Research Interests
Motivated by $G_2$-manifolds with coassociative fibrations in the adiabatic limit, Donaldson and Scaduto conjectured the existence of associative submanifolds homeomorphic to a three-holed $3$-sphere with three asymptotically cylindrical ends in $X \times \mathbb{R}^3$, or equivalently similar special Lagrangians in $X \times \mathbb{R}^2$, where $X$ is an $A_2$-type ALE hyperkähler manifold. We prove this conjecture by solving a real Monge-Amp\`ere equation with a singular right-hand side. The method produces many other asymptotically cylindrical $U(1)$-invariant special Lagrangians in $X\times \mathbb{R}^2$, where $X$ arises from the Gibbons-Hawking construction. You can see Yang Li's talk on the subject here.
2- Towards a Monopole Fueter Floer Homology for 3-Manifolds.
The goal of this program is to define a new Floer homology for 3-manifolds. This invariant is defined by a `count' of Fueter sections on a hyperkahler bundle, where the fibers are modeled on the moduli spaces of monopoles on R3.
There are three main reasons motivating the definition of these invariants.
- Following a program proposed by Donaldson and Segal, counting these Fueter sections on special Lagrangians in Calabi-Yau 3-folds would potentially define a Calabi-Yau invariant.
- This generalizes Taubes' SW=Gr of symplectic 4-manifolds to Calabi-Yau 3-folds, as conjectured by Donaldson and Segal.
- It is closely related to certain generalizations of Seiberg-Witten equations proposed by Taubes.
- This Floer-theory potentially can categorify a Rozansky-Witten invariant.
The main difficulty in defining this Monopole Fueter Floer Homology comes from the compactness problems. In a sequence of papers, we are addressing different sources of non-compactness and wall-crossing phenomena and moving towards compactifying the spaces of Fueter sections of the monopole bundles. I have given partial answer to these questions in my PhD thesis under supervision of Simon Donaldson: Monopoles, Singularities and Hyperkähler Geometry. You can also see my talk on the subject: Towards a Fueter Floer Homology.
3- Moduli Spaces of Monopoles on 3-Manifolds, Calabi-Yau 3-folds, and G2-manifolds.
Monopoles on 3-manifolds appear as dimensional reduction of instantons on 4-manifolds. They contain interesting information about 3-manifolds. The study of monopoles on 3-manifolds is also motivated by the higher-dimensional gauge theory, and specially the study of monopoles on Calabi-Yau 3-folds. As a first step in that direction, we prove the existence of non-trivial irreducible SU(2)-monopoles with Dirac singularities on rational homology 3-spheres using a gluing construction: Singular Monopoles on Closed 3-Manifolds.
Joint with D. Fadel, L. Foscolo, Á. Nagy, and G. Oliveira we are working on different methods of construction of monopoles on Calabi-Yau and G2-manifolds.