Talks

Upcoming talks, mini-symposia, and conferences

• The unreasonably effective interaction between math and applications: A case study on persistence images, 2nd POSTECH MINDS Workshop on Topological Data Analysis and Machine Learning, Seoul, Korea, Sep 26-29, 2022. [Abstract, Slides]
  
  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 persistent images, a stable vector representation of persistent homology. If you combine topology with data, you get persistent homology. If you combine persistent homology with machine learning, you might get persistent landscapes or persistence images or a host of other options. The first attempt at persistence images were not stable (i.e. continuous), but by making them stable, their machine learning performance improves, as we will describe on examples ranging from materials science to biology. This ping-ponging behavior of injecting ideas from mathematics, then injecting ideas from applications, etc, leads to robust applied tools and new mathematical questions. Joint work with Sofya Chepushtanova, Tegan Emerson, Eric Hanson, Michael Kirby, Francis Motta, Rachel Neville, Chris Peterson, Patrick Shipman, and Lori Ziegelmeier.

• Geometry, Topology, Dynamical Systems Seminar, UT Dallas, Oct 24, 2022.
• Evasion paths in mobile sensor networks, Oklahoma University Topology and Data Science Seminar, online, Oct 31, 2022. [Abstract, Slides, Poster, Multimedia]
  
  Suppose ball-shaped sensors wander in a bounded domain. A sensor doesn't know its location but does know when it overlaps a nearby sensor. We say that an evasion path exists in this sensor network if a moving intruder can avoid detection. Relative homology or zigzag persistence give a necessary condition, depending only on the time-varying connectivity data of the sensors. However, no method with time-varying connectivity data (i.e. Čech complexes) as input can give necessary and sufficient conditions for the existence of an evasion path. Indeed, the existence of an evasion path depends on more than just the fibrewise homotopy type of the region covered by sensors. In the setting of planar sensors that also measure weak rotation information, we provide necessary and sufficient conditions for the existence of an evasion path, and we pose an open question concerning Čech and alpha complexes. Joint with Gunnar Carlsson, with recently computed statistics for the probability of mobile coverage under different motion models joint with Deepjyoti Ghosh, Clark Mask, William Ott, and Kyle Williams at the University of Houston.

• Speaking at the CIMAT Applied Geometry and Topology seminar, online, Nov 7, 2022.
• Attending the Dagstuhl Seminar on Topological Data Analysis and Applications, Dagstuhl, Germany, May 7-12, 2023.
• Speaking at a workshop on Computational Topology and Geometry at the Foundations of Computational Mathematics conference, Paris, France, June 12-14, 2023.

Research talks

• What are Gromov-Hausdorff distances?, Department Colloquium, Auburn University, Sep 2022. [Abstract, Slides]
  
  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. In the first part of my talk, I will give an introduction to Gromov-Hausdorff distances. Then, in the second part of my talk, I will use tools from applied topology to give (potentially tight) lower bounds on the Gromov-Hausdorff distance between unit spheres of different dimensions. This is joint work in a polymath-style project with many people who are currently or formerly at Colorado State, Ohio State, Carnegie Mellon, or Freie Universität Berlin.

• An introduction to Vietoris-Rips complexes, Topology Seminar, Auburn University, Sep 2022. [Abstract, Slides]
  
  I will give an introduction to Vietoris-Rips complexes, Vietoris-Rips metric thickenings, and their uses in applied and computational topology. So, this talk will provide a complementary perspective to Krystyna Kuperberg's talk last week, though attendance last week is not assumed, since I wasn't there! If a dataset is sampled from some unknown underlying space (say a manifold), then as more and more samples are drawn, the Vietoris-Rips persistent homology of the dataset converges to the Vietoris-Rips persistent homology of the manifold. But little is known about Vietoris-Rips complexes of manifolds. An exception is the case of the circle: I will describe how as the scale parameter increases, the Vietoris-Rips complexes of the circle obtain the homotopy types of the circle, the 3-sphere, the 5-sphere, ..., until finally they are contractible. Much less is known about Vietoris-Rips complexes of spheres. Throughout my talk, I'll advertise some open questions.

• Gromov-Hausdorff distances, Borsuk-Ulam theorems, and Vietoris-Rips complexes, Algebraic Topology and Topological Data Analysis: A conference in honor of Gunnar Carlsson, Institute for Mathematics and its Applications, Minneapolis, MN, Aug 2022. [Abstract, Slides, Video]
  
  The Gromov-Hausdorff distance between two metric spaces is an important tool in geometry, but it is difficult to compute. For example, the Gromov-Hausdorff distance between unit spheres of different dimensions is unknown in nearly all cases. I will introduce recent work by Lim, Mémoli, and Smith that finds the exact Gromov-Hausdorff distances between S¹, S², and S³, and that lower bounds the Gromov-Hausdorff distance between any two spheres using Borsuk-Ulam theorems. We improve some of these lower bounds by connecting this story to Vietoris-Rips complexes, providing new generalizations of the Borsuk-Ulam theorem. This is joint work in a polymath-style project with many people, most of whom are currently or formerly at Colorado State, Ohio State, Carnegie Mellon, or Freie Universität Berlin.

• The persistent topology of optimal transport based thickenings, Applied, Combinatorial and Toric Topology, Institute for Mathematical Sciences, Singapore, online, July 2022. [Abstract, Slides]
  
  The input to persistent homology is a filtration - an increasing sequence of spaces. We recast filtrations arising in applied topology, including Cech and Vietoris-Rips filtrations, in terms of optimal transport. This perspective faithfully preserves the (undecorated) persistence diagram. Furthermore, these optimal transport filtrations have nicer metric properties enabling simpler proofs of results such as Hausmann's theorem or homotopy types of Vietoris-Rips thickenings of spheres. Joint work with Michał Adamaszek, Florian Frick, Michael Moy, Facundo Mémoli, and Qingsong Wang.

• The persistent topology of optimal transport based thickenings, ATMCS 10, Oxford University, England, June 2022. [Abstract, Slides]
  
  A metric thickening of a given metric space X is any metric space admitting an isometric embedding of X. Thickenings have found use in applications of topology to data analysis, where one may approximate the shape of a dataset via the persistent homology of an increasing sequence of spaces. We introduce two new families of metric thickenings, the p-Vietoris-Rips and p-Čech metric thickenings for all 1≤ p≤ ∞, which include all measures on X whose p-diameter or p-radius is bounded from above, equipped with an optimal transport metric. The p-diameter (resp. p-radius) of a measure is a certain ℓ_p relaxation of the usual notion of diameter (resp. radius) of a subset of a metric space. These families recover the previously studied Vietoris-Rips and Čech metric thickenings when p=∞. As our main contribution, we prove a stability theorem for the persistent homology of p-Vietoris-Rips and p-Čech metric thickenings, which is novel even in the case p=∞. In the specific case p=2, we prove a Hausmann-type theorem for thickenings of manifolds, and we derive the complete list of homotopy types of the 2-Vietoris-Rips thickenings of the n-sphere as the scale increases. Joint with Facundo Mémoli, Michael Moy, and Qingsong Wang.

• The unreasonably effective interaction between math and applications: A case study on persistence images, TORI seminar, Sorbonne Université, Paris, France, June 2022. [Abstract, Slides, Slides as Notability file, Video]
  
  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 persistent images, a stable vector representation of persistent homology. If you combine topology with data, you get persistent homology. If you combine persistent homology with machine learning, you might get persistent landscapes or persistence images or a host of other options. The first attempt at persistence images were not stable (i.e. continuous), but by making them stable, their machine learning performance improves, as we will describe on examples ranging from materials science to biology. This ping-ponging behavior of injecting ideas from mathematics, then injecting ideas from applications, etc, leads to robust applied tools and new mathematical questions. Joint work with Sofya Chepushtanova, Tegan Emerson, Eric Hanson, Michael Kirby, Francis Motta, Rachel Neville, Chris Peterson, Patrick Shipman, and Lori Ziegelmeier.

• Gromov-Hausdorff distances, Borsuk-Ulam theorems, and Vietoris-Rips complexes, 3rd Biennial Meeting of SIAM Pacific Northwest Section, Washington State University, online, May 2022. [Abstract, Slides, Slides as Notability file]
  
  The Gromov-Hausdorff distance between two metric spaces is an important tool in geometry, but it is difficult to compute. For example, the Gromov-Hausdorff distance between unit spheres of different dimensions is unknown in nearly all cases. I will introduce recent work by Lim, Mémoli, and Smith that lower bounds the Gromov-Hausdorff distance between spheres using Borsuk-Ulam theorems. We improve these lower bounds by connecting this story to Vietoris-Rips complexes, providing new generalizations of the Borsuk-Ulam theorem. This is joint work in a polymath-style project with many people, most of whom are currently or formerly at Colorado State, Ohio State, Carnegie Mellon, or Freie Universität Berlin.

