For a manifold $M$, a smooth subbundle of the tangent bundle $TM$ is called a distribution on $M$. A distribution $D$ is said to be bracket generating if successive Lie brackets of (smooth) local vector fields in $D$ generate the whole tangent bundle. By Chow's theorem, any two points of $M$ can be joined by a smooth path which is tangential to the distribution at all points. We shall begin by explaining this theorem and some related questions in the context of $3$-dimensional contact distributions, which are the primary examples of bracket generating distributions. More generally, one may consider smooth immersions or embeddings $f:\Sigma\to (M,D)$ of an arbitrary manifold $\Sigma$ such that the derivative of $f$ maps $T\Sigma$ into $D$. Such maps are called horizontal to $D$. Horizontal immersions to contact distributions are completely understood due to a result of Gromov. We also have a similar classification of smooth horizontal immersions for holomorphic contact structures and their real analogues. This is a joint work with Aritra Bhowmick.

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.

TBA