Details of Grant 

EPSRC Reference:	GR/A10130/01
Title:	SINGULARITIES OF SPECIAL LAGRANGIAN SUBMANFOLDS IN CALABI-YAU MANIFOLDS
Principal Investigator:	Joyce, Professor D
Other Investigators:
Researcher Co-Investigators:
Project Partners:
Department:	Mathematical Institute
Organisation:	University of Oxford
Scheme:	Advanced Fellowship (Pre-FEC)
Starts:	01 October 2001	Ends:	30 September 2006	Value (£):	227,071
EPSRC Research Topic Classifications:	Algebra & Geometry
EPSRC Industrial Sector Classifications:	No relevance to Underpinning Sectors
Related Grants:
Panel History:	Panel Date Panel Name Outcome 16 Nov 2000 Maths Advanced/Senior Fellowship Sifting Panel Deferred
Summary on Grant Application Form
I would like to investigate the singularities that can develop in (generic) families of compact special Lagrangian submanifolds in Calabi-Yau manifolds, especially in complex dimension three. This is a large project requiring a number of different techniques. One must first develop and classify explicit local models of the singularities in Cm, and their resolutions. This can be done using symmetry, evolution equations, and integrable systems techniques. Then one must prove some hard analytic results relating these local models to what happens in compact special Lagrangian submanifolds in Calabi-Yau manifolds. This will probably be easier in low dimensions.Once the basic theory is established, there are several exciting applications. Firstly, 1 want to work out what the correct local model is for generic 3-dimensional special Lagrangian 'fibrations', and prove a stability result for fibrations with this local model under small perturbations. This would be significant progress towards understanding the SYZ conjecture.Secondly, I want to try and define invariants of Calabi-Yau 3-folds by counting special Lagrangian homology 3-spheres with weights. For these to be interesting they must transform in a very rigid way as the special Lagrangian 3-folds become singular, so one needs to understand such singularities very well. If such invariants can be defined, they might reveal a great deal about the hidden structure of Calabi-Yau 3-folds, as Donaldson invariants have about 4-manifolds.
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Organisation Website:	http://www.ox.ac.uk