• Support Vector Machines and Radon's Theorem, Special Session on Computational Topology and Applications, AMS Western Sectional Meeting, online, May 2022. See also this schedule. [Abstract, Slides, Slides as Notability file, Video]
  
  A support vector machine (SVM) is an algorithm which finds a hyperplane that optimally separates labeled data points in R^n into positive and negative classes. The data points on the margin of this separating hyperplane are called support vectors. We connect the possible configurations of support vectors to Radon?s theorem, which provides guarantees for when a set of points can be divided into two classes (positive and negative) whose convex hulls intersect. For example, if the convex hulls of the positive and negative support vectors are projected onto a separating hyperplane, then the projections intersect if and only if the hyperplane is optimal. Joint work with Elin Farnell and Brittany Story.

• Gromov-Hausdorff distances, Borsuk-Ulam theorems, and Vietoris-Rips complexes, TU Graz, Austria, May 2022. [Abstract, Slides, Slides as Notability file]
  
  The Gromov-Hausdorff distance between two metric spaces is an important tool in geometry, but it is difficult to compute. For example, the Gromov-Hausdorff distance between unit spheres of different dimensions is unknown in nearly all cases. I will introduce recent work by Lim, Mémoli, and Smith that lower bounds the Gromov-Hausdorff distance between spheres using Borsuk-Ulam theorems. We improve these lower bounds by connecting this story to Vietoris-Rips complexes, providing new generalizations of the Borsuk-Ulam theorem. This is joint work in a polymath-style project with many people, most of whom are currently or formerly at Colorado State, Ohio State, Carnegie Mellon, or Freie Universität Berlin.

• Topology in Machine Learning, Seminar on TDA and its applications, NTU Singapore, online, May 2022. [Abstract, Slides, Slides as Notability file, Video]
  
  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.

• Bounding Gromov-Hausdorff distances with Borsuk-Ulam theorems and Vietoris-Rips complexes, Oklahoma University Topology and Data Science Seminar, online, Apr 2022. [Abstract, Slides, Slides as Notability file, Video]
  
  The Gromov-Hausdorff distance between two metric spaces is an important tool in geometry, but it is difficult to compute. For example, the Gromov-Hausdorff distance between unit spheres of different dimensions is unknown in nearly all cases. I will introduce recent work by Lim, Mémoli, and Smith that lower bounds the Gromov-Hausdorff distance between spheres using Borsuk-Ulam theorems. We improve these lower bounds by connecting this story to Vietoris-Rips complexes, providing new generalizations of Borsuk-Ulam. This is joint work in a polymath-style project with many people, most of whom are currently or formerly at Colorado State, Ohio State, Carnegie Mellon, or Freie Universität Berlin.

• Gromov-Hausdorff distances, Borsuk-Ulam theorems, and Vietoris-Rips complexes, EPFL Applied Topology Seminar, Lausanne, Switzerland, online, Mar 2022. [Abstract, Slides, Slides as Notability file, Video]
  
  The Gromov-Hausdorff distance between two metric spaces is an important tool in geometry, but it is difficult to compute. For example, the Gromov-Hausdorff distance between unit spheres of different dimensions is unknown in nearly all cases. I will introduce recent work by Lim, Mémoli, and Smith that lower bounds the Gromov-Hausdorff distance between spheres using Borsuk-Ulam theorems. We improve these lower bounds by connecting this story to Vietoris-Rips complexes, providing new generalizations of Borsuk-Ulam. This is joint work in a polymath-style project with many people, most of whom are currently or formerly at Colorado State, Ohio State, Carnegie Mellon, or Freie Universität Berlin.

• An introduction to Vietoris-Rips complexes, Bilkent Topology Seminar, Turkey, online, Mar 2022. [Abstract, Slides, Slides as Notability file, Video]
  
  I will give an introduction to Vietoris-Rips complexes and their uses in applied and computational topology. If a dataset is sampled from some unknown underlying space (say a manifold), then as more and more samples are drawn, the Vietoris-Rips persistent homology of the dataset converges to the Vietoris-Rips persistent homology of the manifold. But little is known about Vietoris-Rips complexes of manifolds. An exception is the case of the circle: I will describe how as the scale parameter increases, the Vietoris-Rips complexes of the circle obtain the homotopy types of the circle, the 3-sphere, the 5-sphere, ..., until finally they are contractible. Much less is known about Vietoris-Rips complexes of spheres. I will also briefly explain how Vietoris-Rips complexes relate to generalizations of the Borsuk-Ulam theorem and to Gromov-Hausdorff distances between spheres.

• The topology of projective codes, Codes and Expansions (CodEx) Seminar, online, Feb 2022. [Abstract, Slides, Slides as Notability file, Video]
  
  The goal of this talk is to explain the relationships between four objects:
  (i) Packings and coverings in projective space,
  (ii) The Borsuk-Ulam theorem (a classic result in topology),
  (iii) Vietoris-Rips complexes (used in topological data analysis), and
  (iv) Gromov-Hausdorff distances (used in shape matching applications).
  The CodEx audience has more experience than I do on topic (i), and I will ask for references you recommend on packings and coverings in projective space. I will assume no familiarity with topics (ii)-(iv), which I will introduce. For an arXiv preprint on some of these topics, see https://arxiv.org/abs/2106.14677.

• Topology in machine learning, IST Austria Institute Colloquium, Jan 2022. [Abstract, Slides, Slides as Notability file]
  
  How do you "vectorize" geometry, i.e., extract it as a feature for use in machine learning? I will give a gentle introduction to 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), and describe how these techniques are related to explainable machine learning.

• Bounding Gromov-Hausdorff distances with Borsuk-Ulam theorems and Vietoris-Rips complexes, Copenhagen-Jerusalem Combinatorics Seminar, Copenhagen, Denmark, Dec 2021. [Abstract, Slides, Slides as Notability file, Video]
  
  The Gromov-Hausdorff distance between two metric spaces is an important tool in geometry, but it is difficult to compute. For example, the Gromov-Hausdorff distance between unit spheres of different dimensions is unknown in nearly all cases. I will introduce recent work by Lim, Mémoli, and Smith that lower bounds the Gromov-Hausdorff distance between spheres using Borsuk-Ulam theorems. We improve these lower bounds by connecting this story to Vietoris-Rips complexes, providing new generalizations of Borsuk-Ulam. This is joint work in a polymath-style project with many people, most of whom are currently or formerly at Colorado State, Ohio State, Carnegie Mellon, or Freie Universität Berlin.

• Open questions on clique complexes of graphs, Facets of Complexity, Freie Universität Berlin, Technische Universität Berlin, and Humboldt-Universität zu Berlin, Germany, Nov 2021. [Abstract, Slides, Slides as Notability file]
  
  The clique complex of a graph is a simplicial complex with a simplex for each clique. Clique complexes are frequently being computed in applications of topology to data, but we do not understand their algorithmic theory or their mathematical theory. I will introduce clique complexes of graphs, explain why applied topologists care about them, and survey open problems about the topology of clique complexes of unit disk graphs, powers of lattice graphs, and powers of hypercube graphs.

• Gromov-Hausdorff distances, Borsuk-Ulam theorems, and Vietoris-Rips complexes, Discrete Geometry and Topological Combinatorics research seminar, Freie Universität, Berlin, Germany, Nov 2021. [Abstract, Slides, Slides as Notability file]
  
  The Gromov-Hausdorff distance between two metric spaces is an important tool in geometry, but it is difficult to compute. For example, the Gromov-Hausdorff distance between unit spheres of different dimensions is unknown in nearly all cases. I will introduce recent work by Lim, Mémoli, and Smith that lower bounds the Gromov-Hausdorff distance between spheres using Borsuk-Ulam theorems. We improve these lower bounds by connecting this story to Vietoris-Rips complexes. This is joint work in a polymath-style project with many people, most of whom are currently or formerly at Colorado State, Ohio State, Carnegie Mellon, or Freie Universität Berlin.

• Open problems in need of computer simulation, Computational Persistence Workshop, online, Nov 2021. [Abstract, Slides, Slides as Notability file]
  
  Though math papers are sometimes presented as "definition, lemma, theorem, proof", the ideas instead often arise via a computational experiment, which leads to a conjecture, which only later perhaps becomes a theorem. Such computations should be explained, advertised, acknowledged, and celebrated. I will share open problems where more computational experiments are needed, and where there is plenty of room for folks with algorithmic expertise to get involved. These open problems are related (for example) to nerve complexes, to Vietoris-Rips complexes, and to the persistent homology of fractals.

