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

Preprints

• Examples of real stable bundles on K3 surfaces (with Dino Festi, Daniel Platt, Yuuji Tanaka ),
  arXiv
  Abstract

Motivated by gauge theory on manifolds with exceptional holonomy, we construct examples of stable bundles on K3 surfaces that are invariant under two involutions: one is holomorphic; and the other is anti-holomorphic. These bundles are obtained via the monad construction, and stability is examined using the Generalised Hoppe Criterion of Jardim-Menet-Prata-Sá Earp, which requires verifying an arithmetic condition for elements in the Picard group of the surfaces. We establish this by using computer aid in two critical steps: first, we construct K3 surfaces with small Picard group-one branched double cover of ℙ¹×ℙ¹ with Picard rank 2 using a new method which may be of independent interest; and second, we verify the arithmetic condition for carefully chosen elements of the Picard group, which provides a systematic approach for constructing further examples.

• Revisting 3-Sasakian and G₂ structures (with Simon Salamon), To appear in a special Springer edition on the occasion of Paul Gauduchon’s 80th birthday
  arXiv
  Abstract

The algebra of exterior differential forms on a regular 3-Sasakian 7-manifold is investigated, with special reference to nearly-parallel G₂ 3-forms. This is applied to the study of 3-forms invariant under cohomogeneity-one actions by SO(4) on the 7-sphere and on Berger's space SO(5)/SO(3).

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