Speaking at the Special Session on Topological Data Analysis, Theory and Applications, AMS Fall Western Sectional Meeting, Arizona State University, Tempe, AZ, Nov, 2026.
Speaking at the Special Session on Advances in Applied Topology: Theory and Applications, AMS Fall Eastern Sectional Meeting, George Washington University, Washington, DC, Oct, 2026.
Gromov-Wasserstein Bound between Reeb and Mapper Graphs. TDA seminar, University of Florida, Jan, 2026. Slides
Abstract: The Mapper graph, a computable approximation of the Reeb graph, is widely used in topological data analysis for visualization and inference. However, finding an appropriate metric for comparing Reeb and Mapper graphs that quantifies the rate of convergence of the Mapper graph to the Reeb graph is a difficult problem. Existing metrics do not incorporate measure information when data points are sampled according to an underlying probability measure. This article handles this issue by treating Reeb and Mapper graphs as metric measure spaces, which allows the authors to use Gromov-Wasserstein metrics to compare these graphs directly in order to better incorporate the probability measures that data points are sampled from. We will describe the geometry that arises from this perspective and derive rates of convergence of the Mapper graph to the Reeb graph in this context.
Anti-Vietoris--Rips metric thickenings and Borsuk graph homomorphisms. TDA seminar, University of Florida, Oct, 2025. Slides
Abstract: For a metric space X and a scale r>0, the anti-Vietoris--Rips metric thickening, AVR^m(X;r), is the set of finitely supported probability measures with spread at least r, equipped with the optimal transport topology. We will discuss the homotopy types of anti-Vietoris--Rips metric thickenings of n-spheres at various scales. Using these homotopy types, we will obstruct the existence of graph homomorphisms from a Borsuk graph on S^k to a Borsuk graph on S^n, for k>n, unless the scale of the latter is sufficiently relaxed. We further explore and make conjectures about how the existence and non-existence of Borsuk graph homomorphisms depend on the parameters of both domain and codomain.
Homotopy connectivity of Cech complexes of spheres. DMS Topology & Geometry seminar, IISER Kolkata, September, 2025. Slides
Abstract: Let S^n be the n-sphere with the geodesic metric. The intrinsic Cech complex of S^n at scale r is the nerve of all open balls of radius r in S^n. In this talk, we will show how to control the homotopy connectivity of Cech complexes of spheres as the scale varies over (0, pi), in terms of coverings of spheres. Our upper bound on the connectivity, which is sharp in the case n=1, comes from the chromatic numbers of Borsuk graphs of spheres. Our lower bound is obtained using the conicity (in the sense of Barmak) of Cech complexes of the sufficiently dense, finite subsets of S^n. These bounds imply that for n > 1, the homotopy type of the Cech complex of S^n at scale r changes infinitely many times as r varies over (0, pi). Additionally, we will lower bound the homological dimension of Cech complexes of finite subsets of S^n, in terms of their packings. This is joint work with Henry Adams and Ekansh Jauhari.
Homotopy connectivity of Cech complexes of spheres. Topology and Computing session, SumTopo 2025, August, 2025. Slides
Abstract: Let S^n be the n-sphere with the geodesic metric. The intrinsic Cech complex of S^n at scale r is the nerve of all open balls of radius r in S^n. In this talk, we will show how to control the homotopy connectivity of Cech complexes of spheres as the scale varies over (0, pi), in terms of coverings of spheres. Our upper bound on the connectivity, which is sharp in the case n=1, comes from the chromatic numbers of Borsuk graphs of spheres. Our lower bound is obtained using the conicity (in the sense of Barmak) of Cech complexes of the sufficiently dense, finite subsets of S^n. These bounds imply that for n > 1, the homotopy type of the Cech complex of S^n at scale r changes infinitely many times as r varies over (0, pi). Additionally, we will lower bound the homological dimension of Cech complexes of finite subsets of S^n, in terms of their packings. This is joint work with Henry Adams and Ekansh Jauhari.
Homotopy connectivity of Cech complexes of spheres. GMA colloquium, University of Florida, Mar, 2025.
Abstract: Let S^n be the n-sphere with the geodesic metric. The intrinsic Cech complex of S^n at scale r is the nerve of all open balls of radius r in S^n. In this talk, we will show how to control the homotopy connectivity of Cech complexes of spheres as the scale varies over (0, pi), in terms of coverings of spheres. Our upper bound on the connectivity, which is sharp in the case n=1, comes from the chromatic numbers of Borsuk graphs of spheres. Our lower bound is obtained using the conicity (in the sense of Barmak) of Cech complexes of the sufficiently dense, finite subsets of S^n. These bounds imply that for n > 1, the homotopy type of the Cech complex of S^n at scale r changes infinitely many times as r varies over (0, pi). Additionally, we will lower bound the homological dimension of Cech complexes of finite subsets of S^n, in terms of their packings. This is joint work with Henry Adams and Ekansh Jauhari.
