homology

Not only did she basically invent homology groups as we know them today, but she revolutionized the way we think about rings.

How should one explain singular homology to someone without strong background in topology, algebra and pure math itself?

Out of curiosity what criteria did you use to indicate a homology element was persistent?

Topologists like simplicial homology because it "only" requires that your input data is given in the form of a simplicial complex.

They start with some basic graph theory and build on that to give a solid overview of algebraic topology, persistent homology, and even some Morse theory.

I don't think you can make a strong statement about homology between genomes using only 65,000 bases of sequence from potentially heavily degraded sample.

Their calculation is usually done by something that is called "simplicial homology" in algebraic topology.

For instance I'm guessing that any mathematician will understand "immersed surfaces in 4-manifolds" and maybe even "homology cylinder" even if the layman won't.

By this metric, singular homology is certainly useful, because it enables us to create clean and elegant proofs of some highly non-trivial and interesting facts, but I cannot imagine anything more useless in real life.

Sage aims to include tools for computations in all parts of mathematics, including all of the above but also more advanced things like group theory, representation theory, homological algebra, commutative algebra, homology of simplicial complexes, graph theory, Galois theory, algebraic number theory, etc.