THREE EXAMPLES OF APPLIED & COMPUTATIONAL HOMOLOGY

Mathematics is limitless in its dual capacity for abstraction and incarnation. To a large degree, many of the modern revolutions in technology and information rest on piers of mathematics that assist, inform, or otherwise catalyze progress. It appears that those branches of mathematics which are most easily understood and communicated are precisely those which find greatest applicability in the modern world. To conclude from this that deeper or more difficult fields are inherently less applicable would be premature. Consider for example the utility of algebraic topology. Long cloistered behind formal and categorical walls, this branch of mathematics has been the source of little in the way of concrete applications, as compares with its more analytic or combinatorial cousins. In this author’s opinion, this is not due to a fundamental lack of applicability so much as to (1) the lack of a motivating exposition of the tools for practitioners; and (2) an historical lack of emphasis on computational features of the theory. These two issues are coupled. Advances which demonstrate the utility of a topological theory spur the need for good computation. Good algorithms for computing topological data spur the search for further applications. Algebraic topology is the mathematics that arises in the attempt to describe the global features of a space via local data. That such tools have utility in applied problems concerning large data sets is not difficult to argue. To give a sense of what is possible, we sketch three recent examples of specific applications of homological tools. This list is neither inclusive nor ranked: these examples were chosen for concreteness, simplicity, and timeliness. This brief and woefully incomplete sketch is meant