Upcoming talks, mini-symposia, and conferences
Please click on the “plus” sign to see the talk abstract, conference link, slides, etc.
- With Hana Dal Poz Kouřimská, Teresa Heiss, Sara Kališnik, Bastian Rieck, co-organized The Geometric Realization of AATRN, Institute for Mathematical and Statistical Innovation (iMSi), Chicago, IL, Aug 18-22, 2025. [Poster]
- With Herbert Edelsbrunner and Yasu Hiraoka, co-organized the Topological Data Analysis (TDA) Week, Kyoto University, Kyoto, Japan, June 16-20, 2025.
- With Ziqin Feng, co-organized a Session on Applied Topology at the 58th Spring Topology and Dynamics Conference, Christopher Newport University, VA, Mar 2025.
- With Evgeniya Lagoda, co-organized a AMS Special Session on the Open Neighborhood of Applied Topology, Joint Meetings, Seattle, WA, Jan 2025.
- With Claudia Landi and Nicolò Zava, co-organized a Special Session on Computational Topology: Foundations, Algorithms, and Applications, Joint Meetings of Unione Matematica Italiana (UMI) and the American Mathematical Society (AMS), Palermo, Italy, July 2024.
- With Johnathan Bush and Hubert Wagner, co-organized a Special Session on Topological Algorithms for Complex Data and Biology, AMS Sectional Meeting, Florida State University, Mar 2024.
- With Ling Zhou, co-organized the AMS Special Session on Bridging Applied and Quantitative Topology, Joint Meetings, San Francisco, CA, Jan 2024.
- Metric reconstruction via optimal transport. SIAM Conference on Applied Algebraic Geometry, July 2017. [Poster]
- The Vietoris-Rips complexes of a circle. ICERM Workshop on Topology and Geometry in a Discrete Setting, Nov 2016. [Poster]
- The Vietoris-Rips complex of the circle. IMA Workshop on Topology and Geometry of Networks and Discrete Metric Spaces, Apr 2014. [Poster]
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Vietoris-Rips and restricted Čech complexes of circular points. IMA Workshop on Topological Systems: Communication, Sensing, and Actuation, Mar 2014. [Poster]
- Evasion paths in mobile sensor networks. [Poster]
- Mobile sensors and pursuit-evasion: Can directed algebraic toplogy help? Aalborg GETCO Workshop, Jan 2010. [Poster]
Special Session on Advances in Applied Topology and Topological Data Analysis
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. 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.
Special Session on Computational Topology and Geometry in Data Science
Abstract: Wigner described the unreasonable effectiveness of mathematics in the natural sciences: ideas from mathematics are unreasonably effective in advancing applications, and ideas from applications are unreasonably effective in advancing mathematics. We describe a case study on “persistence images”, which live at the intersection of topology and machine learning, and which can be thought of as a way to “vectorize” geometry for use in machine learning tasks. I will survey applications arising from materials science, computer vision, and agent-based modeling (modeling a flock of birds or a school of fish), and show how these applications also inspire new topology questions. Joint work with Sofya Chepushtanova, Tegan Emerson, Eric Hanson, Michael Kirby, Francis Motta, Rachel Neville, Chris Peterson, Patrick Shipman, and Lori Ziegelmeier.
Recent research talks
39th Summer Topology and its Applications Conference
Abstract: The goal of this talk is to show how tools from topology can bound or compute 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. Next I will explain the nerve lemma, which says when a cover of a space faithfully encodes the shape of that space. As the main result, I’ll show how when X is a sufficiently dense subset of a closed Riemannian manifold M, we can use the nerve lemma 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, and even obtains the optimal value 1 (meaning the Hausdorff and Gromov-Hausdorff distances coincide) when M is the circle.
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.
