Another tool that comes to mind is the Kohonen map (or rather, the logic behind it).

A lot of people think that applied math means engineering or physics or maybe statistics, but in our graduate program we took the same courses as the students in pure math. I think the program showed us that you have to know the math FIRST, and that provides you with a toolkit. When the applied problem comes along, you’re ready to reach in and pull out the right tool. Of course, some do that better than others, and Rob is one of the best because of his curiosity, insight, and creativity. Way to go, Rob!

Other noteworthy applied topic is in the algebraic geometry of conditional independence, something that grew out of Sturmfel’s study of combinatorial independence in matroids. Turns out every graphical model, tool heavily used in statistical machine learning, can be reasoned about as an algebraic variety and algorithms of algebraic geometry, of which there is a vast, can be applied. Jason Morton who works in this Algebraic Statistics may have also interesting things to say about unrelated field of tensor computing, i.e doing linear algebra with tensor-valued matrices and tensor decompositions (such as Mahoney’s CUR).

As to Dante there is a 1994 book about errors and inconsistencies (also spatial, one might say, topological) in the Comedy, “Mismapping the Underworld” by John Kleiner.

The best book about Computational Topology at the level we’re talking about here is Zomorodian “Topology for Computing” from 2005. Last year came out Edelsbrunner’s book targeted at similar audience, but I haven’t read it yet.