Research Interests

Published articles

• Examples of real stable bundles on K3 surfaces (with Dino Festi, Daniel Platt, Shizhuo Zhang), arXiv, Accepted for publication in Mathematische Zeitschrift
  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. We use our construction to produce examples of G₂ instantons on the Caldararu-Dominic G₂ manifold. We also produce stable bundles that are invariant under the involution defined by antiholomorphic automorphisms via moduli.




• Revisting 3-Sasakian and G₂ structures (with Simon Salamon), Real and Complex Geometry, special edition on the occasion of Paul Gauduchon's 80th birthday
  arXiv Journal
  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 obtain new information about the homogeneous 3-Sasakian structures N_{k,l} of Aloff and Wallach, and the quaternion Kahler orbifold $\mathbb{HP}^n/C_m$ which underlies each such structure. We construct nearly parallel G₂-structures on all N_{k,l} and show that these induce 3-Sasakian structures on $\mathbb{HP}^n /C_m$.




• 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, and we classify the space of such structures on S³ × S³. We also study the deformations of nearly half-flat SU(3)-structures and compute their deformation space.




• Deformation theory of nearly G₂ manifolds (with Shubham Dwivedi), Communications in Analysis and Geometry, Volume 31 (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 manifolds are unobstructed, and compute the dimension of the deformation space. We also show that the deformations of a compact nearly G₂ manifold correspond to the kernel of a twisted spin-Dirac operator.




• 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 the isometry classes of homogeneous nearly G₂ manifolds. For each isometry class, we compute the dimension of the deformation space of G₂-instantons and also describe these deformation spaces. We show that the infinitesimal deformations of G₂-instantons are unobstructed and are in one-to-one correspondence to the kernel of a twisted spin-Dirac operator.