Brief Description:-
(1) Fundamental group schemes: The fundamental group schemes play an important role in the classification of schemes. There is a strategy to define fundamental group schemes, initiated by Madhav Nori, to study certain classes of vector bundles by Tannakian duality. There are many examples of such Tannakian fundamental group schemes. We mainly work on the extended Nori fundamental group scheme described by Otabe. This fundamental group scheme is helpful in classifying semi-essentially finite bundles. We have computed it for abelian varieties and proved the birational invariance property. Currently, we are working on this extended Nori fundamental group scheme in positive characteristic to get arithmetic insights into it.
(2) Connections on a vector bundle: A connection differentiates sections of a vector bundle and tracks how they change from point to point, giving rise to concepts like parallel transport and curvature. It links complex geometry and topology and plays a crucial role in gauge theory. Notably, Atiyah-Weil established conditions for the existence of holomorphic connections on vector bundles over compact Riemann surfaces. We are currently working on related problems in this fascinating area.
(3) Supergeometry: Supergeometry builds upon classical algebraic geometry by introducing both commuting and anticommuting coordinates. In this setting, supermanifolds and superschemes are geometric spaces locally modelled on a mix of even (commuting) and odd (anticommuting) variables. This framework plays a central role in the mathematical formulation of supersymmetry in theoretical physics. We are currently exploring foundational problems in supergeometry with the aim of bridging classical algebraic geometry and modern developments in the field.
I am always open to new ideas and collaborations. Please feel free to contact me for potential collaborations!
Publications:-
(1) On an extension of Nori and local fundamental group schemes, Communications in Algebra, 53(10), 4241-4255, (2025) (with Sanjay Amrutiya).
(2) Notes on fundamental groupoid schemes, (preprints) (with Sanjay Amrutiya).
(3) Semi-finite bundles on Complex tori and Compact Riemann surface, (preprints) (with Sanjay Amrutiya).