nLab isosurface

Idea

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).

Examples

• isobar

References

General

See also

• Wikipedia, Isosurface

Cohomotopy in topological data analysis

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)