Imperial College London

Professor Sir Simon Donaldson FRS

Faculty of Natural SciencesDepartment of Mathematics

Chair in Pure Mathematics
 
 
 
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Contact

 

+44 (0)20 7594 8559s.donaldson Website

 
 
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Location

 

674Huxley BuildingSouth Kensington Campus

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Summary

 

Publications

Publication Type
Year
to

75 results found

Donaldson S, Boundary value problems in dimensions seven, four and three related to exceptional holonomy

In the paper we introduce a boundary value problem for a G_{2} structure on a7-manifold with boundary, with prescribed 3-form on the boundary. We make somegeneral observations about this problem and then study in more detailreductions to 4 and 3 dimensions by imposing symmetry. The 3-dimensionalreduction admits a dual variational formulation in terms of solutions of thereal Monge-Ampere equation

JOURNAL ARTICLE

Donaldson S, Kahler-Einstein metrics and algebraic geometry

This is a survey article, based on the author's lectures in the 2015 Currentdevelopments in Mathematics meeting; published in "Current developments inMathematics". Version 2, references corrected and added.

JOURNAL ARTICLE

Donaldson S, Stability of algebraic varieties and Kahler geometry

This is a survey article, based on the author's lectures in the 2015 AMSSummer Research Institute in Algebraic Geometry, and to appear in theProceedings.

JOURNAL ARTICLE

Donaldson S, Adiabatic limits of co-associative Kovalev-Lefschetz fibrations

We study co-associative fibrations of G_{2}-manifolds. We propose that theadiabatic limit of this structure should be given locally by a maximalsubmanifold in a space of indefinite signature and set up global versions ofthe constructions.

JOURNAL ARTICLE

Donaldson S, Algebraic families of constant scalar curvature Kähler metrics

We give a new proof of the fact that the condition of a Fano manifoldadmitting a K\"ahler-Einstein metric is Zariski-open (provided that theautomorphism group is discrete). This proof does not use the characterisationinvolving stability. The arguments involve estimates of Futaki invariantsobtained from a differential-geometric "volume estimate", and variants of thealgebro-geometric arguments of Stoppa. Many of the ideas apply to constantscalar curvature K\"ahler metrics.

JOURNAL ARTICLE

Donaldson S, The Ding functional, Berndtsson convexity and moment maps

We explain how the formal aspects of the theory of Kahler-Einstein metricscan be developed in the framework of moment maps. The central result we use isthe Berndtsson convexity theorem, which is interpreted as defining a metric onthe space of complex structures. We discuss some applications of these ideas tothe Kahler-Ricci flow.

JOURNAL ARTICLE

Donaldson S, 2016, Kahler-Einstein metrics and algebraic structures on limit spaces, Conference on Geometry and Topology, Publisher: INTERNATIONAL PRESS INC BOSTON, Pages: 85-94, ISSN: 1052-9233

CONFERENCE PAPER

Chen X, Donaldson S, Sun S, 2015, KAHLER-EINSTEIN METRICS ON FANO MANIFOLDS. I: APPROXIMATION OF METRICS WITH CONE SINGULARITIES, JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY, Vol: 28, Pages: 183-197, ISSN: 0894-0347

JOURNAL ARTICLE

Chen X, Donaldson S, Sun S, 2015, KAHLER-EINSTEIN METRICS ON FANO MANIFOLDS. II: LIMITS WITH CONE ANGLE LESS THAN 2 pi, JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY, Vol: 28, Pages: 199-234, ISSN: 0894-0347

JOURNAL ARTICLE

Chen X, Donaldson S, Sun S, 2015, KAHLER-EINSTEIN METRICS ON FANO MANIFOLDS. III: LIMITS AS CONE ANGLE APPROACHES 2 pi AND COMPLETION OF THE MAIN PROOF, JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY, Vol: 28, Pages: 235-278, ISSN: 0894-0347

JOURNAL ARTICLE

Chen X, Donaldson S, Sun S, 2014, Kahler-Einstein Metrics and Stability, INTERNATIONAL MATHEMATICS RESEARCH NOTICES, Pages: 2119-2125, ISSN: 1073-7928

JOURNAL ARTICLE

Chen X-X, Donaldson SK, 2014, Integral bounds on curvature and Gromov-Hausdorff limits, JOURNAL OF TOPOLOGY, Vol: 7, Pages: 543-556, ISSN: 1753-8416

JOURNAL ARTICLE

Donaldson S, Sun S, 2014, Gromov-Hausdorff limits of Kahler manifolds and algebraic geometry, ACTA MATHEMATICA, Vol: 213, Pages: 63-106, ISSN: 0001-5962

