Bartosz Furmanek | Searcharxiv

arXiv2026

Topological simplification guided by forbidden regions

Topological simplification is the process of reducing complexity of a function while maintaining its essential features. Its goal is to find a new filter function, which reorders cells of the input complex in a way which eliminates some persistent homological features, without affecting the rest. We present a new approach to simplification based on the concept of forbidden regions and combinatorial dynamics. It allows us to reorder and cancel critical values, whose cancellation is not possible using existing methods because they are not consecutive in the total order. Each such cancellation takes O(c$\cdot$n) time in the worst case, where c is the number of birth-death pairs and n is the size of the input complex.

math.AT↗

arXiv2025

Linear Relations of Finite Length Modules are Shift Equivalent to Maps

Linear relations, defined as submodules of the direct sum of two modules, can be viewed as objects that carry dynamical information and reflect the inherent uncertainty of sampled dynamics. These objects also provide an algebraic structure that enables the definition of subtle invariants for dynamical systems. In this paper, we prove that linear relations defined on modules of finite length are shift equivalent to bijective mappings.

math.DS↗