| EPSRC Reference: |
GR/S87034/01 |
| Title: |
Complex Floer theories for calibrated submanifolds of special holonomy manifolds. |
| Principal Investigator: |
Donaldson, Professor S |
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| Department: |
Mathematics |
| Organisation: |
Imperial College London |
| Scheme: |
Standard Research (Pre-FEC) |
| Starts: |
01 October 2004 |
Ends: |
30 June 2008 |
Value (£): |
177,086
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| No relevance to Underpinning Sectors |
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Summary on Grant Application Form |
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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 well-known soap films in ordinary 3-space, 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 Picard-Lefschetz theory. Unfortunately, passing from ordinary to partial differential equations introduces many extra problems. The long-term 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.
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Key Findings |
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This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk
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Potential use in non-academic contexts |
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This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk
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Impacts |
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| Description |
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk |
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Sectors submitted by the Researcher |
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This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk
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| Project URL: |
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| Further Information: |
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| Organisation Website: |
http://www.imperial.ac.uk |