• The unreasonably effective interaction between pure and applied mathematics: A case study on persistence images, Workshop on Topology, Algebra and Geometry in Computer Vision at ICCV 2021, online, Oct 2021. [Abstract, Slides, Slides as Notability file]
  
  Wigner described the unreasonable effectiveness of mathematics in the natural sciences. In a similar vein, ideas from pure mathematics are unreasonably effective in advancing applied mathematics, and ideas from applied mathematics are unreasonably effective in advancing pure mathematics. We describe a case study on persistent images, a stable vector representation of persistent homology. If you combine topology with data, you get persistent homology. If you combine persistent homology with machine learning, you might get persistent landscapes or persistence images or a host of other options. The first attempt at persistence images were not stable (i.e. continuous), but by making them stable, their machine learning performance improves, as we will describe on examples ranging from materials science to biology. This ping-ponging behavior of injecting ideas from pure mathematics, then injecting ideas from applied mathematics, etc, leads to robust applied tools and new mathematical questions. Joint work with Sofya Chepushtanova, Tegan Emerson, Eric Hanson, Michael Kirby, Francis Motta, Rachel Neville, Chris Peterson, Patrick Shipman, and Lori Ziegelmeier.

• Vietoris-Rips thickenings of spheres, Geometry Seminar, Pennsylvania State University, online, Oct 2021. [Abstract, Slides, Slides as Notability file]
  
  If a dataset is sampled from a manifold, then as more and more samples are drawn, the persistent homology of the Vietoris-Rips complexes of the dataset converges to the persistent homology of the Vietoris-Rips complexes of the manifold. But little is known about Vietoris-Rips complexes of manifolds. An exception is the case of the circle: as the scale parameter increases, the Vietoris-Rips complexes of the circle obtain the homotopy types of the circle, the 3-sphere, the 5-sphere, ..., until finally they are contractible. We describe what is known about the homotopy types of Vietoris-Rips thickenings of spheres, which have applications for generalizations of the Borsuk-Ulam theorem, for projective codes (packings in projective space), and (conjecturally) for Gromov-Hausdorff distances between spheres. This is joint work with Michał Adamaszek, Johnathan Bush, and Florian Frick.

• Relationships between persistent homology, machine learning, and local geometry, Mini-Conference on Computational Topology and Machine Learning for the Einstein Semester on Geometric and Topological Structure of Materials, online, Sept 2021. [Abstract, Slides, Slides as Notability file]
  
  Persistent homology and its vectorization techniques are popular ways to incorporate geometry in machine learning. I will survey applications arising from machine learning tasks in agent-based modeling, shape recognition, materials science, and biology. The use of topology in machine learning provides additional incentives to study the relationship between persistent homology and local geometry. Joint work with the Pattern Analysis Lab at Colorado State University.

• Vietoris-Rips thickenings of spheres, Algebraic Geometry and Geometric Topology Seminar, Tulane University, online, Sept 2021. [Abstract, Slides, Slides as Notability file]
  
  If a dataset is sampled from a manifold, then as more and more samples are drawn, the persistent homology of the Vietoris-Rips complexes of the dataset converges to the persistent homology of the Vietoris-Rips complexes of the manifold. But little is known about Vietoris-Rips complexes of manifolds. An exception is the case of the circle: as the scale parameter increases, the Vietoris-Rips complexes of the circle obtain the homotopy types of the circle, the 3-sphere, the 5-sphere, ..., until finally they are contractible. The Vietoris-Rips thickenings of the n-sphere first obtain the homotopy type of the n-sphere, and then next the (n+1)-fold suspension of a (topological) quotient of the special orthogonal group SO(n+1) by an alternating group A_n+2. Not much is known at later scales, even though (as we will explain) these homotopy types have applications for generalizations of the Borsuk-Ulam theorem, for projective codes (packings in projective space), and (conjecturally) for Gromov-Hausdorff distances between spheres. This is joint work with Michał Adamaszek, Johnathan Bush, and Florian Frick.

• Vietoris-Rips complexes of hypercube graphs, Workshop on Beyond TDA - Persistent functions and its applications in data sciences, online, Aug 2021. [Abstract, Slides, Slides as Notability file, Video]
  
  Questions about Vietoris-Rips complexes of hypercube graphs arise naturally from problems in genetic recombination, and also from Künneth formulas for persistent homology with the sum metric. We describe the homotopy types of Vietoris-Rips complexes of hypercube graphs at small scale parameters. In more detail, let Q_n be the vertex set of the hypercube graph with 2ⁿ vertices, equipped with the shortest path metric. Equivalently, Q_n is the set of all binary strings of length n, equipped with the Hamming distance. The Vietoris-Rips complex of Q_n at scale zero is 2ⁿ points, and the Vietoris-Rips complex of Q_n at scale one is the hypercube graph, which is homotopy equivalent to a wedge sum of circles. We show that the Vietoris-Rips complex of Q_n at scale two is homotopy equivalent to a wedge sum of 3-spheres, and furthermore we provide a formula for the number of 3-spheres as a function of n. Many questions about the Vietoris-Rips complexes of hypercube graphs at larger scale parameters remain open. We describe these questions, and pose some conjectures motivated by homology computations in Ripser++. Joint work with Michał Adamaszek.

• Applied topology: From global to local, DANGER workshop focused on applications of machine learning to pure mathematics, online, Aug 2021. [Abstract, Slides, Slides as Notability file, Video]
  
  Through the use of examples, I will explain one way in which applied topology has evolved since the birth of persistent homology in the early 2000s. The first applications of topology to data emphasized the global shape of a dataset, such as the three-circle model for 3 x 3 pixel patches from natural images, or the configuration space of the cyclo-octane molecule, which is a sphere with a Klein bottle attached via two circles of singularity. More recently, persistent homology is being used to measure the local geometry of data. How do you vectorize geometry for use in machine learning problems? Persistent homology, and its vectorization techniques including persistence landscapes and persistence images, provide popular techniques for incorporating geometry in machine learning. I will survey applications arising from machine learning tasks in agent-based modeling, shape recognition, materials science, and biology.

• Topology in Machine Learning, Workshop on Knowledge Guided Machine Learning (KGML), online, Aug 2021. [Abstract, Slides, Slides as Notability file, Video]
  
  How do you vectorize geometry for use in machine learning problems? In this talk I will introduce persistent homology, a popular technique for incorporating geometry and topology in machine learning. I will survey applications arising from machine learning tasks in materials science, computer vision, and agent-based modeling, and describe how these techniques are related to explainable machine learning.

• The unreasonably effective interaction between pure and applied mathematics: A case study on persistence images, CIMAT/TRIPODS 2021 Summer Conference, online, Aug 2021. [Abstract, Slides, Slides as Notability file]
  
  Wigner described the unreasonable effectiveness of mathematics in the natural sciences. In a similar vein, ideas from pure mathematics are unreasonably effective in advancing applied mathematics, and ideas from applied mathematics are unreasonably effective in advancing pure mathematics. We describe a case study on persistent images, a stable vector representation of persistent homology. If you combine topology with data, you get persistent homology. If you combine persistent homology with machine learning, you might get persistent landscapes or persistence images or a host of other options. The first attempt at persistence images were not stable (i.e. continuous), but by making them stable, their machine learning performance improves, as we will describe on examples ranging from materials science to biology. This ping-ponging behavior of injecting ideas from pure mathematics, then injecting ideas from applied mathematics, etc, leads to robust applied tools and new mathematical questions.

• Bridging applied and geometric topology, 38th Annual Workshop in Geometric Topology, online, June 2021. [Abstract, Slides, Slides as Notability file, Video]
  
  I will advertise open questions in applied topology for which tools from geometric topology are relevant. If a point cloud is sampled from a manifold, then as more samples are drawn, the persistent homology of the Vietoris-Rips complex of the point cloud converges to the persistent homology of the Vietoris-Rips complex of the manifold. But what are the homotopy types of Vietoris-Rips complexes of manifolds? Essentially nothing is known, except that as the scale parameter increases, the Vietoris-Rips complex of the circle obtains the homotopy types of the circle, the 3-sphere, the 5-sphere, the 7-sphere, ..., until finally it is contractible. I will survey emerging connections between Vietoris-Rips complexes of manifolds and the filling radius (which Gromov used to prove the systolic inequality in 1983), thick-thin decompositions, and sweepouts. This is joint work with Baris Coskunuzer.

