I am postdoctoral researcher at the Université Libre de Bruxelles working in gauge theory and geometric flows on Kähler manifolds. My work in the past has focused on the Yang-Mills flow in this context, a non-linear parabolic (after gauge fixing) PDE for a 1-parameter family of connections on a holomorphic vector bundle. On a Kähler manifold, by a result of Donaldson this flow is known to exist for all time, and I have studied its asymptotic behaviour and singularity formation at infinite time when the bundle in question is unstable (see my publications below).
My current work is a continuation of these ideas in various directions. One project (joint with Richard Wentworth) is a study of compactifictions of the moduli space of holomorphic vector bundles on projective manifolds in higher dimensions. This space can be compactified by means of algebraic geometry, namely by adding torsion-free Gieseker-semi-stable sheaves at the boundary. On the other hand, via the Kobayashi-Hitchin correspondence, it is also the space of irreducible Hermitian-Yang-Mills connections. Due to more recent work of Tian, this space admits a compactification of a kind more familiar to gauge theorists, defined as a space of ideal connections using the compactness theorems of Uhlenbeck. In the case of a projective surface, Jun Li has proven that in fact the Uhlenbeck compactification has the structure of scheme, and there is a natural morphism from the algebraic-geometric compactification. We seek to generalise this result to the higher dimensional case. This is very closely related to recent work of Greb-Toma on the moduli space of slope-semi-stable sheaves in higher dimensions.
A second project (with Joel Fine) is an investigation of the Calabi flow on ruled manifolds. This is a 4th order flow that is designed to find constant scalar curvature Kähler metrics. There are very few examples where even the long-time existence is known. One case where this seems to be tractable is the case of a projective bundle. More specifically, even the case of the projectivisation of a rank 2 non-split extension over a Riemann surface is completely unstudied. It is known that manifolds of this kind admit no constant scalar curvature Kähler metric, so the Calabi flow must blow up either at finite or infinite time. Nevertheless, we expect to be able to prove long-time existence in this case. The Calabi flow can be approximated in an appropriate sense by the Yang-Mills flow. This latter flow, thought of as a flow of moving holomorphic structures starting at the non-split extension, converges at infinity to the split holomorphic structure. Moreover, up to diffeomorphisms, the Calabi flow may be thought of as a flow of complex structures on the projectivisation obtained as the gradient flow of the Calabi functional. By analogy with the asymptotic behaviour of the Yang-Mills flow, one would expect this flow to converge to a limiting complex structure, which is the projectivisation of the split bundle. The next step would be to investigate singularity formation for the corresponding flow of metrics.
Here is a list of my publications:
- “Analytic cycles, Bott-Chern forms, and singular sets for the Yang-Mills flow on Kähler manifolds” (with Richard Wentworth), Advances in Mathematics, 279 (2015), 501-531. See also arXiv:1402.3808.
- “Asymptotics of the Yang-Mills flow for holomorphic vector bundles over Kähler manifolds: the canonical structure of the limit”, Journal für die reine und angewandte Mathematik (Crelles Journal), Volume 2015, Issue 706, Pages 123-191. See also arXiv:1206.5491v4.
Work in progress
cations of the moduli space of stable bundles on a projective manifold” (joint
with Richard Wentworth)
- “Long-time existence and singularity formation for the Calabi flow on a ruled surface”
(joint with Joel Fine)
- “Continuity of the Yang-Mills flow on the space of holomorphic structures”
- Limits and bubbling sets for the Yang-Mills flow on Kähler manifolds, IAP Dygest Annual Meeting, KU Leuven (April 2015)
- Limits and bubbling sets for the Yang-Mills flow on Kähler manifolds, Université Libre de Bruxelles, Differential Geometry Seminar (October 2014)
- Limits and bubbling sets for the Yang-Mills flow on Kähler manifolds, Institute for Mathematical Sciences/National University of Singapore (July 2014)
- Limits and bubbling sets for the Yang-Mills flow on Kähler manifolds, LMU München Oberseminar, (April 2014)
- Asymptotics of the Yang-Mills flow on Kähler manifolds, Princeton University, Differential Geometry Seminar (May 2013)