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Recent years have seen an enormous growth in the applications of topology to other disciplines from the biological sciences to materials science, and from dynamical systems to cosmology and engineering. Many of these are factored through “topological data analysis” (TDA), but not all, with notable exceptions among those from dynamics and from engineering. All of these applications are due to topology’s capacity to define precise invariants from imprecise data: topological invariants (like the winding number of a curve around a point) are usually discrete and have some stability properties (here, to arbitrary perturbations that don’t move points as much as their distance to that point) that make them attractive.
However, topological stability is quite different from ordinary statistical stability. A single outlier can completely change the apparent topology of a space. One of the ways of dealing with this, persistence, has had numerous applications within pure math in recent years (in differential geometry, group theory, and approximation theory, to name three). The study of topology of random processes, and how the randomness perturbs topology is thus arising as an important scientific issue with potentially very wide significance. This workshop will bring together workers who have been dealing with this in different settings and in different ways, which should lead to progress in both application domains, and in a longer run, on the fundamental problems.