• The unreasonably effective interaction between pure and applied mathematics: A case study on persistence images, Minisymposium on Topological Signal Processing at the SIAM conference on Applications of Dynamical Systems, online, May 2021. [Abstract, Slides, Slides as Notability file]
• Representations of energy landscapes by sublevelset persistent homology: An example with n-alkanes, Minisymposium on Multi-Scale Statistical Descriptors of Materials at the SIAM Conference on Mathematics Aspects of Materials Science, online, May 2021. [Abstract, Slides, Slides as Notability file]
  
  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 for DELTA, an NSF Harnessing the Data Revolution project.

• Two-talk workshop on Topology in Data Science at The 54th Spring Topology and Dynamical Systems Conference, online, May 2021. [Abstract, Slides Part I, Slides Part II]
• Bridging applied and quantitative topology at a Virtual Workshop on Topological Data Analysis: Theory and Applications, hosted by the University of Western Ontario and the Tutte Institute, May 2021. [Abstract, Slides, Slides as Notability file]
  
  I will survey emerging connections between applied topology and quantitative topology. Vietoris-Rips complexes were invented by Vietoris in order to define a (co)homology theory for metric spaces, and by Rips for use in geometric group theory. More recently, they have found applications in computational topology for approximating the shape of a dataset. I will describe a recent result by my student Michael Moy showing that Vietoris-Rips simplicial complexes and Vietoris-Rips metric thickenings (defined using optimal transport) have the same persistence diagrams, and based upon this work I will speculate on the homotopy types of Vietoris-Rips complexes of n-spheres.

• Representations of energy landscapes by sublevelset persistent homology: An example with n-alkanes, American Chemical Society symposium on Graph theory underpinning new domains of physical chemistry, online, Apr 2021. [Abstract, Slides, Slides as Notability file]
  
  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 for DELTA, an NSF Harnessing the Data Revolution project.

• Vietoris-Rips thickenings: Problems for birds and frogs, Vietoris-Rips Seminar, online, Mar 2021. [Abstract, Slides, Slides as Notability file, Video]
  
  An artificial distinction is to describe some mathematicians as birds, who from their high vantage point connect disparate areas of mathematics through broad theories, and other mathematicians as frogs, who dig deep into particular problems to solve them one at a time. Neither type of mathematician thrives without the help of the other! In this talk, I will survey open problems related to Vietoris-Rips complexes that are attractive to both birds and frogs. Though Vietoris-Rips complexes are frequently used to approximate the shape of a dataset, many questions remain about their mathematical properties. Frogs may delight in open problems such as the homotopy types of Vietoris-Rips complexes of spheres, ellipsoids, tori, graphs, Cayley graphs of groups, geodesic spaces, subsets of the plane, and even the integer lattice Zⁿ with the taxicab metric for n ≥ 4. Birds may enjoy emerging connections between Vietoris-Rips complexes and a variety of areas in pure mathematics, including metric geometry (Gromov-Hausdorff distances), quantitative topology (Gromov's filling radius), measure theory (optimal transport), topological combinatorics (Borsuk-Ulam theorems), geometric group theory (finiteness properties of groups), and geometric topology (thick-thin decompositions).

• Mini-course on A visual introduction to geometric data analysis with Lara Kassab at the workshop Geometry: Education, Art, and Research, online, Banff International Research Station, Feb 19-21, 2021. [Slides, Slides as Notability file, Video Part 1, Video Part 2]
• Descriptors of Energy Landscapes using Topological Analysis (DELTA) at the AMS Special Session on Combinatorial Approaches to Topological Structures and Applications, Joint Mathematics Meetings, Washington DC, online, Jan 2021. [Abstract, Slides, Slides as Notability file]
• Applied topology: From global to local at the AMS Special Session on Geometry in the Mathematics of Data Science, Joint Mathematics Meetings, Washington DC, online, Jan 2021. [Abstract, Slides, Slides as Notability file]
• Bridging applied and quantitative topology at the Topology and Dynamics Seminar, University of Florida, Gainesville, online, Oct 2020. [Abstract, Slides, Slides with blanks]
  
  I will survey emerging connections between applied topology and quantitative topology, including two theorems studied by Gromov. The first, the systolic inequality, bounds the length of the shortest non-contractible loop in a manifold by a function of the manifold's volume. The second, Gromov's "waist of the sphere" theorem, guarantees a point in the image of Sⁿ → R^k, for k < n, whose preimage has sufficiently large volume. The connection between these theorems and applied topology is through Vietoris-Rips complexes. Given a metric space X and a scale parameter r, the Vietoris-Rips simplicial complex VR(X;r) has X as its vertex set, and contains a finite subset as a simplex if its diameter is at most r. Vietoris-Rips complexes were invented by Vietoris in order to define a (co)homology theory for metric spaces, and by Rips for use in geometric group theory. More recently, they have found applications in computational topology for approximating the shape of a dataset.

• Metric reconstruction via optimal transport, Workshop on Optimal Transport, Topological Data Analysis, and Applications to Shape and Machine Learning, Mathematical Biosciences Institute - The Ohio State University, online conference, July 2020. [Abstract, Slides, Slides with blanks, Video]
  
  Given a sample of points X from a metric space M, the Vietoris-Rips simplicial complex VR(X;r) at scale r > 0 is a standard construction to attempt to recover M from X, up to homotopy type. A deficiency is that VR(X;r) is not metrizable if it is not locally finite, and thus does not recover metric information about M. We remedy this shortcoming by defining the Vietoris-Rips metric thickening VR^m(X;r) via the theory of optimal transport. Vertices are reinterpreted as Dirac delta masses, points in simplices are reinterpreted as convex combinations of Dirac delta masses, and distances are given by the Wasserstein distance between probability measures. When M is a Riemannian manifold, the Vietoris-Rips thickening satisfies Hausmann's theorem (VR^m(M;r) is homotopy equivalent to M for r sufficiently small) with a simpler proof: homotopy equivalence VR^m(M;r) → M is now canonically defined as a center of mass map, and its homotopy inverse is the (now continuous) inclusion M → VR^m(M;r). We discuss Vietoris-Rips thickenings of circles and n-spheres, and relate these constructions to Borsuk-Ulam theorems into higher-dimensional codomains.

  Joint work with Michał Adamaszek, John Bush, and Florian Frick.

• Borsuk-Ulam theorems and Vietoris-Rips complexes, Workshop on Topological Data Analysis, as part of the Thematic Program on Toric Topology and Polyhedral Products, Fields Institute, Toronto, Canada, online conference, June 2020. [Abstract, Slides, Slides with blanks, Video]
  
  I will describe Borsuk-Ulam theorems into higher-dimensional codomains. One formulation of the Borsuk-Ulam theorem is that any odd map from the n-dimensional sphere into Euclidean space Rⁿ hits the origin. We generalize this result by increasing the dimension of the codomain, and by finding a small-diameter subset of the sphere whose image contains the origin in its convex hull. For example, given an odd map from the circle of circumference one into R^2k+1, there is some subset of the circle of diameter at most k/(2k+1) whose image contains the origin in its convex hull, and this diameter bound is sharp. We will pose open questions for the n-sphere, where optimal diameters are known only for odd maps of the n-sphere into Rⁿ⁺¹ and Rⁿ⁺². Furthermore, we will explain how these Borsuk-Ulam theorems are related to Vietoris-Rips complexes of spheres.

  Joint work with Johnathan Bush and Florian Frick.

• Descriptors of Energy Landscapes using Topological Analysis (DELTA) at the Mathematics of Data Science Virtual Lecture Series, Tufts University, June 2020. [Abstract, Slides, Slides as Notability file, Video]
  
  Many of the properties of a chemical system are described by its energy landscape, a real-valued function defined on a high-dimensional domain. I will explain how topology, and in particular persistent homology, can be used in order to describe some of the pertinent features of an energy landscape. Whereas a merge tree encodes how connected components of an energy landscape evolve as the energy level increases, persistent homology can also quantify the shape of these connected components. As a motivating example, we completely describe the sublevelset persistent homology of the n-alkanes. In a recently funded NSF Harnessing the Data Revolution project, the DELTA team is learning how to identify and leverage changing topological features of energy landscapes across a range of chemical conditions in order to predict reactivity.

• Applied topology: From global to local at the Mathematics of Data and Decisions at Davis (MADDD) Seminar, UC Davis, online talk, June 2020. [Abstract, Slides, Slides as Notability file, Video]
  
  Through the use of examples, I will explain one way in which applied topology has evolved since the birth of persistent homology in the early 2000s. The first applications of topology to data emphasized the global shape of a dataset, such as the three-circle model for 3 x 3 pixel patches from natural images, or the configuration space of the cyclo-octane molecule, which is a sphere with a Klein bottle attached via two circles of singularity. More recently, persistent homology is being used to measure the local geometry of data. How do you vectorize geometry for use in machine learning problems? Persistent homology, and its vectorization techniques including persistence landscapes and persistence images, provide popular techniques for incorporating geometry in machine learning. I will survey applications arising from machine learning tasks in agent-based modeling, shape recognition, archaeology, materials science, and biology.

