Research

Research Interests

1. My Ph.D. thesis work can be divided into two parts:
   (i) In the first part, I have studied the theory of connections for vector bundles on Klein surfaces (as a dianalytic manifold) and the criteria for its existence in the spirit of Atiyah-Weil criteria.
   (ii) In the second part, I have studied real points of moduli of parabolic bundles over real algebraic curves using gauge theoretic and GIT theoretic viewpoints.
2. In post-doc tenure, I have explored the theory of Lie algebroids as well as Lie groupoids and proved the gauge theoretic moduli of Lie algebroid connections for a given smooth vector bundle with fixed real (res. quaternionic) structure on a complex manifold with involution has smooth Hilbert Hausdorff manifold structure and space of connections over the moduli space has principal bundle structure. I have also studied the Hodge decomposition theorem for compact $d$-complex manifold with $d$-Kahler structure.
3. For future directions, I am reading about the classification problem of smooth manifolds, some differential geometric invariants, and some algebraic invariants of certain moduli spaces. I am also reading about Hom structures on various types of algebra (Lie algebra, Leibniz algebra, Lie-Rinehart algebra, etc.) as well as algebroids (Lie algebroids, Leibniz algebroids, etc.).

Publications

1. A gauge theoretic aspects of parabolic bundles over Klein surfaces (with Sanjay Amrutiya) (Link)
2. On d-holomorphic connections (with Sanjay Amrutiya) (Link)

Preprints

1. Hodge decomposition theorem on compact d-Kähler manifolds (with Sanjay Amrutiya) (arxiv)
2. Moduli Spaces of Hom-Lie Algebroid Connections (arxiv)

Ongoing

1. Cohomology of certain algebras, their deformation, Rota-baxter and Nijenhuis operators on certain algebras.