Talks

ISQGD Special Session on Topology and Geometry

[Slides] and [Notes] and [Video]

Abstract: I will survey what is known (and mostly unknown) about Vietoris-Rips complexes of manifolds. We might consider a dataset to be 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 comment on Vietoris-Rips complexes of the circle, spheres, ellipsoids, tori, and manifolds, and advertise open questions.

Geometry and Topology Seminar, Florida State University

[Notes]

Abstract: Let X be a sample of points from a metric space M. How do we recover the geometric and topological properties of M from X? One way is to build a metric thickening space P(X;r) of all probability measures on X whose “size” is at most r, where size could be measured as an L^p variance or as an L^p diameter, for 1 ≤ p ≤ infty. Can a Morse theory be developed to describe how P(X;r) changes as r increases? Particular cases of interest are when X is finite (which I will mention) or when X is a Riemannian manifold (which I will focus on).

Mathematics Department Colloquium, Tulane University

[Slides] and [Notes]

Abstract: The Gromov-Hausdorff distance is a notion of dissimilarity between two datasets or between two metric spaces. It is an important tool in geometry, but notoriously difficult to compute. I will show how to provide new lower bounds on the Gromov-Hausdorff distance between unit spheres of different dimensions by combining Borsuk-Ulam theorems with Vietoris-Rips complexes. This joint work with 15 coauthors is available at https://arxiv.org/abs/2301.00246. Many questions remain open!

Mathematics Department Colloquium, West Chester University

Tetrahedral Geometry & Topology Seminar

[Notes]

Abstract: I will survey what is known (and mostly unknown) about Vietoris-Rips complexes of manifolds. We might consider a dataset to be 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 comment on Vietoris-Rips complexes of the circle, spheres, ellipsoids, tori, and manifolds, and advertise open questions.

OneMath World School on Topological Data Analysis and its Applications, Chennai Mathematical Institute

[Slides]

[DatasetLinks]

Workshop on Topological Statistics, Data and Intelligence at the Tsinghua Sanya International Mathematics Forum (TSIMF)

[Slides] and [Notes]

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 Topological and Geometric Shape Reconstruction

[Slides]

Abstract: I will explain how topological ideas can be used to bound quantities arising in metric geometry, such as the Gromov-Hausdorff distance between a manifold and a subset thereof. Joint work with Florian Frick, Sushovan Majhi, and Nicholas McBride.

Special Session on Computational Topology and Geometry in Data Science

[Slides]

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.

Special Session on Advances in Applied Topology and Topological Data Analysis

[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. 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.

39th Summer Topology and its Applications Conference

[Slides]

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.

CECAM Workshop on Advancing simulation, analysis and prediction of complex chemical systems using modern chemical graph theory and computational topology

[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 C_mH_2m+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

[Slides]

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(S^k;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(S^k;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

[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.

[Slides]

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.

20th Annual Topology Workshop

[Notes]

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

[Slides]

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.

OURFA²M² Conference

[Slides]

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

[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.

Joint meeting of the New Zealand, Australian, and American mathematical societies

[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.

Northeastern Topology Seminar

[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.

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

SIAM Sectional Meeting

[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.

Minisymposium on Advances in Algebra, Topology and Geometry with Applications to Data Analysis

SIAM Sectional Meeting

[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 C_mH_2m+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)

[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.

Biomolecular Topology: Modelling and Data Analysis

[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.

19th Annual Topology Workshop

[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.

Mid-Atlantic Topology Conference

[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.

STEMinar Series

[Slides]

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

[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.

SIAM Texas-Louisiana Sectional Meeting, Minisymposium on Explorations in Topological Data Analysis

[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.

Math Department Colloquium

[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.

Frontiers in AI Technology, Artificial Intelligence Days

[Slides, Related Video]

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.

Topology & Geometry Seminar

Notes

Abstract: When does an embedding f : X → ℝ^d exist? I will survey results by Whitney (1944), van Kampen and Flores (1933), and Haefliger and Weber (1962, 1967) when X is a manifold or a simplicial complex. When no such embedding f exists, can we quantify how discontinuous any injective function f : X → ℝ^d must be? Joint work with Florian Frick, Michael Harrison, Nikola Sadovek, Matt Superdock, available at https://arxiv.org/abs/2511.07636.

Topology & Geometry Seminar

Abstract: The Borsuk-Ulam theorem states that a map from Sⁿ to Rⁿ identifies antipodal points, or equivalently, that there is no odd map from Sⁿ to S^n-1. I will survey three quantitative generalizations:
(1) Almgrem and Gromov: The waist inequality says that a continuous map from S^k to Rⁿ has some fiber whose (k-n)-dimensional volume is at least as large as that of S^k-n.
(2) Urysohn: If we relax the codomain to be any n-dimensional simplicial complex, and measure the size of a fiber using diameter, then we ask about the (mostly unknown) Urysohn widths of spheres.
(3) Dubins and Schwarz: For k>n, how discontinuous must an odd function from S^k to Sⁿ be?
We know a tight quantitative bound for (3). I would like to know what new tools are needed to address (2).

Abstract: How do you “vectorize” geometry, i.e., extract it as a feature for use in machine learning? One way is persistent homology, a popular technique for incorporating geometry and topology in data analysis 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). Furthermore, I will explain how these techniques are related to the local geometry of a dataset and to explainable machine learning.

Applied Topology Seminar

[Slides]

Abstract: I will describe the torus model for optical flow. Our model is derived from a database of ground-truth optical flow from the computer-generated video Sintel. Using persistent homology and zigzag persistence, we show that the high-contrast 3×3 optical flow patches are well-modeled by a torus. This optical flow torus is equipped with the structure of a fiber bundle, related to the statistics of range image patches. Joint work with Johnathan Bush, Brittany Carr, Lara Kassab, Joshua Mirth.

Topology & Geometry Seminar

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.

University Math Society

[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.

MMaths High School Math Competition

UF University Math Society

[Slides]

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.

Simple Words Seminar

[Slides]

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.

Topology & Dynamics Seminar

[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.

Topology & Dynamics Seminar

[Notes]

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.

University Math Society

[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

[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.

Topology & Dynamics Seminar

[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.

[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.