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SUMMARY:Geometry and Topology of 3-dimensional spaces - Professor John Mor
 gan\, Columbia University\, USA
DTSTART:20080520T110000Z
DTEND:20080520T120000Z
UID:TALK24124@talks.cam.ac.uk
DESCRIPTION:When he introduced what is now known as Riemannian geometry\, 
 Riemann vastly generalized what had come before. He also explicitly separa
 ted geometry from the topology of the underlying space\; geometry became a
 n extra structure given to a topological space. Around the turn of the 20t
 h century Poincare put topology on an independent footing as a subdiscipli
 ne of mathematics. He also formulated a question which he considered as ce
 ntral. That question was to characterize the simplest 3-dimensional space\
 , the 3-sphere. Poincare's conjecture was generalized to include a classif
 ication of all 3-dimensional spaces\, technically\, compact 3-manifolds\, 
 and even higher dimensional spaces (but that is another story). In the 198
 0s Thurston conjectured that 3-dimensional spaces could be classified\, an
 d Poincare's original conjecture could be resolved\, by uniting homogeneou
 s Riemannian geometry and topology in dimension 3\, undoing\, in a sense f
 or 3-dimensional spaces\, Riemann's separation of topology and geometry. A
 round the same time\, Richard Hamilton proposed a method of attacking Thur
 ston's conjecture. His idea was to use a version of the heat equation for 
 Riemannian metrics to evolve any starting Riemannian metric on the space u
 nder consideration to a nice Riemannian metric. Recently\, Perelman has gi
 ven a complete proof of Thurston's conjecture along the general lines envi
 sioned by Hamilton.\n\nThe talk will introduce ways of thinking about the 
 topology 3-dimensional spaces and the homogeneous geometries that come int
 o play. The talk will describe the version of the heat-type equation\, cal
 led the Ricci flow equation\, for Riemannian metrics. It will then discuss
  the analytic and geometric approaches and ideas and some of the difficult
 ies that one must overcome in order to arrive at a positive resolution of 
 all these conjectures by these methods. 
LOCATION:Room 3\, Mill Lane Lecture Rooms\, 8 Mill Lane\, Cambridge.
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