My area of research is differential geometry, geomteric analysis and gauge theory. I particularly think about

• manifolds with special geometric structures
• moduli spaces of instantons on special manifolds
• deformation theory of special algebraic structures

Published articles

• Nearly half-flat SU(3)-structures on S³×S³, Differential Geometry and Its Applications,Volume 97 (2024) arXiv Journal
  Abstract

We study the SU(3)-structure induced on an oriented hypersurface of a 7-dimensional manifold with a nearly parallel G₂-structure. We call such SU(3)-structures nearly half-flat. We characterise the left invariant nearly half-flat structures on S³×S³. This characterisation then help us to systematically analyse nearly parallel G₂-structures on an interval times S³×S³.




• Deformation theory of nearly G₂ manifolds (with Shubham Dwivedi), Communications in Analysis and Geometry, Volume 31, Number 3(2023), arXiv Journal
  Abstract

We study the deformation theory of nearly G₂ manifolds. These are seven dimensional manifolds admitting real Killing spinors. We show that the infinitesimal deformations of nearly G₂ structures are obstructed in general. Explicitly, we prove that the infinitesimal deformations of the homogeneous nearly G₂ structure on the Aloff—Wallach space are all obstructed to second order. We also completely describe the cohomology of nearly G₂ manifolds.




• Deformations of G₂ instantons on nearly G₂ manifolds, Annals of Global Analysis and Geometry, Volume 62 (2022), arXiv Journal
  Abstract

We study the deformation theory of G₂-instantons on nearly G₂ manifolds. There is a one-to-one correspondence between nearly parallel G₂ structures and real Killing spinors, thus the deformation theory can be formulated in terms of spinors and Dirac operators. We prove that the space of infinitesimal deformations of an instanton is isomorphic to the kernel of an elliptic operator. Using this formulation we prove that abelian instantons are rigid. Then we apply our results to describe the deformation space of the canonical connection on the four normal homogeneous nearly G₂ manifolds.

Thesis

• Deformation theory of nearly G₂-structures and nearly G₂ instantons, PhD thesis, University of Waterloo (2021) Available Online