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CATEGORIES:Rouse Ball Lectures
SUMMARY:Geometry and Topology of 3-dimensional spaces - Pr
ofessor John Morgan\, Columbia University\, USA
DTSTART;TZID=Europe/London:20080520T120000
DTEND;TZID=Europe/London:20080520T130000
UID:TALK24124AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/24124
DESCRIPTION:When he introduced what is now known as Riemannian
geometry\, Riemann vastly generalized what had co
me before. He also explicitly separated geometry f
rom the topology of the underlying space\; geometr
y became an extra structure given to a topological
space. Around the turn of the 20th century Poinca
re put topology on an independent footing as a sub
discipline of mathematics. He also formulated a qu
estion which he considered as central. That questi
on was to characterize the simplest 3-dimensional
space\, the 3-sphere. Poincare's conjecture was ge
neralized to include a classification of all 3-dim
ensional spaces\, technically\, compact 3-manifold
s\, and even higher dimensional spaces (but that i
s another story). In the 1980s Thurston conjecture
d that 3-dimensional spaces could be classified\,
and Poincare's original conjecture could be resolv
ed\, by uniting homogeneous Riemannian geometry an
d topology in dimension 3\, undoing\, in a sense f
or 3-dimensional spaces\, Riemann's separation of
topology and geometry. Around the same time\, Rich
ard Hamilton proposed a method of attacking Thurst
on'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 under
consideration to a nice Riemannian metric. Recent
ly\, Perelman has given a complete proof of Thurst
on's conjecture along the general lines envisioned
by Hamilton.\n\nThe talk will introduce ways of t
hinking about the topology 3-dimensional spaces an
d the homogeneous geometries that come into play.
The talk will describe the version of the heat-typ
e equation\, called the Ricci flow equation\, for
Riemannian metrics. It will then discuss the analy
tic and geometric approaches and ideas and some of
the difficulties that one must overcome in order
to arrive at a positive resolution of all these co
njectures by these methods.
LOCATION:Room 3\, Mill Lane Lecture Rooms\, 8 Mill Lane\, C
ambridge.
CONTACT:
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