Gromov's Filling Radius and Persistent Homology
This course surveys the geometry and topology of metric spaces through the lens of Gromov’s (homological) filling radius and its connections to systolic inequalities and various notions of width. We will develop the basic definitions, prove foundational properties, and explore rigidity and stability phenomena that link Riemannian geometry, quantitative topology, and applied topology.
We begin with a review of the Gromov–Hausdorff distance, the topology it induces on the class of compact metric spaces, and Gromov’s precompactness theorem. We then develop the theory of Gromov’s filling radius $\mathrm{FillRad}(X)$ (via homological fillings in a Banach ambient space) and compare it to systolic invariants (e.g., $\operatorname{sys}_1$ on essential manifolds) and to width-type quantities (e.g., Urysohn width). Along the way we highlight sharp inequalities and extremal examples.
A core theme is rigidity. We study Wilhelm’s rigidity theorem for positively curved manifolds and view it as an analogue—at the level of filling radius—of Bonnet–Myers–type diameter/curvature bounds. We examine equality and near-equality cases and what they force about the underlying metric and topology.
In the second half, we connect classical filling ideas to persistent homology. Working mainly with Vietoris–Rips filtrations, we show how $\mathrm{FillRad}$ controls—and is reflected in—the birth–death scales of homology classes. We discuss interleaving stability, Gromov–Hausdorff continuity of barcodes, and quantitative estimates that relate filling radii, systoles, and Rips persistence. Examples include spheres, projective spaces, flat tori, and selected non-manifold metric spaces.
Some background in differential geometry at the level of basic Riemannian geometry, algebraic topology (homology/cohomology), and metric geometry basics. Familiarity with persistent homology is helpful but not required.