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.

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.

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.

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.

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.

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.

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.

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.

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.

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.