Combinatorial Vector Fields for Dynamical Systems and Shape Similarity

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My general long-term goal has been to develop Computational Topology in the context of analysis and processing of multidimensional data. In the past decade, the concept of combinatorial vector field, also called discrete vector field (abbreviation DVF) introduced by Forman became a highly successful tool for discretization of continuous problems in Computational Mathematics, Imaging and Visualization. My current research proposal is focused on developing combinatorial vector fields along two axes of applications: in dynamical systems and in the study of shape recognition. The applications of DVF to dynamical systems will be based on a modification of Forman's concept of V-paths and the use of a multivalued mapping dynamics introduced in my joint 2014 work with Mrozek and Wanner. This modification permits studying both periodic and asymptotic trajectories, thus giving access to a full specter of dynamical behavior. Our combinatorial model for the Lorenz attractor shows that DVFs also provide a good framework for describing chaotic dynamics. The primary goal is to establish a one-to-one correspondence between continuous and combinatorial vector field dynamics, in particular on the level of Conley index and Morse decomposition. We expect that our first publication is a starting point for a very promising research direction. The applications of DVF to shape recognition are a continuation of my recently initiated research in collaboration with Allili and Landi. Our matching algorithm is the first tentative of adapting the DVF methods to functions with multidimensional values with the goal of speeding up the computation of multi-filtered persistent homology, that is, the subspace homology system filtered by a lattice rather than by a linear order. This structure will in turn be used for computing a refined shape similarity measure, currently bringing computational challenges. This project involves computer programming as well as a theoretical development of an appropriate extension of the Morse theory to multidimensional functions. Finally, I plan to work, jointly with Mischaikow, Mrozek, and Wanner, on a new edition of the Computational Homology book published by Springer in 2004. We were approached by Springer editors about the second edition several times but we postponed the project because the field developed so much since 2004 that this will mean writing practically a new book. As it was the case with the first edition, we expect that the process of writing the book will motivate the progress in current research and prompt for many student research projects.