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Geometric analysis, complex geometry and gauge theory

Objective

The proposal is for work in Geometric Analysis aimed at two different problems. One is to establish necessary and sufficient conditions for the existence of extremal metrics on complex algebraic manifolds. These metrics are characterised by conditions on their curvature tensor a paradigm being the Riemannian version of the Einstein equation of General Relativity The standard conjecture is that the right condition should be the stability of the manifold, a condition defined entirely in the language of algebraic geometry. But there are very few cases where this conjecture has been verified. The problem comes down to proving the existence of a solution to highly nonlinear partial differential equation. The aim is to advance this theory by a detailed study of interesting but more amenable cases, for example where there is a large symmetry group. The second problem is to develop new invariants and structures associated to a particular class of manifolds of dimension 6 and 7 (with holonomy SU(3) and G2). These would be derived from the solutions of versions of the Yang-Mills equation over the manifolds, in a similar manner to familiar theories in 3 and 4 dimensions. In higher dimensions there are fundamental new difficulties to overcome to set up a theory rigorously and the main point of this part of the proposal is to attack these. It is likely that the new structures, if they do exist, will have interesting connections to other developments in this general area, involving string theory and algebraic geometry.

Field of science

  • /natural sciences/physical sciences/string theory
  • /natural sciences/mathematics/pure mathematics/geometry
  • /natural sciences/mathematics/pure mathematics/mathematical analysis/differential equations/partial differential equations
  • /humanities/languages and literature/languages - general
  • /natural sciences/mathematics/pure mathematics/algebra/algebraic geometry

Call for proposal

ERC-2009-AdG
See other projects for this call

Funding Scheme

ERC-AG - ERC Advanced Grant

Host institution

IMPERIAL COLLEGE OF SCIENCE TECHNOLOGY AND MEDICINE
Address
South Kensington Campus Exhibition Road
SW7 2AZ London
United Kingdom
Activity type
Higher or Secondary Education Establishments
EU contribution
€ 1 501 361
Principal investigator
Simon Kirwan Donaldson (Prof.)
Administrative Contact
Brooke Alasya (Ms.)

Beneficiaries (1)

IMPERIAL COLLEGE OF SCIENCE TECHNOLOGY AND MEDICINE
United Kingdom
EU contribution
€ 1 501 361
Address
South Kensington Campus Exhibition Road
SW7 2AZ London
Activity type
Higher or Secondary Education Establishments
Principal investigator
Simon Kirwan Donaldson (Prof.)
Administrative Contact
Brooke Alasya (Ms.)