Representations of Energy Landscapes by sublevelset Persistent Homology. TDA seminar, University of Florida, Nov, 2024. Slides
Abstract: Representation of features (like minima and paths) of energy landscapes is a challenging task due to its high dimensionality. Even though merge trees summarize the local minima and lowest barrier pathways, we will describe how more information can be gained by considering the sublevelset topology of the energy landscapes. This paper uses sublevelset persistent homology on the potential energy landscape (PEL) of n-alkanes. We will also discuss the complete characterization given by the paper, of the sublevelset persistent homology of the PEL for all n-alkanes and for all homological dimensions.
Anti-Vietoris--Rips metric thickenings and Borsuk graphs. Topology and Dynamics seminar, University of Florida, Oct, 2024.
Abstract: For a metric space X and a scale r>0, the anti-Vietoris--Rips metric thickening, AVR^m(X;r), is the set of finitely supported probability measures with spread at least r, equipped with the optimal transport topology. We will discuss the homotopy types of anti-Vietoris--Rips metric thickenings of n-spheres at various scales. Using these homotopy types, we will show the absence of graph homomorphisms from Borsuk graph on S^k to Borsuk graph on S^n, for k>n, unless the scale of the latter is sufficiently relaxed. For a compact, Riemannian n-manifold M, we will upper bound the covering dimension of AVR^m(M;r). This is joint work with Henry Adams, Alex Elchesen and Michael Moy.
Anti-Vietoris-Rips metric thickenings, Borsuk graphs and Chromatic numbers. DMS Geometry & Topology Seminar, IISER Kolkata, Aug, 2024.
Abstract: In this talk, I will introduce Borsuk graphs, (circular) chromatic numbers and anti--Vietoris-Rips complexes. I will discuss the homotopy types of the anti--VR complexes and metric thickennings built on S^n upto certain scales. Using topological contradictions, I will show the absence of graph homomorphisms from the Borsuk graph on S^n to the Borsuk graph on S^1 at a certain scale. This gives us a new proof of the lower bound n+2 of the chromatic numbers of Borsuk graphs on S^n. This is joint work with Henry Adams, Alex Elchesen and Michael Moy.
Kneser conjecture-- history and a topological proof. Graduate Student Topology Seminar, University of Florida, Feb, 2024.
Abstract: In graph theory, Kneser graph, K(n,k) is the graph whose vertices are k-element subsets of an n-element set and two vertices are adjacent if they are disjoint. In 1955, M. Kneser conjectured that the chromatic number of K(n,k) is n-2k+2. In 1978, Lovász gave a proof of it for the first time, using the connectedness of the neighborhood complex of a graph. Soon after, Imre Bárány gave a simpler proof using the Borsuk-Ulam theorem and Gale's lemma. In 2002, Joshua Greene further simplified the proof. In this talk, we will discuss the history and the proof given by Joshua Greene.
Chromatic numbers of Borsuk graphs using anti-Vietoris-Rips thickenings. JMM 2024 AMS Special Session on “Bridging Applied and Quantitative Topology”, Jan, 2024. Slides
Abstract: In this talk, I will introduce Borsuk graphs, (circular) chromatic numbers and anti--Vietoris-Rips complexes. I will discuss the homotopy types of the anti--VR complexes and metric thickennings built on S^n upto certain scales. Using topological contradictions, I will show the absence of graph homomorphisms from the Borsuk graph on S^n to the Borsuk graph on S^1 at a certain scale. This gives us a new proof of the lower bound n+2 of the chromatic numbers of Borsuk graphs on S^n. This is joint work with Henry Adams and Alex Elchesen.
Chromatic numbers of Borsuk graphs using anti-Vietoris-Rips thickenings. UF/FSU Topology and Geometry Meeting, Nov, 2023. Slides
Abstract: In this talk, I will introduce Borsuk graphs, (circular) chromatic numbers, and anti-Vietoris-Rips thickenings. I will discuss the homotopy types of the anti-VR thickenings built on S^n in a range of scales. Using topological obstructions, I will show that for k>n, a graph homomorphisms from the Borsuk graph on S^k to the Borsuk graph on S^n can exist only if the scale for the latter Borsuk graph is sufficiently relaxed. Can similar techniques be used to provide new lower bounds on the chromatic number of Borsuk graphs? This is joint work with Henry Adams, Alex Elchesen and, Michael Moy.