Research

Cut Locus of Submanifolds

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.

Also, with Dr. Aritra Bhowmick, we studied the cut locus of a closed submanifold in a Finsler manifold.

Combinatorial Topology: Independence Complexes

I have developed a strong interest in combinatorial topology, particularly in studying the independence complex of graphs. The independence complex of a graph is a simplicial complex whose faces correspond to independent sets of vertices in the graph. This object encodes rich topological information about the graph structure.

This area bridges discrete mathematics and algebraic topology, revealing surprising connections between combinatorial properties of graphs and their associated topological spaces. Understanding the homotopy type and homology of independence complexes has applications in various areas including theoretical computer science and statistical mechanics.

Lvov-Kaplansky Conjecture

I am investigating the Lvov-Kaplansky conjecture, a fundamental open problem in algebra. The conjecture 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. This deceptively simple statement has profound implications for understanding polynomial identities in matrix algebras and has connections to invariant theory and representation theory.