EPSRC Reference: 
GR/S87034/01 
Title: 
Complex Floer theories for calibrated submanifolds of special holonomy manifolds. 
Principal Investigator: 
Donaldson, Professor S 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

Department: 
Mathematics 
Organisation: 
Imperial College London 
Scheme: 
Standard Research (PreFEC) 
Starts: 
01 October 2004 
Ends: 
30 June 2008 
Value (£): 
177,086

EPSRC Research Topic Classifications: 

EPSRC Industrial Sector Classifications: 
No relevance to Underpinning Sectors 


Related Grants: 

Panel History: 

Summary on Grant Application Form 
We will develop new ways to distinguish various kinds of special geometric structures (calibrated submanifolds) which live inside some special kinds of spaces (special holonomy manifolds). Both these geometric structures and the spaces they live in are of particular interest both to mathematicians and some theoretical physicists (string theorists) because of their unusual geometric properties. String theorists believe that properties of these spaces tell us physical information about our universe, like the number of fundamental particles of various kinds. Studying what kinds of geometric structures can live inside these spaces will also give information about the space itself. The geometric structures themselves have minimal possible volume, and as such are generalizations of the wellknown soap films in ordinary 3space, which have minimal area.The invariants we have in mind will be defined by counting certain kinds of solutions of partial differential equations. There are some simpler invariants in other settings defined by counting solutions of ordinary differential equations. This type of counting is associated with complex numbers and is called PicardLefschetz theory. Unfortunately, passing from ordinary to partial differential equations introduces many extra problems. The longterm goal of this research is overcome these technical difficulties and prove that this sort of counting really makes sense.We will start by focussing on the geometric structures for which these technical problems should be simplest to deal with. Even here there are still problems to overcome, so we focus initially on two particular questions. Other researchers in mathematics would be interested in the answers to these problems. Also the problems seem easier to solve than the more general case because we have a good idea what techniques will be needed to solve these problems based on other related problems that other researchers have already solved.

Key Findings 
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk

Potential use in nonacademic contexts 
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk

Impacts 
Description 
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk 
Summary 

Date Materialised 


Sectors submitted by the Researcher 
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk

Project URL: 

Further Information: 

Organisation Website: 
http://www.imperial.ac.uk 