group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
constructive mathematics, realizability, computability
propositions as types, proofs as programs, computational trinitarianism
Persistent homology is a homology theory adapted to a computational context, for instance, in analysis of large data sets. It keeps track of homology classes which stay ‘persistent’ when the approximate image of a space gets refined to higher resolutions.
We suppose given a ‘data cloud of samples’, $P\subset \mathbb{R}^m$, from some space $X$, yielding a simplicial complex $S_\rho(X)$ for each $\rho \gt 0$ via one of the family of simplicial complex approximation methods that are listed below (TO BE ADDED). For these, the important idea to retain is that if $\rho \lt \rho^\prime$, then
so we get a ‘filtration structure’ on the complex.
The idea of persistent homology is to look for features that persist for some range of parameter values. Typically a feature, such as a hole, will initially not be observed, then will appear, and after a range of values of the parameter it will disappear again. A typical feature will be a Betti number of the complex, $S_\rho(X)$, which then will vary with the parameter $\rho$.
One can compute intervals for homological features algorithmically over field coefficients and software packages are available for this purpose. See for instance Perseus.
A clear introduction to the use of persistent homology in data analysis is
Other references:
Last revised on August 25, 2017 at 16:24:38. See the history of this page for a list of all contributions to it.