We discuss the chromatic configuration space $\text{Conf}_\Gamma(\mathbb{R}^N)$ associated to a simple finite graph $\Gamma$. This is the complement of the so-called graphic subspace arrangement associated to $\Gamma$. Using poset topology, we show that the Poincaré polynomial of the chromatic configuration space is the reciprocal of the chromatic polynomial of $\Gamma$ (with signs). We further show that these spaces split after a single suspension as a wedge of spheres, the number of wedge summands being given in terms of the coefficients of the chromatic polynomial. This splitting is deduced from the description of the homology generators in terms of “forests of spanning trees with no-broken cycles (NBC)”. This description generalizes the theory of classical configuration spaces. As a good application, we deduce the homology of spaces of configurations consisting of “$n$ moving objects in $\mathbb{R}^N$, distinct or not, each avoiding a given subset of $r$ fixed obstacles”

Venue: Zoom Link
Time: 4 PM IST (13th October)
2 PM IST (15th October)

We use poset stratifications to construct a Grothendieck ring for stratifiable spaces. We then use it to compute the topological Euler characteristic of a number of constructions in topology and geometry. The computations are remarkably streamlined. Main applications pertain to orbit spaces and to spaces stratified by configuration spaces, like graph configuration spaces, orbit configuration spaces, or finite subset spaces. Our $K_0$ can be viewed as an analog of the ring of constructible functions in the theory of Euler calculus, valid for a larger class of spaces.

Venue: Zoom Link
Time: 2 PM IST

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.

Venue: LHC 211, IISER Kolkata
Time: 2 PM IST

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.

Venue: Zoom Link
Time: 5 PM IST (7:30 AM EST)
Slides

One method to understand compact finite dimensional complexes is to break it up into simpler pieces. In this context, one of the most useful results is that up to homotopy, these can be built up from disks. Stronger decomposition results can be proved when one considers either the loop space or a suspension. In this talk, I will discuss some of these results and some geometric consequences of them.

Venue: LHC 211, IISER Kolkata
Time: 3 PM IST