Directed Persistence for Irreversible Phenomena: From Dihomotopy to Medical Prognosis

This project proposes a directed persistent homology framework that builds functors from time-indexed data to dihomotopy types (such as d-spaces or posets of execution traces) and derives "directed barcodes" and causal persistence landscapes sensitive to arrow-of-time effects and overlapping processes. It reinterprets filtrations as covers by past and future cones, tracking homology classes that can be created but not annihilated by forward morphisms. The novelty lies in replacing ordinary homology with directed (co)homology and establishing stability theorems tailored to monotone maps. Beyond primary sclerosing cholangitis (PSC), this framework generalizes to any progression data including disease, material fatigue, or learning dynamics, potentially sharpening predictive signals and resolving conflicts in topological data analysis by distinguishing trajectories from their time-reversals, aligning topological summaries with causal structure.

References:

1. Algebraic topology-based machine learning using MRI predicts outcomes in primary sclerosing cholangitis. Yashbir Singh, William Jons, J. Eaton, M. Vesterhus, T. Karlsen, Ida Bjoerk, A. Abildgaard, K. Jørgensen, T. Folseraas, Derek H. W. Little, A. Gulamhusein, K. Petrović, A. Negård, G. Conte, Joseph D. Sobek, J. Jagtap, S. Venkatesh, G. Gores, N. LaRusso, K. Lazaridis, Bradley J. Erickson (2022). European Radiology Experimental.
2. Directed Algebraic Topology and Concurrency. Lisbeth Fajstrup, É. Goubault, E. Haucourt, S. Mimram, M. Raussen (2016). Cambridge International Law Journal.