Thurston has left :( so I figure a geometerization of 3 manifolds post is in order.
This is all in his book Three Dimensional Geometry and Topology.
For surfaces, uniformization is prehistoric. We have the easy invariant of the genus to tell us the covering space and it gets a constant curvature metric.
For 3d manifolds, the situation is trickier. We again want to find model manifolds along with their Lie groups of isometries. We also want these actions to be transitive. None of the points should get any special treatment.
The connected component of the point stabilizers are going to be Lie subgroups of SO(3).
If they are the full SO(3), then all directions you stare into look the same so we have a symmetric space. It must be one of euclidean, spherical or hyperbolic depending on the curvature.
If the component of the stabilizer is SO(2), then look in the leftover direction at each point to construct a vector field. The flow lines of the vector field give a foliation of the manifold where every leaf is 1D. The leaves are either circles or lines.
The easiest example of such would be surface cross R. This gives two new geometeries and euclidean 3-space over again.
There are two more of these, the cover of SL2 and the Heisenberg group.
This leaves when the point stabilizer is 1D. The only possibility is Sol geometry.
Friday, August 24, 2012
Thursday, August 9, 2012
Don't let go of money you don't have to
http://www.chrisstucchio.com/blog/2012/hft_apology.html
I would suspect a high frequency trader to be with the Alliance.
I would suspect a high frequency trader to be with the Alliance.
Thursday, August 2, 2012
(2,0)
“The relation between 4D N=4 SYM and the 6D (2, 0) theory is just like that between Darth Vader and the Emperor. You see Darth Vader and you think “Isn’t he just great? How can anyone be greater than that? No way’.Then you meet the Emperor”. - Nima
It all comes down to the distinction between string theory and M theory. Instead of thinking about the worldvolume theory of a short stack of D3 branes, you know think about stacked M5 branes compactified on a torus. Since we are compactifying on a torus, that most BORING of all the Calabi-Yaus, we keep all that nice supersymmetry. It explains S duality nicely as the mapping class group of the torus. If we compactify on a general Riemann surface, we get other S-duality groups corresponding to that surface.
It all comes down to the distinction between string theory and M theory. Instead of thinking about the worldvolume theory of a short stack of D3 branes, you know think about stacked M5 branes compactified on a torus. Since we are compactifying on a torus, that most BORING of all the Calabi-Yaus, we keep all that nice supersymmetry. It explains S duality nicely as the mapping class group of the torus. If we compactify on a general Riemann surface, we get other S-duality groups corresponding to that surface.
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