Speaker
Description
The homotopic distance D(f,g) between two continuous maps measures how far they are from being homotopic. It provides a common framework for classical invariants such as the Lusternik–Schnirelmann category and topological complexity, and it admits a natural interpretation in terms of sectional category. In this talk, I will discuss recent developments on homotopic distance, with special emphasis on its combinatorial and computational aspects. After recalling its basic properties and motivating examples, I will explain why the direct computation of D(f,g) is often difficult and how simplicial methods help to address this problem. This leads to cohomological and simplicial versions of the theory: cohomological distances provide computable lower bounds that refine the classical cup-length estimates, while simplicial cohomological distance yields a discrete model with good approximation properties. In particular, after sufficiently many barycentric subdivisions, it recovers the cohomological distance of the corresponding continuous maps. Time permitting, I will illustrate these ideas with examples related to simplicial complexity and motion-planning-type problems.