JOURNAL ARTICLE

Donaldson SK, 2014, b-STABILITY AND BLOW-UPS, PROCEEDINGS OF THE EDINBURGH MATHEMATICAL SOCIETY, Vol: 57, Pages: 125-137, ISSN: 0013-0915

JOURNAL ARTICLE

Chen X-X, Donaldson SK, 2013, VOLUME ESTIMATES FOR KAHLER-EINSTEIN METRICS: THE THREE-DIMENSIONAL CASE, JOURNAL OF DIFFERENTIAL GEOMETRY, Vol: 93, Pages: 175-189, ISSN: 0022-040X

JOURNAL ARTICLE

Chen X-X, Donaldson SK, 2013, VOLUME ESTIMATES FOR KAHLER-EINSTEIN METRICS AND RIGIDITY OF COMPLEX STRUCTURES, JOURNAL OF DIFFERENTIAL GEOMETRY, Vol: 93, Pages: 191-201, ISSN: 0022-040X

JOURNAL ARTICLE

Donaldson S, 2012, Kahler metrics with cone singularities along a divisor

We develop some foundations for the study of Kahler-Einstein metrics withcone singularities transverse to a divisor. The main goal is a treatment of thedeformation of the cone angle.

BOOK CHAPTER

Donaldson S, 2010, Calabi-Yau metrics on Kummer surfaces as a model glueing problem

This is an expository paper which aims to give a simple proof of theexistence of Ricci-flat metrics on certain K3 surfaces, as an illustration ofgeneral "glueing" techniques.

JOURNAL ARTICLE

Donaldson S, 2010, Stability, birational transformations and the Kahler-Einstein problem

We define a new notion of "b-stability" for a polarised algebraic variety,adapted to the existence problem for Kahler-Einstein metrics on Fano manifolds.

JOURNAL ARTICLE

Donaldson SK, 2009, Constant Scalar Curvature Metrics on Toric Surfaces, GEOMETRIC AND FUNCTIONAL ANALYSIS, Vol: 19, Pages: 83-136, ISSN: 1016-443X

JOURNAL ARTICLE

Donaldson SK, 2009, Some Numerical Results in Complex Differential Geometry, PURE AND APPLIED MATHEMATICS QUARTERLY, Vol: 5, Pages: 571-618, ISSN: 1558-8599

JOURNAL ARTICLE

Bunch RS, Donaldson SK, 2008, Numerical approximations to extremal metrics on toric surfaces

We give some detailed numerical information about extremal metrics on fourdifferent toric surfaces. These are sample of many other cases which can betreated using a computer programme outlined in the paper.

JOURNAL ARTICLE

Donaldson SK, 2008, Some problems in differential geometry and topology, NONLINEARITY, Vol: 21, Pages: T157-T164, ISSN: 0951-7715

JOURNAL ARTICLE

Donaldson SK, 2008, Extremal metrics on toric surfaces: A continuity method, JOURNAL OF DIFFERENTIAL GEOMETRY, Vol: 79, Pages: 389-432, ISSN: 0022-040X

JOURNAL ARTICLE

Donaldson SK, 2008, Kahler geometry on toric manifolds, and some other manifolds with large symmetry

This is an expository article. Among other topics, we discuss the existenceof Kahler-Ricci soliton metrics on toric Fano manifolds, and Kahler-Einsteinmetrics on deformations of the Mukai-Umemura 3-fold

JOURNAL ARTICLE

Donaldson SK, 2008, A generalised Joyce construction for a family of nonlinear partial differential equations

We explain a simple construction of solutions to a family of PDE's in twodimensions which includes that defining zero scalar curvature Kahler metrics,with two Killing fields, and the affine maximal equation.

WORKING PAPER

Donaldson SK, 2007, Nahm's equations and free-boundary problems

This paper is a discussion of relations between some free-boundary problemsand infinite dimensional Lie groups; particularly a version of Nahm's equationsfor the group of Hamiltonian diffeomorphisms in two dimensions.

JOURNAL ARTICLE

Donaldson SK, 2007, A note on the α-invariant of the Mukai-Umemura 3-fold

We give an elementary argument to compute the $\alpha$-invariant of this Fano3-fold, which implies the existence of a Kahler-Einstein metric.

JOURNAL ARTICLE

Donaldson SK, 2007, Lie algebra theory without algebra

This is an expository paper in which we explain how basic, standard, resultsabout simple Lie algebras can be obtained by geometric arguments, followingideas of Cartan, Richardson and others.

JOURNAL ARTICLE

Donaldson S, Fine J, 2006, Tonic anti-self-dual 4-manifolds via complex geometry, MATHEMATISCHE ANNALEN, Vol: 336, Pages: 281-309, ISSN: 0025-5831

JOURNAL ARTICLE

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