• From persistent homology to machine learning, for the Minisymposium on Topological Image Analysis of the SIAM Conference on Mathematics of Data Science (MDS20). This event was rescheduled virtually for June, 2020. [Abstract, Slides, Video, Conference Poster]
  
  I will give an overview of a variety of ways to turn persistent homology output into input for machine learning tasks, including a discussion of the stability and interpretability properties of these methods. Persistent homology is a reasonable summary not only of the global topology but also of the local geometry present in a dataset. When persistent homology is interpreted by humans, the perspective is that the most persistent features matter the most, but machine learning algorithms heavily use low-persistent features measuring local geometry.

• Vietoris-Rips complexes and Borsuk-Ulam theorems, Online Algebraic Topology Seminar (OATS), May 2020. [Abstract, Slides, Video]
  
  Given a metric space X and a scale parameter r, the Vietoris-Rips simplicial complex VR(X;r) has X as its vertex set, and contains a finite subset as a simplex if its diameter is at most r. Vietoris-Rips complexes were invented by Vietoris in order to define a (co)homology theory for metric spaces, and by Rips for use in geometric group theory. More recently, they have found applications in computational topology for approximating of the shape of a dataset. I will explain how the Vietoris-Rips complexes of the circle, as the scale parameter r increases, obtain the homotopy types of the circle, the 3-sphere, the 5-sphere, the 7-sphere, ..., until they are finally contractible. Only very little is understood about the homotopy types of the Vietoris-Rips complexes of the n-sphere. Knowing the homotopy connectivities of Vietoris-Rips complexes of spheres allows one to prove generalizations of the Borsuk-Ulam theorem for maps from the n-sphere into k-dimensional Euclidean space with k > n.

  Joint work with Michał Adamaszek, John Bush, and Florian Frick.

• Computational topology and energy landscapes, National Meeting of the American Chemical Society, for the Graph Theory Underpinning New Domains of Physical Chemistry symposium, Philadelphia, PA, Mar 2020. Talk changed to an online slide submission due to COVID-19. [Abstract, Slides as PDF, Slides on conference webpage]
  
  Many of the properties of a chemical system are described by its energy landscape, a real-valued function defined on a high-dimensional domain. I will explain how topology, and in particular persistent homology, can be used in order to describe some of the pertinent features of an energy landscape. As a motivating example, the low-energy conformation space of an aluminate molecular anion interacting with a simple potassium cation can be well-modeled by the permutohedron (joint work with Aurora Clark and Hung Le). In a recently funded NSF Harnessing the Data Revolution project, the DELTA team is learning how to identify and leverage changing topological features of energy landscapes across a range of chemical conditions in order to predict reactivity.

• Gave a mini-course on Geometric complexes in applied topology at the Winter School on Geometric and Topological Data Analysis at CIMAT, in Guanajuato, Mexico, Jan 2020. [Abstract, Notes]
  
  The course will begin with an introduction to applied an computational topology. The shape of a dataset often reflects important patterns within. One such dataset with an interesting shape is the conformation space of the cyclo-octane molecule, which is a Klein bottle glued to a 2-sphere along two circles. I will introduce persistent homology as a topological tool for learning properties of a space from only a finite sample, and hence for visualizing, understanding, and performing machine learning tasks on high-dimensional datasets.

  Equipped with this motivation, we will move to a mathematical study of geometric complexes arising in applied topology, including nerve, Cech, alpha, and Vietoris-Rips complexes. These complexes thicken a finite sampling into a simplicial complex approximating the underlying space. We will begin by studying these complexes with first principles, but will advertise many open questions and current research directions throughout.

• Vietoris-Rips thickenings of spheres. AMS Special Session on Vietoris-Borsuk-Rips Homotopy, Joint Mathematics Meetings, Denver, Jan 2020. [Abstract, Slides]
  
  If a dataset is sampled from a manifold, then as more and more samples are drawn, the persistent homology of the Vietoris-Rips complex of the dataset converges to the persistent homology of the Vietoris-Rips complex of the manifold. But very little is known about the homotopy types of Vietoris-Rips complexes of manifolds. An exception is the case of the circle: as the scale parameter increases, the Vietoris-Rips complex of the circle obtains the homotopy types of the circle, the 3-sphere, the 5-sphere, the 7-sphere, ..., until finally it is contractible. The Vietoris-Rips complex of the n-sphere first obtains the homotopy type of the n-sphere, and then next the (n+1)-fold suspension of a (topological) quotient of the special orthogonal group SO(n+1) by an alternating group A_n+2. Nothing is known at later scales, although one could conjecture that the critical scale parameters of Vietoris-Rips complexes of spheres are related to Lovász' strongly self-dual polytopes. This is joint work with Michał Adamaszek and Florian Frick.

• Metric reconstruction via optimal transport. [Abstract, Slides, Notes]
  
  Given a metric space X and a distance threshold r > 0, the Vietoris-Rips simplicial complex has as its simplices the finite subsets of X of diameter less than r. A theorem of Jean-Claude Hausmann states that if X is a Riemannian manifold and r is sufficiently small, then the Vietoris-Rips complex is homotopy equivalent to the original manifold. Janko Latschev proves an analogous theorem for sufficiently dense samplings. Little is known about the behavior of Vietoris-Rips complexes for larger values of r, even though these complexes arise naturally when using persistent homology. We describe how as r increases, the Vietoris-Rips complex of the circle obtains the homotopy types of the circle, the 3-sphere, the 5-sphere, the 7-sphere, ..., until finally it is contractible. These homotopy types are related to cyclic and centrally symmetric polytopes and orbitopes. We argue that infinite Vietoris-Rips complexes should be equipped with a different topology: an optimal transport or Wasserstein metric thickening the metric on X. With this new metric, we describe the first change in homotopy type (as r increases) of Vietoris-Rips complexes of higher-dimensional spheres. Joint work with Michał Adamaszek and Florian Frick.

  • Arches Topology Conference, Hurricane, Utah, Apr 2019.
  • AMS Special Session on Topological Data Analysis: Theory and Applications, Joint Mathematics Meetings, Baltimore, Jan 2019. [Slides with an application to Borsuk-Ulam theorems]
  • Topology Seminar, Texas State University, Nov 2018. [Abstract]
  • Upstate New York Topology Seminar (UNYTS), University of Albany, Nov 2018. [Notes]
  • Lafayette-Lehigh Geometry-Topology Seminar, Apr 2018. [Notes]
  • AATRN Online Seminar, Oct 2017. [Video]
  • 58th Cascade Topology Seminar, University of British Columbia, May 2017.
  • Florida International University Winter Conference on Geometry, Topology, and Applications, Jan 2017. [Abstract, Old Notes]
• An introduction to applied topology. [Abstract]
  
  This talk is an introduction to computational topology, as applied to data analysis and to sensor networks. The shape of a dataset often reflects important patterns within. Two such datasets with interesting shapes are a space of 3x3 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. As a second application, I will describe how topology has been applied to coverage problems in mobile sensor networks.

  • Applied Mathematics Colloquium, CU Boulder, Sept 2019. [Abstract, Slides]
  • Three talk lecture series, along with a software tutorial on persistent homology, at the Summer School on Data Science for Dynamical Systems, Lorentz Center, Leiden University, Netherlands, July 2019. [Slides]
  • The Symposium of Physics and Mathematics FCFM-IFM at the University of Michoacan, Morelia, Mexico, Nov 2018. [Abstract, Slides]
  • Department Colloquium, Texas State University, Nov 2018. [Abstract, Slides]
  • Data Science Seminar, University of Tennessee, Nov 2018. [Slides]
  • Undergraduate colloquium at Lafayette College, Apr 2018. [Abstract, Slides]
  • Florida International University Winter Conference on Geometry, Topology, and Applications, Jan 2017. [Abstract, Slides]
  • CSU Computer Science Colloquium, Oct 2016. [Abstract, Slides, Video]
• An introduction to applied topology software. NSF-CBMS Conference and Software Day on Topological Methods in Machine Learning and Artificial Intelligence, College of Charleston, South Carolina, May 2019. Also led a week-long research conversation group, and an afternoon-long coding sprint on real-life applied topology examples for beginners. [Abstract, Slides, Video]
  
  I will survey of a wide variety of software packages for applied and computational topology. We learn software best by running it on examples, and I will describe some of my favorite exercises for students, including sampling points from a torus, Klein bottle, or projective plane. Two real-world datasets that can be analyzed with applied topology software include the three-circle model for optical image patches, and the conformation space of the cyclo-octane molecule (which is a Klein bottle glued to a sphere along two circles of singularities). I'll conclude by describing two applied topology online research seminars and YouTube channels, which allow one to remain connected to the community.

