I'm a Royal Society University Research Fellow at the University of Cambridge, and a Research Fellow at Gonville & Caius College. I've previously spent time in École Polytechnique, Université libre de Bruxelles and Trinity College Dublin.

My first name is pronounced "Rui" (the "dh" is silent, it's Irish).

I research complex geometry. I'm interested in both the algebro-geometric and differential-geometric sides of complex geometry, and am particularly interested in the interplay between algebraic geometry and geometric analysis. Most of my research focuses on the existence of canonical metrics in complex geometry (e.g. Kähler-Einstein, extremal), notions of algebro-geometric stability (e.g. K-stability), and more general aspects of moduli theory and geometric analysis. My work is mostly motivated by the "Yau-Tian-Donaldson" conjecture, which links stability notions to the existence of canonical metrics; see Donaldson's survey for an introduction.

I taught Part III Complex Manifolds in Lent Term 2020 and Lent Term 2019.

My CV. My email address is rd430(at)cam.ac.uk.

Papers and preprints:

- Valuative stability of polarised varieties
*(with Eveline Legendre)*arXiv:2010.04023 (abstract) - Uniqueness of optimal symplectic connections
*(with Lars Sektnan)*arXiv:2003.13626 (video) (abstract) - Moduli theory, stability of fibrations and optimal symplectic connections
*(with Lars Sektnan)*arXiv:1911.12701*(to appear in Geom. Topol.)*(video) (abstract) - Optimal symplectic connections on holomorphic submersions
*(with Lars Sektnan)*arXiv:1907.11014*(to appear in Comm. Pure Appl. Math.)*(abstract) - K-semistability of optimal degenerations arXiv:1905.11334
*(Q. J. Math. Vol. 71 (2020) Issue 3, pp. 989-995)*(abstract) - Moduli of polarised manifolds via canonical Kähler metrics
*(with Philipp Naumann)*arXiv:1810.02576 (video) (abstract) - Extremal metrics on fibrations
*(with Lars Sektnan)*arXiv:1712.05374*(Proc. Lond. Math. Soc. (3) 120 (2020) pp. 587-616)*(video) (abstract) - Stable maps in higher dimensions
*(with Julius Ross)*arXiv:1708.09750*(Math. Ann. Vol. 374 (2019) no. 3-4, pp. 1033-1073)*(abstract) - Hermitian Yang-Mills connections on blowups
*(with Lars Sektnan)*arXiv:1707.07638*(to appear in J. Geom. Anal.)*(abstract) - The Kähler-Ricci flow and optimal degenerations
*(with Gábor Székelyhidi)*arXiv:1612.07299*(J. Differential Geom. Vol. 116 (2020), no. 1 pp. 187-203)*(abstract) - Relative K-stability for Kähler manifolds arXiv:1611.00569
*(Math. Ann. Vol. 372 (2018), no. 3-4, pp. 859-889)*(addendum) (abstract) - K-stability for Kähler manifolds
*(with Julius Ross)*arXiv:1602.08983*(Math. Res. Lett. Vol. 24, No. 3 (2017), pp. 689-739)*(abstract) - A finite dimensional approach to Donaldson's J-flow
*(with Julien Keller)*arXiv:1507.03461*(Comm. Anal. Geom. (2019) Vol. 27, no. 5. pp. 1025-1085)*(abstract) - On K-stability of finite covers arXiv:1505.07754
*(Bull. London Math. Soc. (2016) 48 (4) pp. 717-728)*(abstract) - Non-reductive automorphism groups, the Loewy filtration and K-stability
*(with Giulio Codogni)*arXiv:1501.03372*(Annales de l'institut Fourier 66 no. 5 (2016) pp. 1895-1921)*(corrigendum) (abstract) - Alpha invariants and coercivity of the Mabuchi functional on Fano
manifolds arXiv:1412.1426
*(Ann. Fac. Sci. Toulouse Sér. 6, 25 no. 4 (2016), p. 919-934)*(abstract) - Uniform stability of twisted constant scalar curvature Kähler metrics arXiv:1412.0648
*(Int. Math. Res. Notices (2016) Vol 15 pp. 4728-4783)*(abstract) - Alpha invariants and K-stability for general polarisations of Fano varieties arXiv:1307.6527
*(Int. Math. Res. Notices (2015) Vol 16 pp. 7162-7189)*(abstract)

- Michael Hallam, DPhil student at the University of Oxford, co-supervised by Frances Kirwan (2019 - present)
- John McCarthy, PhD student at Imperial College, co-supervised by Simon Donaldson (2019 - present)
- Qiangru Kuang, undergraduate summer research student at the University of Cambridge (Summer 2019)

I organised conferences in Cambridge (Postgraduate Conference in Complex Geometry, 2015), Rome (Moduli of K-stable varieties, 2017) and Newcastle (Newcastle Complex Geometry Workshop, 2018). The Rome conference has an associated conference proceedings, also edited by Codogni and Viviani.

I ran the Cambridge Junior Geometry Seminar (called "Geometry Tea") in 2013-2014. It continues in 2019-2020.

Slightly outdated list of conferences I've been to or am going to: