arXiv subjects

Atish Mitra

Publications and source records attributed to Atish Mitra.

23 records · Page 2Linked to original sources

arXiv2021

The Space of Persistence Diagrams on $n$ Points Coarsely Embeds into Hilbert Space

We prove that the space of persistence diagrams on $n$ points (with the bottleneck or a Wasserstein distance) coarsely embeds into Hilbert space by showing it is of asymptotic dimension $2n$. Such an embedding enables utilisation of Hilbert space techniques on the space of persistence diagrams. We also prove that when the number of points is not bounded, the corresponding spaces of persistence diagrams do not have finite asymptotic dimension. Furthermore, in the case of the bottleneck distance, the corresponding space does not coarsely embed into Hilbert space.

math.MG↗

arXiv2020

On maps with continuous path lifting

We study a natural generalization of covering projections defined in terms of unique lifting properties. A map $p:E\to X$ has the "continuous path-covering property" if all paths in $X$ lift uniquely and continuously (rel. basepoint) with respect to the compact-open topology. We show that maps with this property are closely related to fibrations with totally path-disconnected fibers and to the natural quotient topology on the homotopy groups. In particular, the class of maps with the continuous path-covering property lies properly between Hurewicz fibrations and Serre fibrations with totally path-disconnected fibers. We extend the usual classification of covering projections to a classification of maps with the continuous path-covering property in terms of topological $π_1$: for any path-connected Hausdorff space $X$, maps $E\to X$ with the continuous path-covering property are classified up to weak equivalence by subgroups $H\leq π_1(X,x_0)$ with totally path-disconnected coset space $π_1(X,x_0)/H$. Here, "weak equivalence" refers to an equivalence relation generated by formally inverting bijective weak homotopy equivalences.

math.AT↗

arXiv2018

Groupoid Models of $C^*$-algebras and Gelfand Duality

We construct a large class of morphisms, which we call partial morphisms, of groupoids that induce $*$-morphisms of maximal and minimal groupoid $C^*$-algebras. We show that the association of a groupoid to its maximal (minimal) groupoid $C^*$-algebra and the association of a partial morphism to its induced morphism are functors (both of which extend the Gelfand functor). We show how to geometrically visualize lots of $*$-morphisms between groupoid $C^*$-algebras. As an application, we construct a groupoid models of the entire inductive systems of the Jiang-Su algebra $\mathcal{Z}$ and the Razak-Jacelon algebra $\mathcal{W}$.

math.OA↗

arXiv2016

Large scale absolute extensors

This paper is devoted to dualization of dimension-theoretical results from the small scale to the large scale. So far there are two approaches for such dualization: one consisting of creating analogs of small scale concepts and the other amounting to the covering dimension of the Higson corona $ν(X)$ of $X$. The first approach was used by M.Gromov when defining the asymptotic dimension $\asdim(X)$ of metric spaces $X$. The second approach was implicitly contained in the paper \cite{Dran AsyTop} by Dranishnikov on asymptotic topology. It is not known if the two approaches yield the same concept. However, Dranishnikov-Keesling-Uspenskiy proved $\dim(ν(X)\leq \asdim(X)$ and Dranishnikov established that $\dim(ν(X)= \asdim(X)$ provided $\asdim(X) < \infty$. We characterize asymptotic dimension (for spaces of finite asymptotic dimension) in terms of extensions of slowly oscillating functions to spheres. Our approach is specifically designed to relate asymptotic dimension to the covering dimension of the Higson corona $ν(X)$ in case of proper metric spaces $X$. As an application, we recover the results of Dranishnikov-Keesling-Uspenskiy and Dranishnikov.

math.MG↗

arXiv2005

Cohomological dimension with respect to nonabelian groups

Cencelj and Dranishnikov showed that for certain nilpotent groups $G$, $K(G_{ab},1) \in \text{AE}(X)$ is equivalent to $K(G,1) \in \text{AE}(X)$ for any compacta $X$ (here $G_{ab}$ is the abelianization of $G$). We examine the same problem for solvable groups. We also give an elementary proof of this fact for any nilpotent group and any 2-dimensional metric space.

math.GT↗