• Survey: From persistent homology to machine learning feature vectors. TRIPODS Summer Bootcamp on Topology and Machine Learning, ICERM, Aug 2018. [Abstract, Slides]
  
  I will give an overview of a variety of ways to turn persistent homology output into input for machine learning tasks, including their stability and interpretability properties. Persistent homology is a reasonable summary not only of the global topology but also of the local geometry present in a dataset. When persistent homology is interpreted by humans, the perspective is that the most persistent features matter the most, but machine learning algorithms heavily use low-persistent features measuring local geometry.

• Connections between Čech complexes and Carathéodory orbitopes. Special Session on Algebraic, Geometric, and Topological Methods in Combinatorics at the AMS Sectional Meeting, Northeastern University, Boston, Apr 2018. [Slides]
• The theory of Vietoris-Rips complexes.
  • AMS Special Session on Topological Data Analysis, Joint Mathematics Meetings in San Diego, Jan 2018. [Abstract, Slides]
  • Applied Topology Seminar, Brown University, Mar 2017. [Abstract, Notes]
    
    Given a metric space X and a distance threshold r > 0, the Vietoris-Rips simplicial complex has as its simplices the finite subsets of X of diameter less than r. A theorem of Jean-Claude Hausmann states that if X is a Riemannian manifold and r is sufficiently small, then the Vietoris-Rips complex is homotopy equivalent to the original manifold. Janko Latschev proves an analogous theorem for sufficiently dense samplings. Little is known about the behavior of Vietoris-Rips complexes for larger values of r, even though these complexes arise naturally in applications of using persistent homology. We describe how as r increases, the Vietoris-Rips complex of the circle obtains the homotopy types of the circle, the 3-sphere, the 5-sphere, the 7-sphere, ..., until finally it is contractible. These homotopy types are related to cyclic and centrally symmetric polytopes and orbitopes. Interestingly, certain Vietoris-Rips complexes of ellipses contain ephemeral summands in their persistent homology modules. We argue that infinite Vietoris-Rips complexes should be equipped with a different topology: an optimal transport or Wasserstein metric thickening the metric on X. With this new metric, we describe the first change in homotopy type (as r increases) of Vietoris-Rips complexes of higher-dimensional spheres. Joint work with Michał Adamaszek, Florian Frick, and Samadwara Reddy.

  • Mini-symposium on Applied and Computational Topology at the SIAM Central States Section Meeting, Oct 2016. [Slides]
• Vietoris-Rips complexes of circles, ellipses, and higher-dimensional spheres.
  • Topology, Geometry, and Data Analysis seminar at The Ohio State University, Feb 2017. [Abstract, Notes]
    
    Given a metric space X and a distance threshold r > 0, the Vietoris-Rips simplicial complex has as its simplices the finite subsets of X of diameter less than r. A theorem of Jean-Claude Hausmann states that if X is a Riemannian manifold and r is sufficiently small, then the Vietoris-Rips complex is homotopy equivalent to the original manifold. Janko Latschev proves an analogous theorem for sufficiently dense samplings. Little is known about the behavior of Vietoris-Rips complexes for larger values of r, even though these complexes arise naturally in applied topology and persistent homology. We describe how as r increases, the Vietoris-Rips complex of the circle obtains the homotopy types of the circle, the 3-sphere, the 5-sphere, the 7-sphere, ..., until finally it is contractible. These homotopy types are connected to cyclic and centrally symmetric polytopes and orbitopes. Interestingly, Latschev's result fails for the ellipse with larger r values, and certain Vietoris-Rips complexes of ellipses contain ephemeral summands in their persistent homology modules. We argue that infinite Vietoris-Rips complexes should be equipped with a different topology: an optimal transport or Wasserstein metric thickening the metric on X. Using this new metric, we describe the first change in homotopy type (as r increases) of Vietoris-Rips complexes of higher-dimensional spheres. Joint work with Michał Adamaszek, Florian Frick, and Samadwara Reddy.

  • Joint Mathematics Meetings, AMS Special Session on Applied and Computational Topology, Jan 2016. [Abstract, Slides]
• What is topology, and how is it applied to data analysis? Front Range Computational & Systems Biology Symposium, July 2016.
• The Vietoris-Rips complexes of a circle. [Abstract, Slides, Poster]
  
  Given a metric space X and a distance threshold r > 0, the Vietoris-Rips simplicial complex has as its simplices the finite subsets of X of diameter less than r. A theorem of Jean-Claude Hausmann states that if X is a Riemannian manifold and r is sufficiently small, then the Vietoris-Rips complex is homotopy equivalent to the original manifold. Little is known about the behavior of Vietoris-Rips complexes for larger values of r, even though these complexes arise naturally in applications using persistent homology. We show that as r increases, the Vietoris-Rips complex of the circle obtains the homotopy types of the circle, the 3-sphere, the 5-sphere, the 7-sphere, ..., until finally it is contractible. As our main tool we introduce a directed graph invariant, the winding fraction, which in some sense is dual to the circular chromatic number. Using the winding fraction we classify the homotopy types of the Vietoris-Rips complex of an arbitrary (possibly infinite) subset of the circle, and we study the expected homotopy type of the Vietoris-Rips complex of a uniformly random sample from the circle. Moreover, we show that as the distance parameter increases, the ambient Čech complex of the circle also obtains the homotopy types of the circle, the 3-sphere, the 5-sphere, the 7-sphere, ..., until finally it is contractible. Joint with Michał Adamaszek.

  • University of Rochester Data Science Colloquium, Apr 2015.
  • Applied Algebraic Topology Research Network, Online Seminar Series, Mar 2015. [Video]
  • Colloquium at Colorado State University, Jan 2015.
  • Geometry and Topology Seminar at Tulane University, Nov 2014.
  • Applied Topology Seminar at the University of Pennsylvania, Nov 2014.
  • Graduate and Faculty Seminar at Duke University, Oct 2014.
  • Colloquium at UNC Greensboro, Oct 2014.
  • Geometry and Topology Seminar at North Carolina State University, Sept 2014.
  • IMA Postdoc Seminar, May 2014.
  • IMA Postdoc Seminar, Oct 2013.
• Evasion paths in mobile sensor networks. [Abstract, Slides, SlidesForUndergrads, Poster, Multimedia]
  
  Suppose that ball-shaped sensors wander in a bounded domain. A sensor doesn't know its location but does know when it overlaps a nearby sensor. We say that an evasion path exists in this sensor network if a moving intruder can avoid detection. In "Coordinate-free coverage in sensor networks with controlled boundaries via homology", Vin de Silva and Robert Ghrist give a necessary condition, depending only on the time-varying connectivity data of the sensors, for an evasion path to exist. Using zigzag persistent homology, we provide an equivalent condition that moreover can be computed in a streaming fashion. However, no method with time-varying connectivity data as input can give necessary and sufficient conditions for the existence of an evasion path. Indeed, we show that the existence of an evasion path depends not only on the fibrewise homotopy type of the region covered by sensors but also on its embedding in spacetime. For planar sensors that also measure weak rotation and distance information, we provide necessary and sufficient conditions for the existence of an evasion path. Joint with Gunnar Carlsson.

  • AMS Special Session on Applied Topology, Joint Mathematics Meetings, online, Jan 2021. [Abstract, Slides]
  • Williams College Colloquium, Mar 2018.
  • Topological data analysis of exclusion zones, Edinburgh, Scotland, Dec 2017.
  • CSU Pattern Analysis Lab, Oct 2015.
  • Duke Graduate & Faculty Seminar, Feb 2015.
  • IMA Workshop on Topological Systems: Communication, Sensing, and Actuation, Mar 2014. [Video]
  • Rocky Mountain Algebraic Combinatorics Seminar, Nov 2013.
  • SIAM Conference on Applied Algebraic Geometry, Aug 2013.
  • Applied Topology in Będlewo, July 2013.
  • MSRI Workshop on Algebraic Topology, June 2013. [Video]
  • Ayasdi Topology Day, June 2013.
  • Stanford CompTop Seminar, May 2013.
  • Special Session on Applied and Computational Topology at MAA MathFest, Aug 2012.
  • Algebraic Topology: Applications and New Directions, July 2012.
  • Minisymposium on Applied Algebraic Topology at SIAM Annual Meetings, July 2012. [Video]
  • Schloss Dagstuhl Seminar on Applications of Combinatorial Topology to Computer Science, Mar 2012.
  • AMS Special Session on Computational and Applied Topology, Joint Meetings, Jan 2012.
  • SIAM Conference on Applied Algebraic Geometry, Oct 2011.
• Nudged elastic band in topological data analysis. [Abstract]
  
  We use the nudged elastic band method from computational chemistry to analyze high-dimensional data. Our approach is inspired by Morse theory, and as output we produce an increasing sequence of small cell complexes modeling the dense regions of the data. We test the method on data sets arising in social networks and in image processing. Furthermore, we apply the method to identify new topological structure in a data set of optical flow patches. Joint with Atanas Atanasov and Gunnar Carlsson.

