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 topological quandles. I am reading Alexander quandles. 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 $N\subseteq \mathrm{Cr}(f)$ (critcal 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 budle. Here we have chosen a Euclidean metric on $E$.

In one of my paper (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. The image of polynomials on upper triangular matrix algebras (joint with Saikat Panja), June 2022
   arXiv link

   Abstract: Let $p$ be a polynomial in non-commutative variables $x_1,x_2,\cdots,x_s$ 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 $T_n(K)$. We introduce a family of polynomial, to be called multi-index $p$-inductive polynomials for a given polynomial $p$. Using this we will show that, if $p$ is a polynomial identity of $T_m(K)$ but not of $T_{m+1}(K)$, then $p \left(T_n(K)\right)\subseteq T_n(K)^{(m-1)}$. Equality is achieved in the case $m=1$ and $n-1$. In the other cases we specify some structure of the matrices which are member of $p \left(T_n(K)\right)$. We also prove that image of $T_m(K)^\times$ under a word map is Zariski dense in $T_m(K)^\times$.

2. Counterexample to a conjecture about dihedral quandle (joint with Saikat Panja), May 2022
   arXiv 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.

3. A connection between cut locus, Thom spaces and Morse-Bott functions (joint with Dr Somnath Basu), June 2021
   To appear in the Algebraic & Geometric Topology
   arXiv 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.