My area of interest includes differential geometry, differential topology, algebraic topology and polynomial maps. More specifically, I am working on the cut locus of a submanifold. Below, I have given a brief overview of cut locus of a submanifold. Recently, I also made an interest towards combinatorial topology. I am studying independence complex of graphs. I am also working on Lvov-Kaplansky conjecture, which says that image of $n\times n$ matrices over any field $K$ under the multilinear polynomial in non-commutative variables is a vector space.

For a given Riemannian manifold $M$ and $N\subset M$ the cut locus of $N$, $\mathrm{Cu}(N)$, is the collection of points $q\in M$ such that there exists a distance minimal geodesic $\gamma$ joining $N$ to $q$ such that any extension of $\gamma$ beyond $q$ is no longer a distance minimal geodesic. Here, by the distance minimal geodesic $\gamma$ joining $N$ to $q$ we mean that there exists $p\in N$ such that the length of $\gamma$ from $p$ to $q$ is same as the distance from $N$ to $q$.

If $N$ is a smooth submanifold of $M$, then we say it is non-degenerate critical submanifold of $f:M\to \mathbb{R}$ if $N\subseteq \mathrm{Cr}(f)$ (critical points of $f$) and for any $p\in N$, the Hessian of $f$ at $p$ is non-degenerate in the direction normal to $N$ at $p$. The function $f$ is said to be Morse-Bott if the connected components of $\mathrm{Cr}(f)$ are non-degenerate critical submanifolds.

The Thom space $\mathrm{Th}(E)$ of a real vector bundle $E\to B$ of rank $k$ is $D(E)/S(E)$, where $D(E)$ is the unit disk bundle and $S(E)$ is the unit sphere bundle. Here we have chosen a Euclidean metric on $E$.

In one of my papers (joint with Dr Somnath Basu), we discussed the cut locus of a closed submanifold and described the relation between it with Thom spaces and Morse-Bott functions.

Currently, I am working on the cut locus of a quotient manifold and its applications to classifying spaces.

Publications and Preprints

1. Distance from a Finsler submanifold to its cut locus and the existence of a tubular neighborhood (joint with A. Bhowmick), November 2024
   arXiv link

   Abstract:In this article we prove that for a closed, not necessarily compact, submanifold $N$ of a possibly non-complete Finsler manifold $(M,F)$, the cut time map is always positive. As a consequence, we prove the existence of a tubular neighborhood of such a submanifold. When $N$ is compact, it then follows that there exists an $\epsilon>0$ such that the distance between $N$ and its cut locus $\mathrm{Cu}(N)$ is at least $\epsilon$. This was originally proved by B. Alves and M. A. Javaloyes (Proc. Amer. Math. Soc. 2019). We have given an alternative, rather geometric proof of the same, which is novel even in the Riemannian setup. We also obtain easier proofs of some results from N. Innami et al. (Trans. Amer. Math. Soc., 2019), under weaker hypothesis.

2. On the focal locus of submanifolds of a Finsler manifold (joint with A. Bhowmick), September 2024
   arXiv link

   Abstract: In this article, we investigate the focal locus of closed (not necessarily compact) submanifolds in a forward complete Finsler manifold. The main goal is to show that the associated normal exponential map is regular in the sense of F.W. Warner (Am. J. of Math., 87, 1965). This leads to the proof of the fact that the normal exponential is non-injective near tangent focal points. As an application, following R.L. Bishop's work (Proc. Amer. Math. Soc., 65, 1977), we express the tangent cut locus as a closure of a certain set of points, called separating tangent cut points. This strengthens the results from the present authors' previous work (J. Geom. Anal., 34, 2024).

3. Six problems in Intuitive Geometry (joint with S. Honda, N. Horio, J. Itoh, N. Nomura), January 2024
   Bulletin Mathematique de la Société des Sciences Mathématiques de Roumanie Volume 67, No. 2, April 2024, Pages 223-229
   Journal link

   Abstract: In this note we present several open, yet easy to state, geometric problems.

