In this talk, I will present the theory of combinatorial multivector fields for finite topological spaces, the main subject of my thesis. The idea of combinatorial vector fields came from Forman and emerged naturally from discrete Morse theory. Lately, Mrozek generalized it to the multivector fields theory for Lefschetz complexes. In our work, we simplified and extended it to the finite topological spaces settings. We developed a combinatorial counterpart for dynamical objects, such as isolated invariant sets, isolating neighbourhoods, Conley index, limit sets, and Morse decomposition. We proved the additivity property of the Conley index and the Morse inequalities. Furthermore, we applied persistence homology to study the evolution and the stability of Morse decomposition. In the last part of the talk, I will show numerical results and potential future directions from a data-analysis perspective.
