set methods (LSM) are a conceptual framework for using level sets as a tool for numerical analysis of surfaces and shapes. The advantage of the level-set model is that one can perform numerical computations involving curves and surfaces o ...
References
DJVU file on Goresky's page
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Generalized manifolds
Morse theory
Singularity theory
Stratifications
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