Thickenings of a metric space capture local geometric properties of the space. Here, we exhibit applications of lower bounding the topology of thickenings of the circle and more generally the sphere. We explain interconnections with the geometry of circle actions on Euclidean space, the structure of zeros of trigonometric polynomials, and theorems of Borsuk–Ulam type. We use the combinatorial and geometric structure of the convex hull of orbits of circle actions on Euclidean space to give geometric proofs of the homotopy type of metric thickenings of the circle. Homotopical connectivity bounds of thickenings of the sphere allow us to prove that a weighted average of function values of odd maps Sn→double-struckRn+2 on a small diameter set is zero. We prove an additional generalization of the Borsuk–Ulam theorem for odd maps S2n−1→double-struckR2kn+2n−1. We prove such results for odd maps from the circle to any Euclidean space with optimal quantitative bounds. This in turn implies that any raked homogeneous trigonometric polynomial has a zero on a subset of the circle of a specific diameter; these results are optimal.

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., two- or three-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 much information to be gained by also considering the topology of each connected component at different energy thresholds (or sublevelsets). We propose sublevelset persistent homology as an appropriate tool for this purpose. Our computations on the configuration phase space of n-alkanes from butane to octane allow us to conjecture, and then prove, a complete characterization of the sublevelset persistent homology of the alkane CmH2m+2 Potential Energy Landscapes (PELs), for all m, in all homological dimensions. We further compare both the analytical configurational PELs and sampled data from molecular dynamics simulation using the united and all-atom descriptions of the intramolecular interactions. In turn, this supports the application of distance metrics to quantify sampling fidelity and lays the foundation for future work regarding new metrics that quantify differences between the topological features of high-dimensional energy landscapes.

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We propose a torus model for high-contrast patches of optical flow. Our model is derived from a database of ground-truth optical flow from the computer-generated video Sintel, collected by Butler et al. in A naturalistic open source movie for optical flow evaluation. Using persistent homology and zigzag persistence, popular tools from the field of computational topology, we show that the highcontrast 3 × 3 patches from this video are well-modeled by a torus, a nonlinear 2-dimensional manifold. Furthermore, we show that the optical flow torus model is naturally equipped with the structure of a fiber bundle, related to the statistics of range image patches.We study the nonlinear statistics of a space of high-contrast 3 × 3 optical flow patches from the Sintel dataset using the topological machinery of [19] and [3]. We identify the topologies of dense subsets of this space using Vietoris-Rips complexes and persistent homology. The densest patches lie near a circle, the horizontal flow circle [1]. In a more refined analysis, we select out the optical flow patches Key words and phrases. Optical flow, computational topology, persistent homology, fiber bundle, zigzag persistence.

We show that the size of codes in projective space controls structural results for zeros of odd maps from spheres to Euclidean space. In fact, this relation is given through the topology of the space of probability measures on the sphere whose supports have diameter bounded by some specific parameter. Our main result is a generalization of the Borsuk-Ulam theorem, and we derive four consequences of it: (i) We give a new proof of a result of Simonyi and Tardos on topological lower bounds for the circular chromatic number of a graph; (ii) we study generic embeddings of spheres into Euclidean space, and show that projective codes give quantitative bounds for a measure of genericity of sphere embeddings; and we prove generalizations of (iii) the Ham Sandwich theorem and (iv) the Lyusternik-Shnirel'man-Borsuk covering theorem for the case where the number of measures or sets in a covering, respectively, may exceed the ambient dimension.