In physics and data analysis, where an observable is typically a function from a given data set to the set of real numbers , or to an n-tuple of such (varying in a Cartesian space ), an iso-hypersurface or level-hypersuface is a level set of this observable, hence the subset of consisting all those points on which this observable takes the same given value .
For example, if is a model for the Earth’s atmosphere, and if assigns to each point the atmospheric pressure at this point – in some suitable physical units and to some suitable approximation –, then an iso-surface for is an isobar: a surface of constant pressure.
For such a level set to actually be a hypersurface, hence a differentiable/smooth submanifold, some regularity conditions on need to be satisfied, such as that is a differentiable function to suitable degree, and, crucially, that its value (whose pre-image is formed) is a regular value.
Phrased this way, the construction of iso-hypersurfaces turns out to be a central topic also in areas of pure mathematics, such as in differential topology and cobordism theory, where the formation of pre-images of regular values inside is known as part of the Pontryagin construction.
Curiously, this means thatPontryagin's theorem applies to iso-hypersurfaces, saying here that, as the value of the observable varies, the shape (topology) of the corresponding iso-hypersurfaces changes (at most) by a cobordism, and that the resulting (normally framed) cobordism class of all these hypersurfaces corresponds to the class of in the -Cohomotopy theory of the data set .
While Pontryagin’s theorem is ancient, the idea that it thus implies the possibility of topological data analysis via Cohomotopy theory of observables and cobordism theory of their iso-hypersurfaces appears only recently (Franek-Krčál 16, Franek-Krčál 17).
See also
Introducing persistent cohomotopy as a tool in topological data analysis, improving on the use of well groups from persistent homology:
Peter Franek, Marek Krčál, On Computability and Triviality of Well Groups, Discrete Comput Geom 56 (2016) 126 (arXiv:1501.03641, doi:10.1007/s00454-016-9794-2)
Peter Franek, Marek Krčál, Persistence of Zero Sets, Homology, Homotopy and Applications, 19 2 (2017) (arXiv:1507.04310, doi:10.4310/HHA.2017.v19.n2.a16)
Peter Franek, Marek Krčál, Hubert Wagner, Solving equations and optimization problems with uncertainty, J Appl. and Comput. Topology 1 (2018) 297 (arxiv:1607.06344, doi:10.1007/s41468-017-0009-6)
Review:
Peter Franek, Marek Krčál, Cohomotopy groups capture robust Properties of Zero Sets via Homotopy Theory, talk at ACAT meeting 2015 pdf
Urs Schreiber on joint work with Hisham Sati: New Foundations for TDA – Cohomotopy, (May 2022)
Created on February 12, 2021 at 11:13:12. See the history of this page for a list of all contributions to it.