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Henry Adams

Department of Mathematics

Talks

Upcoming talks, mini-symposia, and conferences

Recent invited talks

  • Vietoris-Rips complexes of manifolds, AMS Special Session on Geometric and Topological Combinatorics, Joint Meetings, Seattle, WA, Jan 2025. [Slides]

    Abstract: I will survey what is known (and mostly unknown) about Vietoris-Rips complexes of manifolds. We might consider a dataset to be quite nice if it were densely sampled from a manifold. As more and more data points are sampled, the persistent homology of the dataset converges to the persistent homology of the manifold, by the stability of persistent homology. But we do not know what the Vietoris-Rips persistent homology of manifolds looks like, typically! I will describe what is known about Vietoris-Rips complexes of the circle, spheres, ellipsoids, tori, and manifolds, and advertise many open questions.

  • Persistent equivariant cohomology, Special Session on Applied and Computational Topology, Joint meeting of the New Zealand, Australian, and American mathematical societies, Auckland, New Zealand, Dec 2024. [Slides]

    Abstract: Persistent equivariant cohomology measures not only the shape of the filtration, but also attributes of a group action on the filtration, including in particular its fixed points. We give an explicit description of the persistent equivariant cohomology of the circle action on the Vietoris-Rips metric thickenings of the circle. Our computation relies on the Serre spectral sequence and the Gysin homomorphism. Joint with Evgeniya Lagoda, Michael Moy, Nikola Sadovek, and Aditya De Saha, available at https://arxiv.org/abs/2408.17331.

  • Persistent equivariant cohomology, Northeastern Topology Seminar, Boston, MA, Nov 2024. [Notes]

    Abstract: Persistent equivariant cohomology measures not only the shape of the filtration, but also attributes of a group action on the filtration, including in particular its fixed points. We give an explicit description of the persistent equivariant cohomology of the circle action on the Vietoris-Rips metric thickenings of the circle. Our computation relies on the Serre spectral sequence and the Gysin homomorphism. Joint with Evgeniya Lagoda, Michael Moy, Nikola Sadovek, and Aditya De Saha, available at https://arxiv.org/abs/2408.17331.

  • Evasion Paths in Mobile Sensor Networks, Minisymposium on Compositional Foundations for Optimization and Data Science, SIAM Conference on Mathematics of Data Science, Atlanta, GA, Oct 2024. [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!

  • The connectivity of Vietoris-Rips complexes of spheres, Minisymposium on Explorations in Topological Data Analysis, SIAM Sectional Meeting, Baylor University, Waco, TX, Oct 2024. [Slides]

    Abstract: Though Vietoris-Rips complexes are frequently built in applied topology to approximate the “shape” of a dataset, their theoretical properties are poorly understood. Interestingly, the Vietoris-Rips complexes of the circle obtain the homotopy types of the circle, the 3-sphere, the 5-sphere, the 7-sphere, …, as the scale parameter increases. But little is known about Vietoris-Rips complexes of the n-sphere Sn for n ≥ 2, which we equip with the geodesic metric of diameter π. We show how to control the homotopy connectivity of Vietoris-Rips complexes of spheres in terms of coverings of spheres and projective spaces. For t > 0, suppose that the first nontrivial homotopy group of the Vietoris-Rips complex of the n-sphere at scale π-t occurs in dimension k. Then there exist 2k+2 balls of radius t that cover Sn, and no set of k balls of radius t/2 cover the projective space RPn. Joint work with Johnathan Bush and Žiga Virk.

  • Representations of Energy Landscapes by Sublevelset Persistent Homology, Minisymposium on Advances in Algebra, Topology and Geometry with Applications to Data Analysis, SIAM Sectional Meeting, Baylor University, Waco, TX, Oct 2024. [Slides]

    Abstract: Encoding the complex features of an energy landscape is a challenging task, and often chemists pursue the most salient features (minima and barriers) along a highly reduced space, i.e. 2- or 3-dimensions. Even though disconnectivity graphs or merge trees summarize the connectivity of the local minima of an energy landscape via the lowest-barrier pathways, there is more information to be gained by also considering the topology of each connected component at different energy thresholds (or sublevelsets). We propose sublevelset persistent homology as an appropriate tool for this purpose. Our computations on the configuration phase space of n-alkanes from butane to octane allow us to conjecture, and then prove, a complete characterization of the sublevelset persistent homology of the alkane CmH2m+2 potential energy landscapes, for all m, and in all homological dimensions. We further compare both the analytical configurational potential energy landscapes and sampled data from molecular dynamics simulation, using the united and all-atom descriptions of the intramolecular interactions. Joint work with Joshua Mirth, Yanqin Zhai, Johnathan Bush, Enrique Alvarado, Howie Jordan, Mark Heim, Bala Krishnamoorthy, Markus Pflaum, Aurora Clark, Yang Zang.

