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Grisha got it!by marcus
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#19
Aug2806, 01:07 PM

P: 83

I want to understand why 3D case is totaly different from 1D2D and nD,
n > 3 for which analoques of Poincare conjecture were proof a long time ago. Why the dimention 3D is really matter? 


#20
Aug2806, 01:38 PM

P: 2,046




#21
Aug2806, 02:01 PM

P: 1,636

Supposedly, in 3d there isn't enough dimensions (only 3) to do their math stuff. Where as in higher dimensions you can utilize alot of different techniques because you have lots of room to move around.



#22
Aug2806, 03:20 PM

Emeritus
PF Gold
P: 8,147

You got it. The one and two dimensional cases are trivial, and the n greater than three results, while difficult, were solved by Smale a long time ago. The reason threemanifolds are so interesting is that they are right on this boundary between being so limited they're trivial and being so unlimited they're trivial in a different way. Back when I was a beginning topology student I was not very interested in threemanifolds (which had been an active subtopic since Poincare's day), thinking, "Well then, when they get that all resolved they'll just have to go on to fourmanifolds, and then 5, 6, ..., google, ...". It seemed like a mug's game to me. But it ain't so; three is, even at the most abstract level, a very special, very important case. 


#23
Aug2806, 04:18 PM

P: 83

Thanks, selfAjoint.
Now, more specific question: if the Ricciflow technic works just fine for 3D case (on the boundary), it can work for nD case (n>3). Is that correct? And what does mean the parameter t in Ricciflow eq.: d (g_{i,j})/dt =2*Rici? My guess: we issue any parametric curve on manifold from point P and take derivitive along it. Correct? Sorry, I don't have any original Hamilton paper to look details. 


#24
Aug2906, 03:43 AM

P: 255

In other words, they are looking at the (infinitedimensional) space of all Riemannian metrics living on the given 3manifold and studying paths in that space that satisfy the Ricciflow equation, which is actually just a differential equation. Regarding the strangeness of 3 versus higherdimensions, dimensions 3 and 4 seem to be  in general  harder to work with than higher dimensions. 4manifold theory is quite active these days, although it got a little "easier" when the SeibergWitten equations were discovered in the early 90s. 


#25
Aug3006, 11:57 AM

P: 83

Thanks, Doodle Bob.
It seems that the application of RicciFlow equation to the dimentions n>3 is not straightforward because Riemann curvature for n>3 is not totaly defined by Ricci tensor, which is the right side of RicciFlow equation. Is that correct? 


#26
Aug3006, 08:10 PM

Sci Advisor
HW Helper
P: 9,470

the two dimensional case follows from a clasification of all 2 manifolds S OBTAINED BY joining handles to spheres. the three dimensional case wouldfollow aND APPRENTLY DOES, FROM AN analogous conjecture of thurston on how to obtain three manifolds from joining various basic types of objects.
the definition of "trivial" is of cousre open to discussion, but m,y definition is nythign I MYSELF KNOW HOW TO DO, AND I CANNOT SAY I NKOWHOW TO PROVE THE CLKASIFICATION OF 2 MANIFOLDS. help!! i should have taken typing asa high school studentbut in thsio days it was only offered to secretaries!!!@ 


#27
Aug3006, 08:17 PM

Sci Advisor
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P: 9,470

recalling my ancient history courses in 2 manifolds, i seem to recall that clasificaion of 2 manifolds is ropecedded by a proof that all 2 manifolds are triangulable, itself perhaps not so easy.
or maybe there is an easier morse theory proof? 


#28
Sep206, 08:23 AM

P: 83

http://en.wikipedia.org/wiki/Grigori_Perelman 


#29
Sep406, 03:21 AM

P: 1,157

A very impressive CV, that Tao guy's got.
It's amazing when you see people your own age with results like that! 


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