  • SIAM Conference on Applied Algebraic Geometry, Oct 2011. [Slides]
  • Stanford CompTop Seminar, Jan 2010. [Slides]
  • AMS-SIAM Special Session on Applications of Algebraic Geometry, Joint Meetings, Jan 2010.
• Topological data analysis: Understanding optical flow. IMA Short Course on Applied Algebraic Topology, June 2009. [Slides]

Software talks and demonstrations

• Introduction to Javaplex software for persistent homology. [Abstract]
  
  This exercise session will be a hands-on introduction to persistent homology in topological data analysis. We will use the Javaplex software package, which comes equipped with a tutorial with examples, exercises, and solutions. Two of the real datasets included in the tutorial are the three-circle model for optical image patches, and the conformation space of the cyclo-octane molecule (which is a Klein bottle glued to a sphere along two circles of singularities).

  • Applied Algebraic Topology Research Network, Student Online Seminar Series, Oct 2015.
  • Young Topologists' Meeting, July 2015.
  • AMS Mathematical Research Community on Computational and Applied Topology, June 2011.
• Introduction to JPlex software for persistent homology.
  • AMS Short Course on Computational Topology, Joint Meetings, Jan 2011. [Slides]
  • CSRI Workshop on Combinatorial Algebraic Topology, Aug 2009. [Slides]
  • IMA Short Course on Applied Algebraic Topology, June 2009. [Notes]

Departmental talks and Expository talks

• An introduction to topological combinatorics, Rocky Mountain Algebraic Combinatorics Seminar, Sept 2022. [Abstract, Notes1, Notes2]
  
  The chromatic number of a graph is the number of vertex colors needed so that adjacent vertices have different colors. In 1955, Kneser made a conjecture about the chromatic number of a certain family of graphs (now called Kneser graphs). This conjecture remained unproven for 23 years until Lovász gave a topological proof in 1978. This marked the beginning of a new field, topological combinatorics, in which tools from algebraic topology are used to solve problems in combinatorics. In the first part of my talk, I will share a proof of Kneser's conjecture.

  The second part of my talk will fast-forward to the present day. The Gromov-Hausdorff distance is a notion of dissimilarity between two metric spaces. It is an important tool in geometry, but notoriously difficult to compute. I will show how topological combinatorics provides (potentially tight) lower bounds on the Gromov-Hausdorff distance between unit spheres of different dimensions. This is joint work in a polymath-style project with many people who are currently or formerly at Colorado State, Ohio State, Carnegie Mellon, or Freie Universität Berlin.

• An introduction to applied topology, The Euler Circle, Palo Alto, CA, online, June 2022. [Abstract, Slides]
  
  This talk is an introduction to computational topology, as applied to data analysis and to sensor networks. The shape of a dataset often reflects important patterns within. Two such datasets with interesting shapes are a space of 3x3 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. As a second application, I will describe how topology has been applied to coverage problems in mobile sensor networks.

• An introduction to cyclic polytopes, Barvinok-Novik orbitopes, and the sparsity-promoting L¹ norm, Vietoris-Rips - Tight Span Joint CSU-OSU Seminar, online, June 2021.
• What could Math+YouTube become?, for the CSU Math Department Who is ... what is ...? seminar, online, Mar 2021. [Abstract, Slides, Video]
  
  What could Math+YouTube become? For links corresponding to various sections of my talk:
  (I) Online seminar, see https://www.youtube.com/c/AppliedAlgebraicTopologyNetwork and https://www.aatrn.net/seminar and https://www.ams.org/journals/notices/202008/rnoti-p1138.pdf.
  (II) Interviews, see https://www.aatrn.net/interviews.
  (III) Tutorial-a-thon, see https://sites.google.com/view/aatrn-tutorial-a-thon.
  (IV) Public-facing class segments, see the videos at the top of the webpages https://www.math.colostate.edu/~adams/teaching/dsci475spr2021/ and https://www.math.colostate.edu/~adams/teaching/math366spr2021/.
  (VI) CSU Math YouTube channel, see https://www.youtube.com/channel/UCTYQkbaADo3jk_jiAfkjT2w.

• Coffee break chat on data science, Data Science Seminar, Colorado State University, online, Jan 2021.
• Fair Division, for the Data Science Seminar, Colorado State University, Sept 2020. [Abstract, Notes, Video]
  
  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.

• Borsuk-Ulam theorems into higher-dimensional codomains, for the Rocky Mountain Algebraic Combinatorics Seminar, May 2020. [Abstract, Slides, Slides with blanks, Video]
  
  I will describe Borsuk-Ulam theorems into higher-dimensional codomains. One formulation of the Borsuk-Ulam theorem is that any odd map from the n-dimensional sphere into Euclidean space Rⁿ hits the origin. We generalize this result by increasing the dimension of the codomain, and by finding a small-diameter subset of the sphere whose image contains the origin in its convex hull. For example, given an odd map from the circle into R^2k+1, there is some subset of the circle of diameter at most 2πk/(2k+1) whose image contains the origin in its convex hull, and this diameter bound is sharp. We will pose open questions for the n-sphere, where optimal diameters are known only for odd maps of the n-sphere into Rⁿ⁺¹ and Rⁿ⁺², and relate these questions to Schur polynomials. Joint work with Johnathan Bush and Florian Frick.

• Applying for jobs after math graduate school, Student Chapter of the AMS at CSU, Apr 2020. [Abstract]

  I will give a professional development session on applying for jobs after math graduate school. My session will emphasize how to apply for academic jobs. Many mathematicians instead go into non-academic jobs, including industry, finance, tech, national laboratories, government, defense, and the NSA, and at CSU we are lucky that several faculty either have years of hands-on experience outside of academia, or alternatively are well-trained to advise students towards non-academic jobs.

  Regarding applying to academic jobs, I will give my perspectives on writing research statements, teaching statements, cover letters, diversity statements, and on obtaining letters of recommendation. Now is a good time to begin thinking about these materials in order to go on the job market in the Fall. Sample materials from the year 2012-2013 when I got my PhD are available on my webpage: https://www.math.colostate.edu/~adams/advising/academicJobs/. This webpage also includes links to further information on applying to jobs, academic or otherwise; please feel free to suggest more resources for me to link there!

  The professional development session will be recorded so that if you can't attend it live, you will later be able to find a YouTube recording linked to the webpage above. In addition to public questions, we will also leave time for private (non-recorded) questions at the end.

• Persistent homology, zigzag persistence, and quiver representations, CSU Topology Seminar, Feb 2020. [Abstract]
  
  My talk will be divided into four short vignettes:

  1. From a high level I will briefly explain how one can compute persistent homology using Gaussian elimination.
  2. We will compute one persistent homology barcode via the more efficient (vectorized) data structure used in software. This improved data structure relies on the fact that boundary matrices are quite structured (and sparse).
  3. I will introduce quiver representations and their decompositions.
  4. As a specific case of quiver representations, we will learn about zigzag persistent homology and some of its counterintuitive properties.

• Neighborly polytopes and the sparsity-promoting L¹ norm, CSU Data Science Seminar, Feb 2020. [Abstract, Notes]
  
  I will survey David Donoho's work connecting the sparsity-promoting L¹ norm to neighborly polytopes. Given an underdetermined system of linear equations y=Ax, it is NP-hard to find a solution x with minimal L⁰ norm, i.e., with as few nonzero entries as possible. By contrast, the convex relaxation of this problem, replacing the L⁰ norm of x with the L¹ norm, can be solved efficiently. Interestingly, for many matrices A, the solution to the L¹ problem and the L⁰ problem coincide if the sparsest solution x is sufficiently sparse. A version of this is true, for example, with random matrices A. I will survey how this "equivalence of L⁰ and L¹ optimization" is related to polytopes, cross-polytopes (L¹ balls), centrally-symmetric polytopes, and neighborly polytopes.

