Research
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 the cut locus of a submanifold. Recently, I have also developed an interest in combinatorial topology. I am studying the independence complex of graphs. I am also working on the Lvov-Kaplansky conjecture, which states that the image of $n \times n$ matrices over any field $K$ under a 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 the same as the distance from $N$ to $q$.
If $N$ is a smooth submanifold of $M$, then we say it is a 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 its relation 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.