Topological and Combinatorial Models of Directed Path Spaces

Explore topological and combinatorial models of directed path spaces in this lecture from the Hausdorff Trimester Program on Applied and Computational Algebraic Topology. Delve into concurrency theory in Computer Science, examining methods to address the "state space explosion problem" using combinatorial and topological models. Investigate how execution paths correspond to directed paths in time-flow directed state spaces, and how d-homotopies result in equivalent computations. Learn about the importance of understanding non-reversible time-flow effects and the need to adapt algebraic topology methods. Discover techniques for inferring information about execution spaces between states, with a focus on determining path components. Examine Higher Dimensional Automata (HDA) and methods for identifying the homotopy type of execution spaces, including prodsimplicial complexes and configuration spaces. Explore the Alexander dual of configuration spaces and its implications for calculating homology groups and topological invariants. Gain insights into Ziemiański's method for identifying directed path spaces with prodpermutahedral complexes in general HDAs.