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EPSRC Reference: GR/M98159/01
Title: MORSE THEORY OF CIRCLE-VALUED FUNCTIONS
Principal Investigator: Hitchin, Professor NJ
Other Investigators:
Researcher Co-Investigators:
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Department: Mathematical Institute
Organisation: University of Oxford
Scheme: Standard Research (Pre-FEC)
Starts: 02 November 1999 Ends: 01 January 2000 Value (£): 3,500
EPSRC Research Topic Classifications:
Algebra & Geometry
EPSRC Industrial Sector Classifications:
No relevance to Underpinning Sectors
Related Grants:
Panel History:  
Summary on Grant Application Form
The VF has a number of achievements to his credit in adapting analytical and topological approaches to Morse theory to the case when the Morse function takes values in a circle. This situation occurs in many places, notable where symplectic group actions do not have a moment map and in an infinite dimensional context in Floer theory. There are areas on the interface between physics and geometry where such techniques may well be useful. One of these lies in the deformation theory of special Lagrangian submanifolds of a Calabi-Yau maniforld, which is fundamental in the Strominger-Yau-Zaslow approach to mirror symmetry. Here deformations are governed by harmonic 1-forms on the submanifold and so intersection properties of nearby submanifolds depend on the zeros of closed 1-forms with periods, which can be thought of in terms of circle valued Morse functions. Also the S-duality conjectures for monopole moduli spaces concern harmonic forms with local coefficient systems and a Morse-theoretic viewpoint may shed some light on these.
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Organisation Website: http://www.ox.ac.uk