My research interests are within algebraic geometry and complex algebraic geometry, and links between the two. In particular I am interested notions of stability (in the sense of Geometric invariant theory) both for sheaves and for varieties. In particular I work on the relationship between stability and Kahler geometry, via the existence of special metrics (for example Kahler-Einstein metrics or Hermitian-Einstein metrics). I am also interested in the theory of singular hermitian metrics and some aspects of pluripotential theory as well as the inverse moment problem in the plane and solutions to the Hele-Shaw flow.

You can find more details in a description of papers that I have produced that gives context to my work, as well as some references to relevant things that have since appeared.

Here is a copy of my CV

David Witt Nystrom and I have written a MATLAB package to simulate the hele-shaw flow in the plane (with varying permeability). You can find an example output heleshaw_60frames.avi. There is some documentation and the code and documentation is available: version 1.

I organised workshop on Kahler Geometry, April 10-13th 2012 at Cambridge University with Ivan Cheltsov. Videos for most of the talks can be found here.

- (Joint with Ruadhaí Dervan)
**Stable maps in higher dimension**arXiv:1709.09750 - (Joint with David Witt Nystrom)
**On the Maximal Rank Problem for the Complex Homogeneous Monge-Ampere Equation**arXiv:1610.02280 - (Joint with Daniel Greb and Matei Toma)
**A Master Space for Moduli Spaces of Geiseker-Stable sheaves**Submitted arXiv:1605.06642 - (Joint with Ruadhaí Dervan)
**K-stability for Kahler Manifolds**Math Research Letters,, Vol. 24, No. 3 (2017), pp. 689-739 arXiv:1602.08983 - (Joint with Mario Garcia-Fernandez and Julien Keller)
**Quantization of Hitchin's Equations for Higgs Bundles, I**Preprint arXiv:1601.04960 - (Joint with David Witt Nystrom)
**Applications of the duality between the Complex Monge-Ampere Equation and the Hele-Shaw flow**Submitted arXiv:1509.02665 - (Joint with Daniel Greb and Matei Toma)
**Moduli of Vector Bundles on Higher-Dimensional Base Manifolds -- Construction and Variation**International Journal of Mathematics, Vol 27 No 6 (2016) arXiv:1503.00319 - (Joint with Daniel Greb and Matei Toma)
**Semi-continuity of Stability for Sheaves and Variation of Gieseker Moduli Spaces**To appear in Crelles arXiv:1501.04440 - (Joint with Daniel Greb and Matei Toma)
**Variation of Gieseker Moduli Spaces of Sheaves via Quiver GIT**Geometry & Topology 20-3 (2016), 1539--1610 arXiv:1409.7564 - (Joint with David Witt Nystrom)
**Harmonic Discs of Solutions to the Complex Homogeneous Monge-Ampere Equation**Publ. l'IHES, Volume 122, Issue 1, 315--335 arXiv:1408.6663 - (Joint with David Witt-Nystrom)
**Homogeneous Monge-Ampere Equations and Canonical Tubular Neighbourhoods in Kahler Geometry**To appear in IMRN arXiv:1403.3282 - (Joint with Mario Garcia-Fernandez)
**Balanced Metrics on twisted Higgs bundles**To appear in Math Annalen arxiv:1401.7108 - (Joint with Michael Singer)
**Asymptotics of Partial Bergman kernels for Divisors**Journal of Geometric Analysis, 27(3), 1803-1854 arxiv:1312.1145 - (Joint with David Witt-Nystrom)
**The Hele-Shaw Flow and Moduli of Holomorphic Discs**Compositio Math Volume 151 / Issue 12 / December 2015, pp 2301- 2328 arxiv:1212.2337 - (Joint with David Witt-Nystrom)
**Envelopes of plurisubharmonic metrics with prescribed singularities**To appear in Les Annales de la Faculte des Sciences de Toulouse arXiv:1210.2220 - (Joint with Julien Keller)
**A note on Chow stability of the projectivisation of Gieseker stable bundles**Journal of Geometric Analysis July 2014, Volume 24, Issue 3, pp 1526--1546 arXiv:1110.4489 - (Joint with Mario Garcia-Fernandez)
**Limits of balanced metrics on vector bundles and polarised manifolds**Proc. London Math. Soc. (2013) 106(5): 1143-1156 arXiv:1111.2819 - (Joint with David Witt-Nystrom)
**Analytic test configurations and geodesic rays.**Journal of Symplectic Geometry Volume 12, Number 1 (2014), 125--169 arXiv:1101.1612 - (Joint with R.P. Thomas)
**Weighted Bergman kernels on orbifolds**. Jour. Diff. Geom.**88**, p.87-108, 2011. arXiv:0907.5215 - (Joint with R.P. Thomas)
**Weighted projective embeddings, stability of orbifolds and constant scalar curvature Kähler metrics**Jour. Diff. Geom.**88**, p.109-160, 2011. arXiv:0907.5214 - (Joint with J. Roé)
**An inequality between multipoint Seshadri Constants**. Geometriae Dedicata Volume**140**, Number 1. pp 175-181. arXiv:0804.1662 - (Joint with D. Panov)
**Slope Stability and Exceptional Divisors of High Genus**. Math. Ann. 343 No 1, 2009. arXiv:0710.4078 - (Joint with D.H. Phong and J. Sturm)
**Deligne pairings and the Knudsen-Mumford expansion**Jour. Diff. Geom.**78**, p.475-496, 2008.math.DG/0612555 **Seshadri constants on symmetric products of curves**Math Res. Let. Vol**14**No 1. pp 63-75, 2007. math.AG/0608224.- (Joint with J. Fine)
**A note on Positivity of the CM line bundle**IMRN, vol.**2006**, Article ID 95875, 14 pages, 2006. math.AG/0605302 **Unstable products of smooth curves**. Invent Math. Vol.**165**No. 1 pp 153-162, 2006 math.AG/0506447.- (Joint with R.P. Thomas)
**A study of the Hilbert-Mumford criterion for the stability of projective varieties**. Journal of Algebraic Geometry**16**(2007), 201-255. math.AG/0412519. - (Joint with R.P. Thomas)
**An obstruction to the existence of constant scalar curvature Kahler metrics**, Jour. Diff. Geom.**72**, 429-466, 2006. math.DG/0412518

**Constant scalar curvature metrics on toric manifolds**(pdf) Lecture notes for a minicourse part of MACK5 workshop, Rome (Nov 2012)**Canonical metrics and stability of projective varieties**(Notes written for the workshop on ''Numerical invariants of singularities and higher-dimensional algebraic varieties'' at the American Institute of Mathematics, July 2006)**Instability of Polarised Algebraic Varieties.**Thesis (ps file)**Stability of polarised varieties**(Slides of seminar given at Seattle 2005)

- David Witt Nystrom (Marie Curie, 2013-4)

- Cristiano Spotti (Funded by EPSRC CAF Grant, 2014-5)

- Andrea Tirelli (ERASMUS, with Jacopo Stoppa)
- Rui Dervan (Funded by EPSRC CAF Grant, 2012-present)

- Anton Isopoussu (2010-2015)

I edit a Journal

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