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Henry Adams

Department of Mathematics

Talks

Upcoming talks, mini-symposia, and conferences

Recent talks

  • Connectivity of Vietoris-Rips complexes of spheres at the Topology & Dynamics Seminar, University of Florida, Apr 2024. [Notes]

    We show how to control the homotopy connectivity of Vietoris-Rips complexes of spheres in terms of coverings of spheres and projective spaces. Let Sn be the n-sphere of diameter π, equipped with the geodesic metric, and let δ>0. Suppose that the first nontrivial homotopy group of the Vietoris-Rips complex of the n-sphere at scale π-δ occurs in dimension k, i.e., suppose that the connectivity is k-1. Then there exist 2k+2 balls of radius δ that cover Sn, and no set of k balls of radius δ/2 cover the projective space RPn. Joint work with Johnathan Bush and Žiga Virk.

  • Hausdorff vs Gromov-Hausdorff distances, Mid-Atlantic Topology Conference, Northeastern University, Boston, MA, Mar 2024.
    [Slides, Related Video]

    Abstract: The goal of this talk is to show how tools from topology can be used to bound quantities arising in metric geometry. I’ll begin by introducing the Hausdorff and Gromov-Hausdorff distances, which are ways to measure the “distance” between two metric spaces. Though Hausdorff distances are easy to compute, Gromov-Hausdorff distances are not. In the setting when X is a sufficiently dense subset of a closed Riemannian manifold M, we show how to lower bound the Gromov-Hausdorff distance between X and M by 1/2 the Hausdorff distance between them. The constant 1/2 can be improved depending on the dimension and curvature of the manifold, and obtains the optimal value 1 in the case of the circle. Our proofs begin by converting discontinuous functions between metric spaces into simplicial maps between nerve complexes. We then produce topological obstructions to the existence of such maps using the nerve lemma and the fundamental class of the manifold. Joint with Florian Frick, Sushovan Majhi, Nicholas McBride, available at https://arxiv.org/abs/2309.16648.

  • An introduction to applied topology, STEMinar Series, Daytona State College, Daytona Beach, FL, Feb 2024. [Slides]

  • Hausdorff vs Gromov-Hausdorff distances, AMS Special Session on Discrete Homotopy Theory, Joint Meetings, San Francisco, CA, Jan 2024. [Slides, Related Video]

    Abstract: Though Gromov-Hausdorff distances between metric spaces are a common concept in geometry and data analysis, these distances are hard to compute. If X is a sufficiently dense subset of a closed Riemannian manifold M, then we lower bound the Gromov-Hausdorff distance between X and M by 1/2 the Hausdorff distance between them. The constant 1/2 can be improved depending on the dimension and curvature of the manifold, and obtains the optimal value 1 in the case of the circle. We also provide versions lower bounding the Gromov-Hausdorff distance between two subsets X and Y of the manifold M. Our proofs begin by converting discontinuous functions between metric spaces into simplicial maps between Čech complexes. We then produce topological obstructions to the existence of such maps using the nerve lemma and the fundamental class of the manifold. Joint with Florian Frick, Sushovan Majhi, Nicholas McBride.

  • Hausdorff vs Gromov-Hausdorff distances, SIAM Texas-Louisiana Sectional Meeting, Minisymposium on Explorations in Topological Data Analysis, online, Nov 2023. [Slides, Related Video]

    Abstract: Though Gromov-Hausdorff distances between metric spaces are a common concept in geometry and data analysis, these distances are hard to compute. If X is a sufficiently dense subset of a closed Riemannian manifold M, then we lower bound the Gromov-Hausdorff distance between X and M by 1/2 the Hausdorff distance between them. The constant 1/2 can be improved depending on the dimension and curvature of the manifold, and obtains the optimal value 1 in the case of the circle. We also provide versions lower bounding the Gromov-Hausdorff distance between two subsets X and Y of the manifold M. Our proofs begin by converting discontinuous functions between metric spaces into simplicial maps between Čech complexes. We then produce topological obstructions to the existence of such maps using the nerve lemma and the fundamental class of the manifold. Joint with Florian Frick, Sushovan Majhi, Nicholas McBride.

