Our experience in theoretical work in computational topology is shown in [1], where we adapted the AT-model to calculate persistent homology. We were also concerned in [2] with the treatment of time-varying sequences of 2D binary images and the tracking of connected components over time using persistent homology. More concretely, we designed an algorithm to compute the spatiotemporal 0-dimensional persistence barcode of a spatiotemporal data by encoding it in a cell complex respecting the principle that it is not possible to move backwards in time. To this end, we modify the algorithm presented in [3] for computing AT-models and persistent homology.
Recently we have worked on block functions induced by morphisms between persistence modules [7]. This gives an invariant for understanding the relation between a pair of persistence modules by drawing relations between their corresponding barcodes. We are currently working on adapting this block function to a well-defined induced partial matching. Also, we are working on applications of this new invariant.
Persistence modules are the main algebraic structure of topological data analysis. They can be described using persistence barcodes, a discrete invariant. Morphisms between persistence barcodes are partial matchings. Understanding how morphisms between persistence modules can induce partial matchings between persistence modules satisfying that the induced partial matchings are independent of the basis fixed in the persistence modules, linear respect to sum of morphism between persistence modules, robust respect to perturbations of the morphism, and computable in polynomial time, is an open problem. Our aim is to define such induced basis-independent partial matching. Our specific goalsare the following:
Our team has a long trajectory in persistent homology and cohomology aspects and applications. Previous studies of cohomology generators in 2D images can be found in [3,4]. Computation of the cup product of cohomology generators in 3D images can be found in [3,5]. In [6] we computed a graph-with-loop structure to capture the homology based on Reeb graphs and using chain contractions (AT-models).
We plan to extend the definition of homological scaffolds to its dual cohomological version and will study its properties. We plan to prove that such definition will overcome the drawbacks presented in the current definition of homological scaffold: its arbitrary choice of representative cycles, lack of uniqueness and high complexity. Since homological scaffold has been used to highlight the geometry of obstruction patterns and this is a natural information provided by cohomology, we plan to prove that cohomological scaffold will be a complementary and efficient tool to compute key bridges for the flow information in a network. Specific goals are listed below: