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| | In mathematics, '''stratified Morse theory''' is an analogue to [[Morse theory]] for general [[topologically stratified space|stratified spaces]], originally developed by [[Mark Goresky]] and [[Robert MacPherson (mathematician)|Robert MacPherson]]. The main point of the theory is to consider functions <math>f : M \to \mathbb R</math> and consider how the stratified space <math>f^{-1}(-\infty,c]</math> changes as the real number <math>c \in \mathbb R</math> 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. |
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| | ==See also== |
| | |
| | * [[Digital Morse theory]] |
| | * [[Discrete Morse theory]] |
| | * [[Morse theory]] |
| | * [[Level set method]] |
| | |
| | ==References== |
| | * "Stratified Morse theory", by M. Goresky and R. MacPherson Springer-Verlag, Berlin, Heidelberg, New York, 1988, xiv + 272 pp. [http://www.math.ias.edu/~goresky/pdf/SMT.djvu DJVU file on Goresky's page] |
| | * D. Handron, Generalized billiard paths and Morse theory on manifolds with corners. Topology and its Applications, Volume 126, Number 1, 30 November 2002, pp. 83–118(36) |
| | * S.A. Vakhrameev, Morse lemmas for smooth functions on manifolds with corners. Dynamical systems, 8. J. Math. Sci. (New York) 100 (2000), no. 4, 2428–2445. |
| | |
| | [[Category:Singularity theory]] |
| | [[Category:Generalized manifolds]] |
| | [[Category:Morse theory]] |
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| | {{topology-stub}} |
In mathematics, stratified Morse theory is an analogue to Morse theory for general stratified spaces, originally developed by Mark Goresky 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 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
References
- "Stratified Morse theory", by M. Goresky and R. MacPherson Springer-Verlag, Berlin, Heidelberg, New York, 1988, xiv + 272 pp. DJVU file on Goresky's page
- D. Handron, Generalized billiard paths and Morse theory on manifolds with corners. Topology and its Applications, Volume 126, Number 1, 30 November 2002, pp. 83–118(36)
- S.A. Vakhrameev, Morse lemmas for smooth functions on manifolds with corners. Dynamical systems, 8. J. Math. Sci. (New York) 100 (2000), no. 4, 2428–2445.
Template:Topology-stub