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On this page you'll find a couple of paragraphs about my current research interests and links to my publications. I may eventually put some research-expository type material here as well.

Current collaborators

Of course, I enjoy discussing mathematics with lots of people. These are the ones that I'm currently writing papers with!

Current research topics

Kaehler geometry With Richard Melrose I am writing a semi-expository article which revisits the Kaehler gluing theorems for constant-scalar-curvature and extremal metrics of Arezzo, Pacard myself and Szekelyhidi. The novelty of our approach is to do everything as smoothly as possible: in particular we obtain rather precise information about the one-parameter family of cscK metrics as the gluing parameter goes to zero.

Partial density functions Also in the realm of Kaehler geometry, I am working with Julius Ross on understanding as precisely as possible the large-k asymptotic behaviour of partial density function defined for an ample line bundle over a smooth projective variety, with vanishing conditions imposed along some complex submanifold. We have a pretty good understanding if there is a little bit of symmetry, but going beyond that is a very interesting challenge.

Four-dimensional metrics Special metrics, for example anti-self-dual, anti-self-dual Einstein and scalar-flat Kaehler metrics have been part of my research agenda for many years. Currently I have a couple of projects on boundary value problems for 4-dimensional special-holonomy metrics. This is joint work with Joel Fine and Jason Lotay and uses a gauge-theoretic reformulation of the problem which Joel has been working on with Dima Panov and Kiril Krasnov.

Non-abelian monopoles This is an EPSRC-funded project which involves Richard Melrose, Chris Kottke and Karsten Fritzsch. We're starting off by redoing Taubes's gluing theorem for monopoles from the smooth point of view - the method will be quite analogous to the above-mentioned project on gluing cscK metrics. The advantage of this is that by working smoothly, it is relatively easy to keep control of what's happening to the metrics on the moduli spaces. We are then going to soup up the construction to give a smooth compactification of the moduli space of monopoles as a compact manifold with corners, again, keeping control of the metric as we do so. Closely related work is currently being pursued by Roger Bielawski.


From the arXiv

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