Workshop on Geometric and Topological Graph Theory
Abstract: The chromatic number of a graph can be defined in terms of graph homomorphisms into complete graphs, and the circular chromatic number (which refines the chromatic number) can be defined using graph homomorphisms into the Borsuk graph of the circle. More generally, the Borsuk graph Bor(Sk;r) of the k-sphere at scale r has each point of the k-sphere as a vertex and an edge between two points if they are at distance at least r apart. Using a topological obstruction, we prove that for k > n, there is no graph homomorphism Bor(Sk;r) → Bor(Sn;t) unless t > 2π/3. This is joint work with Alex Elchesen, Sucharita Mallick, and Michael Moy: https://arxiv.org/abs/2503.08862.
Topological Data Analysis (TDA) Week
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.
Abstract: 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 configuration space of the cyclo-octane molecule, which is a Klein bottle glued to a 2-sphere along two circles. I will introduce topological tools (such as persistent homology) for visualizing, understanding, and performing machine learning tasks on high-dimensional datasets.
Abstract: Hausdorff and Gromov-Hausdorff distances measure the distance between metric spaces. 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 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 (meaning the Hausdorff and Gromov-Hausdorff distances coincide) in the case of the circle. Our proofs convert discontinuous functions between metric spaces into simplicial maps between Vietoris-Rips and Čech complexes, and obstruct 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
Abstract: 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 configuration space of the cyclo-octane molecule, which is a Klein bottle glued to a 2-sphere along two circles. I will introduce topological tools (such as persistent homology) for visualizing, understanding, and performing machine learning tasks on high-dimensional datasets.
Abstract: 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 for visualizing, understanding, and performing machine learning tasks on high-dimensional datasets.
AMS Special Session on Geometric and Topological Combinatorics
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.
Joint meeting of the New Zealand, Australian, and American mathematical societies
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.
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.
Minisymposium on Compositional Foundations for Optimization and Data Science
[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!
Minisymposium on Explorations in Topological Data Analysis
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.
Minisymposium on Advances in Algebra, Topology and Geometry with Applications to Data Analysis
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.
Special Session on Discrete and Combinatorial Algebraic Topology
Joint Meetings of Unione Matematica Italiana (UMI) and the American Mathematical Society (AMS)
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.
Biomolecular Topology: Modelling and Data Analysis
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.
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.
Mid-Atlantic Topology Conference
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.
Abstract: 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 for visualizing, understanding, and performing machine learning tasks on high-dimensional datasets.
AMS Special Session on Discrete Homotopy Theory
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.
SIAM Texas-Louisiana Sectional Meeting, Minisymposium on Explorations in Topological Data Analysis
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.
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.
Frontiers in AI Technology, Artificial Intelligence Days
Abstract: Wigner described the unreasonable effectiveness of mathematics in the natural sciences: ideas from mathematics are unreasonably effective in advancing applications, and ideas from applications are unreasonably effective in advancing mathematics. We describe a case study on “persistence images”, which live at the intersection of topology and machine learning, and which can be thought of as a way to “vectorize” geometry for use in machine learning tasks. I will survey applications arising from materials science, computer vision, and agent-based modeling (modeling a flock of birds or a school of fish), and show how these applications also inspire new topology questions. Joint work with Sofya Chepushtanova, Tegan Emerson, Eric Hanson, Michael Kirby, Francis Motta, Rachel Neville, Chris Peterson, Patrick Shipman, and Lori Ziegelmeier.
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
Abstract: We previously showed that if either the Hausdorff or Gromov-Hausdorff distance between the circle and a subset thereof is less than r = π/6, then these two distances coincide. A weaker version of this result holds in any compact manifold, proven using a topological obstruction based on the fundamental class of the manifold. We give an improved result on the circle, showing r = π/3 suffices, using a topological obstruction based on the simpler notion of connectedness. Furthermore, r = π/3 is optimal. What other topological invariants can be converted into bounds on Gromov-Hausdorff distances? Joint work with Sushovan Majhi, Fedor Manin, Žiga Virk, Nicolò Zava.
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.
MMaths High School Math Competition
Abstract: 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.
Abstract: 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.
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.
Abstract: 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.
[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!
Applied Topology Reading Group
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.
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.
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.