  • The connectivity of Vietoris-Rips complexes of spheres, Special Session on Discrete and Combinatorial Algebraic Topology, Joint Meetings of Unione Matematica Italiana (UMI) and the American Mathematical Society (AMS), Palermo, Italy, July 2024. [Slides]

    Abstract: For X a metric space and r>0, the Vietoris-Rips simplicial complex VR(X;r) contains X as its vertex set, and a finite subset of X as a simplex if its diameter is less than r. Some versions of discrete homotopy groups are closely related to the standard homotopy groups of Vietoris-Rips complexes. Interestingly, the Vietoris-Rips complexes of the circle obtain the homotopy types of the circle, the 3-sphere, the 5-sphere, the 7-sphere, …, as the scale parameter increases. But little is known about Vietoris–Rips complexes of the n-sphere Sn for n≥2. We show how to control the homotopy connectivity of Vietoris-Rips complexes of spheres in terms of coverings of spheres and projective spaces. For δ>0, suppose that the first nontrivial homotopy group of VR(Sn;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 at Biomolecular Topology: Modelling and Data Analysis, Institute for Mathematical Sciences (IMS), Singapore, June 2024. [Notes]

    Abstract: Hausdorff and Gromov-Hausdorff distances are two ways to measure the “distance” between datasets, say datasets of alkane or cycloalkane molecule conformations. Though Hausdorff distances are easy to compute, Gromov-Hausdorff distances are not. 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 (in which case the Hausdorff and Gromov-Hausdorff distance coincide). Joint with Florian Frick, Sushovan Majhi, Nicholas McBride, available at https://arxiv.org/abs/2309.16648.

  • Connectivity of Vietoris-Rips complexes of spheres at the 19th Annual Topology Workshop, Nipissing University, North Bay, Canada, May 2024. [Notes]

    Abstract: For X a metric space and r>0, the Vietoris-Rips simplicial complex VR(X;r) contains X as its vertex set, and a finite subset of X as a simplex if its diameter is less than r. Though these complexes are frequently built in applied topology to approximate the “shape” of a dataset, their theoretical properties are poorly understood. Interestingly, the Vietoris-Rips complexes of the circle obtain the homotopy types of the circle, the 3-sphere, the 5-sphere, the 7-sphere, …, as the scale parameter increases. But little is known about Vietoris–Rips complexes of the n-sphere Sn for n≥2. We show how to control the homotopy connectivity of Vietoris-Rips complexes of spheres in terms of coverings of spheres and projective spaces. For δ>0, suppose that the first nontrivial homotopy group of VR(Sn;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]

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 departmental talks

  • Fair Division, University Math Society, University of Florida, Oct 2024. [Slides, Related Video]

    Abstract: Suppose five roommates need to pay $3,000 dollars of rent per month for their five-bedroom apartment. The five bedrooms are not equivalent: one is bigger, one is smaller, one has more windows, one is closer to the kitchen, one is painted neon green. So it is not necessarily fair to have each room cost the same amount. Furthermore, each roommate has a different opinion on the relative desirability of each room. How should the roommates fairly divide the rent between the rooms to cover the $3,000 apartment total, and how should they decide who gets which room? I will describe how Sperner’s lemma, related to combinatorics and topology, can be used to find a fair division of rent. This talk will survey the New York Times article To Divide the Rent, Start With a Triangle by Albert Sun, and the paper Rental Harmony: Sperner’s Lemma in Fair Division by Francis Su.

  • An introduction to applied topology, MMaths High School Math Competition, UF University Math Society, Oct 2024. [Slides]

    This talk is an introduction to applied and computational topology. The shape of a dataset often reflects important patterns within. Two such datasets with interesting shapes are a space of 3×3 pixel patches from optical images, which can be well-modeled by a Klein bottle, and the conformation space of the cyclo-octane molecule, which is a Klein bottle glued to a 2-sphere along two circles. I will introduce topological tools (persistent homology) for visualizing, understanding, and performing machine learning tasks on high-dimensional datasets.

  • An introduction to applied topology, Simple Words Seminar, UF Mathematics Department, Sep 2024. [Slides]

    This talk is an introduction to applied and computational topology. The shape of a dataset often reflects important patterns within. Two such datasets with interesting shapes are a space of 3×3 pixel patches from optical images, which can be well-modeled by a Klein bottle, and the conformation space of the cyclo-octane molecule, which is a Klein bottle glued to a 2-sphere along two circles. I will introduce topological tools (persistent homology) for visualizing, understanding, and performing machine learning tasks on high-dimensional datasets.

  • Persistent equivariant cohomology at the Topology & Dynamics Seminar, University of Florida, Sep 2024. [Notes]

    Abstract: Persistent equivariant cohomology measures not only the shape of the filtration, but also attributes of a group action on the filtration, including in particular its fixed points. This talk will cover parts of Sections 1-5 of the recent paper arxiv.org/pdf/2408.17331 on the persistent equivariant cohomology of the circle action on the Vietoris-Rips metric thickenings of the circle, joint with Evgeniya Lagoda, Michael Moy, Nikola Sadovek, and Aditya De Saha.

  • 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.

  • 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.

Recent mini-symposium or conference organization

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