4. On the cut locus of submanifolds of a Finsler manifold (joint with Aritra Bhowmick), July 2023
   Journal of Geometric Analysis Volume 38, August 2024, Pages 1-38
   arXiv link   Journal link

   Abstract: In this article, we investigate the cut locus of closed (not necessarily compact) submanifolds in a forward complete Finsler manifold. We explore the deformation and characterization of the cut locus, extending the results of Basu and the second author (Algebraic and Geometric Topology, 2023). Given a submanifold $N$, we consider an $N$-geodesic loop as an $N$-geodesic starting and ending in $N$, possibly at different points. This class of geodesics were studied by Omori (Journal of Differential Geometry, 1968). We obtain a generalization of Klingenberg's lemma for closed geodesics (Annals of Mathematics, 1959) for $N$-geodesic loops in the reversible Finsler setting.

5. Independence complexes of wedge of graphs (joint with Saikat Panja and Navnath Daundkar), March 2023
   arXiv link

   Abstract: In this article, we introduce the notion of a wedge of graphs and provide detailed computations for the independence complex of a wedge of path and cycle graphs. In particular, we show that these complexes are either contractible or wedges of spheres.

6. The image of polynomials and Waring type problems on upper triangular matrix algebras (joint with Saikat Panja), June 2022
   Journal of Algebra Volume 631, October 2023, Pages 148-193
   arXiv link   Journal link

   Abstract: Let $p$ be a polynomial in non-commutative variables $x_1,x_2,\ldots,x_n$ with constant term zero over an algebraically closed field $K$. The object of study in this paper is the image of this kind of polynomial over the algebra of upper triangular matrices $T_m(K)$. We introduce a family of polynomials called multi-index $p$-inductive polynomials for a given polynomial $p$. Using this family we will show that, if $p$ is a polynomial identity of $T_t(K)$ but not of $T_{t+1}(K)$, then $p \left(T_m(K)\right)\subseteq T_m(K)^{(t-1)}$. Equality is achieved in the case $t=1,~m-1$ and an example has been provided to show that equality does not hold in general. We further prove existence of $d$ such that each element of $T_m(K)^{(t-1)}$ can be written as sum of $d$ many elements of $p\left( T_m(K) \right)$. It has also been shown that the image of $T_m(K)^\times$ under a word map is Zariski dense in $T_m(K)^\times$.

7. Counterexample to a conjecture about dihedral quandle (joint with Saikat Panja), May 2022
   Miskolc Mathematical Notes Volume 25, January 2024, No. 1, Pages 425-428
   arXiv link   Journal link

   Abstract: It was conjectured that the augmentation ideal of a dihedral quandle of even order $n>2$ satisfies $\left|\Delta^k(\textup{R}_n)/\Delta^{k+1}(\textup{R}_{n})\right|=n$ for all $k\ge 2$. In this article we provide a counterexample against this conjecture.

8. A connection between cut locus, Thom spaces and Morse-Bott functions (joint with Dr Somnath Basu), June 2021
   Algebraic & Geometric Topology Volume 23, November 2023, Pages 4185-4233
   arXiv link   Journal link

   Abstract: Associated to every closed, embedded submanifold $N$ in a connected Riemannian manifold $M$, there is the distance function $d_N$ which measures the distance of a point in $M$ from $N$. We analyze the square of this function and show that it is Morse-Bott on the complement of the cut locus $\mathrm{Cu}(N)$ of $N$, provided $M$ is complete. Moreover, the gradient flow lines provide a deformation retraction of $M-\mathrm{Cu}(N)$ to $N$. If $M$ is a closed manifold, then we prove that the Thom space of the normal bundle of $N$ is homeomorphic to $M/\mathrm{N}$. We also discuss several interesting results which are either applications of these or related observations regarding the theory of cut locus. These results include, but are not limited to, a computation of the local homology of singular matrices, a classification of the homotopy type of the cut locus of a homology sphere inside a sphere, a deformation of the indefinite unitary group $U(p,q)$ to $U(p)\times U(q)$ and a geometric deformation of $GL(n,\mathbb{R})$ to $O(n,\mathbb{R})$ which is different from the Gram-Schmidt retraction.