In physics and data analysis, where an observable is typically a function $X \overset{f}{\longrightarrow} \mathbb{R}^n$ from a given data set $X$ to the set of real numbers $\mathbb{R}^1$, or to an n-tuple of such (varying in a Cartesian space $\mathbb{R}^n$), an iso-hypersurface or level-hypersuface is a level set of this observable, hence the subset of $X$ consisting all those points $x \in X$ on which this observable $f$ takes the same given value $f(x) = c$.
For example, if $X$ is a model for the Earth’s atmosphere, and if $f$ assigns to each point $x \in X$ the atmospheric pressure $p(x)$ at this point – in some suitable physical units and to some suitable approximation –, then an iso-surface for $p$ 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 $f$ need to be satisfied, such as that $f$ is a differentiable function to suitable degree, and, crucially, that its value $c$ (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 $\mathbb{R}^n$ is known as part of the Pontryagin construction.
Curiously, this means thatPontryagin's theorem applies to iso-hypersurfaces, saying here that, as the value $c$ of the observable $f$ 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 $f$ in the $n$-Cohomotopy theory of the data set $X$.
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
The suggestion to regard cobordism theory of iso-hypersurfaces and thus Pontryagin's theorem in Cohomotopy as a tool in (persistent) topological data analysis (improving on homologuical well groups):
Peter Franek, Marek Krčál, On Computability and Triviality of Well Groups, Discrete Comput Geom (2016) 56: 126 (arXiv:1501.03641, doi:10.1007/s00454-016-9794-2)
Peter Franek, Marek Krčál, Persistence of Zero Sets, Homology, Homotopy and Applications, Volume 19 (2017) Number 2 (arXiv:1507.04310, doi:10.4310/HHA.2017.v19.n2.a16)
Peter Franek, Marek Krčál, Cohomotopy groups capture robust Properties of Zero Sets via Homotopy Theory, talk at ACAT meeting 2015 (pfd slides)
Peter Franek, Marek Krčál, Hubert Wagner, Solving equations and optimization problems with uncertainty, J Appl. and Comput. Topology (2018) 1: 297 (arxiv:1607.06344, doi:10.1007/s41468-017-0009-6)
Created on February 12, 2021 at 06:13:12. See the history of this page for a list of all contributions to it.