Details of Grant 

EPSRC Reference:	EP/D075130/1
Title:	Warwick Symposium on Low Dimensional Geometry and Topology
Principal Investigator:	Series, Professor C
Other Investigators:	Topping, Professor P Lackenby, Professor M
Researcher Co-Investigators:
Project Partners:
Department:	Mathematics
Organisation:	University of Warwick
Scheme:	Standard Research
Starts:	01 September 2006	Ends:	31 August 2009	Value (£):	121,775
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 last few years have seen dramatic progress in three dimensional-geometry and topology, with spectacular advances coming from several different directions. This proposal is for a Symposium on these topics run by the University of Warwick Mathematics Research Centre during the academic year 2006-7. The principal organiser is Professor Caroline Series. The symposium will be structured around workshops run by leading British experts and advised by an international scientific committee.Topology provides a language for studying features such as tunnels and linkages in complicated objects. Geometry adds measurement. There are a few classic geometries, Euclidean and spherical being the most familiar, which are homogeneous and symmetrical in every direction. It has long been understood that a good way to study the topology of two dimensional surfaces is to add suitable geometry. It is crucial that the geometry `fits' the surface precisely, without awkward breaks. Working with such a geometry is a powerful tool. However the three dimensional analogue, a 3-manifold, has proved much less tractable.About thirty years ago, W. Thurston conjectured that all 3-manifolds can be assembled out of a few pieces each carrying one of eight types of geometry, of which hyperbolic (non-Euclidean) geometry is the most common. Introducing many remarkable new ideas which revolutionised the subject, Thurston and his students proved many of his conjectures. However there remained some major questions about the classification of all possible hyperbolic manifolds, known as the Ending Lamination Conjecture and the Tameness Conjecture, which have only been resolved very recently with far reaching proofs.Other crucial parts of Thurston's Geometrization Conjectures also remained intractable, notably the celebrated Poincare conjecture that any 3-manifold in which all loops can be contracted to a point is the three dimensional sphere. Despite its apparent simplicity, this problem defeated the best efforts of mathematicians throughout the last century and in 2000 was listed as one of the Clay Millennium problems attracting a million dollar prize. The last few years have seen great excitement with G. Perelman's announcement of a solution to the Poincare conjecture using novel techniques based on R. Hamilton's work on geometric flows. The starting point is any way of measuring distance, technically a Riemannian metric, on the manifold. Continuously modifying this metric as prescribed by a differential equation called the Ricci flow appears to make it more homogeneous. Carefully ironing out singularities along the way, Perelman's claim is that the metric eventually becomes so symmetrical that the underlying manifold must be the 3-sphere. His work is currently being examined by experts, most of who now believe the essentials are correct. Perelman's claim of a proof of the remaining parts of Thurston's geometrization conjectures also appears likely to be correct.All this activity has radically transformed the field. The main aims of this symposium are to advance, synthesize and disseminate knowledge in the light of the recent discoveries.
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Organisation Website:	http://www.warwick.ac.uk