  • Bridging metric geometry and topology, Math Department Colloquium, Florida State University, Tallahassee, FL, Nov 2023. [Slides, Related Video]

    Abstract: The goal of this talk is to show how tools from topology can be used to bound quantities arising in metric geometry. I’ll begin by introducing the Gromov-Hausdorff distance, which is a way to measure the “distance” between two metric spaces. Next, I will explain the nerve lemma, which says when a cover of a space faithfully encodes the shape of that space. Then, I’ll use the nerve lemma to lower bound the Gromov-Hausdorff distance between a manifold and a finite subset thereof. I’ll conclude by advertising a few other problems at the intersection of metric geometry and topology.

  • Topology in Machine Learning, Frontiers in AI Technology, Artificial Intelligence Days, University of Florida, Oct 2023. [Slides, Related Video]

  • Evasion paths in mobile sensor networks, University Math Society, University of Florida, Oct 2023. [Slides, Related Video I, II, III, IV]

    Abstract: Suppose ball-shaped sensors are scattered in a bounded domain. Unfortunately the sensors don’t know their locations (they’re not equipped with GPS), and instead only measure which sensors overlap each other. Can you use this connectivity data to determine if the sensors cover the entire domain? I will explain how tools from topology allow you to address this coverage problem. Suppose now that the sensors are moving; an evasion path exists if a moving intruder can avoid overlapping with any sensor. Can you use the time-varying connectivity data of the sensor network to decide if an evasion path exists? Interestingly, there is no method that gives an if-and-only-if condition for the existence of an evasion path, but I will advertise follow-up questions that remain open!

  • Fair Division, Applied Topology Reading Group, University of Florida, Sep 2023. [Slides, Related Video]

  • Hausdorff vs Gromov-Hausdorff distances at the Topology & Dynamics Seminar, University of Florida, Sep 2023. [Notes, Related Video]

    Abstract: Though Gromov-Hausdorff distances between metric spaces are a common concept in geometry and data analysis, these distances are hard to compute. If X is a sufficiently dense subset of a closed Riemannian manifold M, then we lower bound the Gromov-Hausdorff distance between X and M by 1/2 the Hausdorff distance between them. The constant 1/2 can be improved depending on the dimension and curvature of the manifold, and obtains the optimal value 1 in the case of the circle. We also provide versions lower bounding the Gromov-Hausdorff distance between two subsets X and Y of the manifold M. Our proofs begin by converting discontinuous functions between metric spaces into simplicial maps between Čech complexes. We then produce topological obstructions to the existence of such maps using the nerve lemma and the fundamental class of the manifold. Joint with Florian Frick, Sushovan Majhi, Nicholas McBride.

  • Bridging metric geometry and topology at the Simple Words Seminar, University of Florida, Sep 2023. [Slides, Related Video]

    Abstract: I will describe how tools from topology can be used to bound quantities arising in metric geometry. I’ll begin by introducing the Gromov-Hausdorff distance, which is a way to measure the “distance” between two metric spaces. Next, I will explain the nerve lemma, which says when a cover of a space faithfully encodes the shape of that space. Then, I’ll use the nerve lemma to lower bound the Gromov-Hausdorff distance between a manifold and a finite subset thereof. I’ll conclude by advertising a few other problems at the intersection of metric geometry and topology.

Past talks

Until my University of Florida webpage is setup, please see my prior webpage with information about my past talks: https://www.math.colostate.edu/~adams/talks.

Recent mini-symposium or conference organization

Past mini-symposium or conference organization

Until my University of Florida webpage is setup, please see my prior webpage with information about mini-symposium and conference organization: https://www.math.colostate.edu/~adams/talks.