Details of Grant 

EPSRC Reference:	EP/F006705/1
Title:	Derived categories of coherent sheaves on hyperkahler manifolds
Principal Investigator:	Thomas, Professor R
Other Investigators:
Researcher Co-Investigators:
Project Partners:
Department:	Mathematics
Organisation:	Imperial College London
Scheme:	Standard Research
Starts:	05 October 2007	Ends:	04 May 2008	Value (£):	39,904
EPSRC Research Topic Classifications:	Algebra & Geometry Mathematical Physics
EPSRC Industrial Sector Classifications:	No relevance to Underpinning Sectors
Related Grants:
Panel History:	Panel Date Panel Name Outcome 06 Jun 2007 Mathematics Prioritisation Panel (Science) Announced
Summary on Grant Application Form
Algebraic geometry, and particularly the geometry of HyperKahler manifolds, studies a highly constrained type of geometry which is very rigid with very few global symmetries. Mysterious predictions coming from dualities in string theory in physics, however, relate these spaces to symplectic manifolds: these are much more flexible geometries with many symmetries. (For instance the symmetry groups of low dimensional topology -- braid groups, mapping class groups -- occur as the symmetry groups of many symplectic manifolds.) The reason is that while the symmetries do not act on the HyperKahler manifolds themselves, they act on their categories of D-branes .This connection between algebraic geometry and low dimensional topology is a surprise and allows one to use the sophisticated and deep structures of algebraic geometry to construct invariants in low dimensional topology -- i.e. algebro-geometric data encoding properties of the low dimensional topology that can be used to distinguish different topologies or shapes . The simplest example is to produce knot invariants from algebraic geometry, giving data which can distinguish between knotted loops of string embedded into normal 3-dimensional space in different ways.
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Organisation Website:	http://www.imperial.ac.uk