Research
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.
Publications
From the arXiv
- (with Julius Ross), Asymptotics of partial density functions,
arXiv:1312.1145. A continuation of the work started with Florian Pokorny. The difference is that we obtain more precise information about the asymptotics of the partial density function at the boundary of the so-called `forbidden region' and only need to assume a circle-action, rather than the full toric symmetry. Of course, one would like to lose the symmetry assumption altogether...
- (with Florian Pokorny), Toric partial density functions and stability of toric varieties,
arXiv1111.5259. A
distributional asymptotic expansion of partial density functions
is written down, partly based on Florian's Edinburgh PhD thesis.
The relation with K- and slope stability was suggested in
unpublished work by Richard Thomas. (Published by
Math. Annalen, 2014.)
- (with Richard Melrose) Scattering configuration spaces,
arXiv:0808.2022. We
discuss the `stretched scattering product' of n copies of a
scattering manifold. We give the definition as a blow-up of the
n-fold product of X and prove the crucial fact that
the n-fold scattering product maps back to the (n-1)-fold
scattering product via a composite of blow-down maps and
projection. There are a very large number of blow-ups to keep
track of and proving the relevant commutation properties is
somewhat non-trivial.
- (with Rafe Mazzeo) Some remarks on conic degeneration and
bending of Poincare-Einstein metrics, arXiv:0709.1498.
This little note arose from our attempts to resolve
singularities of orbifold Poincare-Einstein metrics. The problem
we were interested in was eventually solved by Olivier
Biquard.
- (with Yann Rollin) Constant scalar curvature Kaehler metrics
and parabolic polystability, arXiv:0703212.
- (with Claudio Arezzo and Frank Pacard) Extremal metrics on
blow-ups, arXiv:0701028.
- (with Yann Rollin) Construction of Kaehler metrics with
constant scalar curvature, arXiv:0412405.
- (with Yann Rollin) Non-minimal scalar-flat Kaehler surfaces
and parabolic stability, arXiv:0404423.
- (with David Calderbank) Toric selfdual Einstein metrics on
compact orbifolds, arXiv:0405020.
- (with David Calderbank) Einstein metrics and complex
singularities, arXiv:0206229.
- (with Michael Murray) Gerbes, Clifford modules and the
index theorem, arXiv:0302096.
- (with Michael Murray) A note on monopole moduli
spaces, arXiv:0302020.
- (with Nicholas Manton and Bernd Schroers) The interaction
energy of well-separated Skyrme solitons, arXiv:0212075.
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