• An introduction to persistent homology. Descriptors of Energy Landscape by Topological Analysis (DELTA), Online Seminar, Jan 2020. [Abstract, Slides, Video]
  
  This talk is an introduction to applied and computational topology, in particular as related to the study of energy landscapes arising in chemistry. We give a visual introduction to topology and homology groups. The focus of our talk is on persistent homology, which summarizes the changes in topological features across any increasing sequence of spaces. We emphasize sublevelset persistent homology, which provides a topological summary of real-valued functions, in particular energy functions. We also briefly introduce persistent homology applied to point cloud data, perhaps sampled from the low-energy conformations of a chemical system. Joint with Johnathan Bush, Mark Heim, and Joshua Mirth.

• An introduction to applied topology. CSU DSCI 100: First Year Seminar in Data Science, Nov 2019. [Slides]
• An introduction to Morse theory. Descriptors of Energy Landscape by Topological Analysis (DELTA), Online Seminar, Nov 2019. [Abstract, Video]
  
  We give an introduction to Morse theory. Given a space equipped with a real-valued function, one can use Morse theory to produce a compact cellular model for that space. Furthermore, the cellular model reflects important properties of the function. We describe CW cell complexes, the Morse lemma, the two main theorems of Morse theory, Morse complexes, Morse-Smale complexes, and simplifications of Morse complexes. Joint with Enrique Alvarado.

• An introduction to applied topology. CSU Data Science Seminar, Oct 2019. [Slides]
• An introduction to matroids. CSU Topology Seminar, Sept 2019. [Notes]
• Applying to academic jobs. CSU Math Department, Sept 2019.
• An introduction to applied topology. Talk for an REU at CU Boulder, June 2019. [Slides]
• The waist inequality. CSU Topology Seminar, Mar 2019. [Notes/Slides, Reference]
• An introduction to applied topology. CSU DSCI 100: First Year Seminar in Data Science, Dec 2018. [Slides]
• Lovász' proof of the Kneser conjecture. CSU Topology Seminar, Nov 2018.
• Using homotopy colimits to understand infinite simplicial complexes. CSU Topology Seminar, May 2018. [Abstract, Notes]
  
  Last week we gave an introduction to homotopy colimits. Building on this, we will describe how to use homotopy colimits to understand infinite simplicial complexes. The main tools we will describe are the Projection Lemma, the Quasifibration Lemma, and the Cofinality Theorem from Section 3 of Homotopy colimits - Comparison lemmas for combinatorial applications by Welker, Ziegler, and Zivaljevic. The running example we will use is the Vietoris-Rips simplicial complex of the circle.

• An introduction to homotopy colimits. CSU Topology Seminar, Apr 2018. [Abstract, Notes]
  
  Though colimits exist in the category of topological spaces, they are often not well-behaved from the perspective of homotopy theory. Homotopy colimits are a tool to fix this. I will give examples of homotopy colimits and describe some of their useful properties. Time permitting, I may describe where homotopy colimits have appeared in my research, or describe their relationship to simplicial sets, or describe their relationship to model categories.

• Vietoris-Rips complexes of the circle. CSU Topology Seminar, Oct 2017.
• Sperner's lemma and fair division. CSU Math Club, Oct 2017. [Abstract, Notes]
  
  Suppose three roommates need to pay $3,000 dollars of rent per month for their three-bedroom apartment. The three bedrooms are not equivalent: some are bigger, some are smaller, some are closer to the kitchen, etc, and so it's not necessarily fair to have each room cost $1,000. How should the roommates fairly divide the rent between the rooms so that the total rent for the apartment is $3,000, and how should they decide who gets which room? I will describe how Sperner's lemma, a mathematical lemma related to combinatorics and topology, can be used to find a fair division of rent. This topic has appeared in a New York Times article and in a paper by Francis Su.

• An introduction to Vietoris-Rips complexes. CSU Topology Seminar, Sept 2017.
• Cyclic polytopes and nerve complexes. Rocky Mountain Algebraic Combinatorics Seminar, Oct 2016. [Abstract, Notes]
• Random cyclic dynamical systems. Rocky Mountain Algebraic Combinatorics Seminar, Sept 2015. [Abstract]
• Introduction to discrete Morse theory. University of Minnesota Student Topology Seminar, Dec 2013. [Notes]
• Applied topology. Stanford University Mathematics Camp Guest Lecture Series, July 2013. [Slides]
• Applied topology. Stanford University Mathematics Camp Guest Lecture Series, July 2011.
• Coverage problems in sensor networks. Stanford Undergraduate Mathematical Organization Speaker Series, May 2010. [Abstract]

Talks for undergraduate students

Conferences organized

• With Johnathan Bush, Sunhyuk Lim, Facundo Mémoli, co-organized the Bridging applied and quantitative topology workshop, online, May 9-13, 2022.
• With Maria D'Orsogna, Rachel Neville, Jose Perea, and Chad Topaz, co-organized an ICERM Topical Workshop on Applied mathematical modeling with topological techniques, Aug 5-9, 2019. [Poster]
• With Jeffrey Brock, Melissa McGuirl, Bjorn Sandstede, and Yitzchak Solomon, co-organized an ICERM TRIPODS Summer Bootcamp on Topology and Machine Learning, Aug 6-10, 2018. Software tutorial available. [Photo]

Mini-symposia organized

• With Sara Kalisnik and Elchanan Solomon, co-organizing the AATRN Interview Series, online, 2020-present.
• With Facundo Mémoli, co-organizing the AATRN Vietoris-Rips Seminar, online, 2021-present.
• With Justin O'Connor, Brittany Story, Kyle Salois, and Ciera Street, co-organized an AMS Special Session on Presenting Research Mathematics Through Visual Storytelling: Slides Without Words and Equations, Joint Mathematics Meetings, online, Apr 2022.
• With Hana Dal Poz Kouřimská, Teresa Heiss, and Antonio Rieser, co-organized the inaugural AATRN Poster Session, online, Oct 2021.
• With Hana Dal Poz Kouřimská, Teresa Heiss, Sarah Percival, Lori Ziegelmeier, co-organized the inaugural AATRN+WinCompTop Tutorial-a-thons, online, Mar 2021 and Nov 2021.
• With Mikael Vejdemo-Johansson, co-organized an AMS Special Session on Applied Topology, Joint Mathematics Meetings, Denver, CO, Jan 17, 2020.
• With Mehmet Aktas, Wenwen Li, Murad Ozaydin, co-organized a minisymposium on Applied and Computational Topology at the SIAM Central States Section Meeting, University of Oklahoma, Oct 5-7, 2018
• With Ellen Gasparovic and Katharine Turner, co-organized the 7th Annual Minisymposium on Computational Topology at Computational Geometry Week, Budapest, June 2018.
• With Mikael Vejdemo-Johansson, co-organized an AMS Special Session on Topological Data Analysis at the Joint Mathematics Meetings in San Diego, Jan 2018.
• With Florian Frick, co-organized a session on Symmetric Simplicial Complexes and Polytopes at the SIAM Conference on Applied Algebraic Geometry, Georgia Institute of Technology, Jul-Aug 2017.
• With Bala Krishnamoorthy, co-organized a session on Recent Advances in Applied Algebraic Topology at the AMS Western Sectional Meeting, Washington State University, Apr 2017.
• With Patrick Shipman, co-organized a minisymposium on Applied and Computational Topology at the SIAM Central States Section Meeting, University of Arkansas at Little Rock, Oct 2016.

Conference program committees

• Spring Topology and Dynamical Systems Conferences, Steering Committee Member, 2022-2025.
• NeurIPS Workshop in Symmetry and Geometry in Neural Representations, Program Committee Member, New Orleans, LA, Dec 2022.
• DANGER workshop on interactions between data science and pure mathematics, Oversight Committee Member, online, Aug 2022.
• Symposium on Computational Geometry (SoCG), Program Committee Member, Berlin, Germany, June 2022.
• Symposium on Computational Geometry (SoCG), Multimedia Exposition Track Committee Member, Zürich, Switzerland, June 2020.
• Program committee member for the IEEE ICMLA Special Session on Topological Data Analysis in Machine Learning, Boca Raton, Florida, Dec 2019.

Panels

• Panelist for the Virtual Department of Energy (DOE) Basic Energy Sciences (BES) Chemical Sciences, Geosciences, and Biosciences (CGSB) Data Science Workshop. On Panel 3: Understanding Topology of Chemical Data, Sept 2020.
• Panelist for Next Steps: Mathematics Departments and the Explosive Growth of Computational and Quantitative Offerings in Higher Education, AMS Committee on Education Panel, Joint Mathematics Meetings, Denver, CO, Jan 2020.

YouTube tutorials

Poster presentations

• 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]
• 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]
  • IMA Workshop on Modern Applications of Homology and Cohomology, Oct 2013.
  • IMA Workshop on Topological Data Analysis, Oct 2013.
• Mobile sensors and pursuit-evasion: Can directed algebraic toplogy help? Aalborg GETCO Workshop, Jan 2010. [Poster]