In physics and data analysis, where an observable is typically a function Xf nX \overset{f}{\longrightarrow} \mathbb{R}^n from a given data set XX to the set of real numbers 1\mathbb{R}^1, or to an n-tuple of such (varying in a Cartesian space n\mathbb{R}^n), an iso-hypersurface or level-hypersuface is a level set of this observable, hence the subset of XX consisting all those points xXx \in X on which this observable ff takes the same given value f(x)=cf(x) = c.

For example, if XX is a model for the Earth’s atmosphere, and if ff assigns to each point xXx \in X the atmospheric pressure p(x)p(x) at this point – in some suitable physical units and to some suitable approximation –, then an iso-surface for pp 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 ff need to be satisfied, such as that ff is a differentiable function to suitable degree, and, crucially, that its value cc (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 n\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 cc of the observable ff 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 ff in the nn-Cohomotopy theory of the data set XX.

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

Cohomotopy in topological data analysis

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):

Created on February 12, 2021 at 06:13:12. See the history of this page for a list of all contributions to it.