Applications of homological algebra in the sciences and engineering

Question

I am looking for elementary applications/examples of the usage of homological algebra in the sciences/engineering. Ideally I am looking for examples that would be accessible to bright undergraduate students with a bit of background in the relevant area.

For example, John Baez gives a nice application to circuits in his blog posts here, here, and here, and there is a Youtube video touching on this as well.

Answer 1

Persistent homology in topological data analysis might be such an example.

For example, given a finite point cloud $X$ in a metric space $M$ (for example $\mathbb{R}^d$), we can approximate the point cloud with an $\mathbb{R}_{\geq 0}$-filtration of (abstract) simplicial complexes called the Vietoris-Rips complexes which we denote with $(\text{Rips}_r(X))_{r\in\mathbb{R_{\geq 0}}}$. $\text{Rips}_r(X)$ is defined by saying that there is an abstract $n$-simplex between points $x_1,...,x_n\in X$ if the distance between any two of those points is less or equal to $r$.

We then consider the simplicial homology $H^*(\text{Rips}_r(X);A)$ as $r$ varies from $0$ to $\infty$. During this, cycles will appear and disappear as the simplicial complex goes from being a discrete point cloud to a contractible simplex. For each of those cycles, we could draw a timeline of their appearance and disappearance yielding a so-called persistence barcode diagram.

As this simple example illustrates, we can use homology to gain some probabilistic information on the underlying topology of a data set. Of course, this is only scratching the surface of persistent homology, and there are many good resources on the internet for further study.

Answer 2

Tony Phillips' article Topology and Elementary Electric Circuit Theory, I explains how homology can be used to understand laws of electrical circuits.

Answer 3

Homological algebra of commutative rings --- Koszul complexes and pieces of spectral sequences and all that, plus a generous amount of the part of commutative algebra that deals with Exts and Tors --- is used in constructing algorithms for finding implicit equations for curves and surfaces given in parametric form. This is used in computer aided graphical industrial design and such worldly things.

Google should find Marc Dohm's PhD thesis, which deals with the case of surfaces, and if I recall correctly the introduction talks also about curves and surely has references to the relevant work.

This is very nice mathematics.