The diversity of biological forms and the nature of their variation lend themselves to geometric data analysis by topological methods. These methods quantify shape by recording the values of parameters (such as height, thickness, time, distance, temperature, or curvature) across which the topology of the geometric object changes: holes emerge or collapse; connected components join or diverge; cavities form or fill. Current geometric methods of this sort can handle one varying parameter, but data often call for more than one. This project develops an algebraic framework to encode and compute in the context of this multiparameter topological data analysis. The proposed methodology is general, applicable to datasets from any scientific inquiry, but it is being developed here in service to a fundamental question in evolutionary biology: what mechanism drives the generation of topological variants in sufficient quantity for selection to act? The model organism for this investigation is the fruit fly, Drosophila melanogaster, specifically the pattern of veins in its wings.
This project develops mathematical foundations for multiparameter persistent homology. This investigation is in service to a specific question in evolutionary biology, using fruit fly wings as the model system, but the proposed methodology is general, applying homological algebra, real algebraic geometry, combinatorics, and computational techniques to encode, control, and provide insight theoretically as well as for applied and algorithmic purposes. The parameters are allowed to vary continuously on stratified spaces, instead of the usual discrete setup with one parameter on a simplicial complex. As such, the project transforms the way we think about commutative algebra in two independent ways simultaneously: by considering real vectors or an arbitrary poset instead of integer vectors for multigradings, and by splicing together the augmentation maps of free and injective resolutions to get "fringe presentations" of modules. The combination of techniques from free resolutions and injective resolutions also transforms how we think about topology and geometry, since it renders the new conception of multiparameter persistent homology transparent topologically -- features are described in terms of birth and death (generators and cogenerators) rather than by birth and relations between births (generators and relations) -- while greatly expanding the potential for effective computation.