Comments on: Complex Structures on the Six-sphere

By: Peter

Peter — Wed, 31 Dec 1969 19:00:00 +0000

The section of the paper on “Chern’s Last Theorem” is almost entirely devoted to explaining well-known facts about G_2/SU(3)=S^6, including the well-known fact that the almost complex structure you get from the octonions is not integrable. I’m pretty sure all of this material goes back to Elie Cartan in the early part of the last century. The new argument due to Chern is only dealt with in the last paragraph of the section (I said it was only one paragraph, not one sentence. String theorists have a weird inability to quote me correctly…). It’s the argument in this last paragraph that is new, but also has a hole in it.

By: DMS

DMS — Wed, 31 Dec 1969 19:00:00 +0000

Peter,

You have captured what I tried to say. And in the process you have given me a glimpse of the complexities and subtleties of (dis-)proving statements in mathematics. Thanks!

By: Lubos Motl

Lubos Motl — Wed, 31 Dec 1969 19:00:00 +0000

It is not true that there is just one sentence dedicated to Chern’s proof in Marshakov et al. Look at their conclusions – and you will see that they say that “we have explained his proof in detail”. I also think that there’s a lot of it over there.

By: Peter

Peter — Wed, 31 Dec 1969 19:00:00 +0000

I think what you mean by “physicist’s proof” is something along the lines of what a mathematician would call an outline of a proof, with some steps in the outline possibly justified by appeal to a physical argument that such a step has to somehow work out correctly.

There certainly are some reasonably good examples of this kind of thing. One would be Witten’s work on supersymmetry and Morse theory. There he outlined an argument relating Morse theory, Hodge theory and index theory, using at crucial points expectations of what should happen based on semi-classical approximation techniques in quantum mechanics.

But the Ricci-flow case is very different. Based on personal experience, I can tell you that claims by some physicists that the RG provides a “physicist’s proof” of geometrization have convinced some mathematicians that said physicists are arrogant idiots. In this subject you can’t assume that metrics are well-behaved, since they aren’t. Singularities develop when you follow the Ricci flow, and understanding what happens then is what the whole subject is about. There is no “physical” argument for ignoring this phenemenon. If there were it would imply an infinite number of untrue mathematical theorems.

By: DMS

DMS — Wed, 31 Dec 1969 19:00:00 +0000

Someone wrote:
“If it’s wrong for a mathematician, it must also be wrong for the physicist.”

Of course that is obvious: don’t need a Harvard education to see that.

The question was: is there a strong physical reason for one to believe a particular statement to be true. Of course, such physical reasons may turn out to be wrong. Alternatively, the statement may be correct, but may require newer mathematics to demonstrate that rigorously (like Dirac delta functions —> distribution theory).

“A difference between a physicist’s proof and a mathematician’s proof can only exist in physics where the physicists know what they mean and mathematicians are slower.”

Well, the recent proof of Thurston geometrization conjecture relied on Hamilton’s Ricci flows, reminiscent of RG.
http://www.math.columbia.edu/~woit/blog/archives/000077.html
A physicist would be happy if he could show it for “well-behaved metrics”, not a (”slow”) mathematician.

By: plato

plato — Wed, 31 Dec 1969 19:00:00 +0000

It’s important that we can see where you are both focused instead of having the ramble of content that deters from appropriate discussion.

So we now have you both fixed in the appropriate direction, so that you can philosphically:) discuss the idea being expounded at the forefront of our knowledge base.

Is one, going to reject the mathematics of another? On what grounds? Physics?:)

By: Lubo� Motl

Lubo� Motl — Wed, 31 Dec 1969 19:00:00 +0000

Impressive guy (Chern), even it it’s not right.

Someone asked whether the proof is valid as a “physicist’s proof”. That’s a ludicrous question. A difference between a physicist’s proof and a mathematician’s proof can only exist in physics where the physicists know what they mean and mathematicians are slower.

In mathematics, for example in discussions about the existence of a complex structure, the physicist’s proof and a mathematician’s proof is the same thing. If it’s wrong for a mathematician, it must also be wrong for the physicist.

By: plato

plato — Wed, 31 Dec 1969 19:00:00 +0000

At planck scale with topological considerations, no determination can ever be foolproof, as it is conjectured math already based, unless, you willingly admit and move into abstract spaces, detached from reality:) But its still fun to conjecture?

By: Peter

Peter — Wed, 31 Dec 1969 19:00:00 +0000

The Marshakov-Niemi paper just gives an outline of Chern’s argument, it doesn’t in any sense provide a “physicist’s proof”.

By: DMS

DMS — Wed, 31 Dec 1969 19:00:00 +0000

What about Marshakov-Niemmi paper itself? Is it valid as a “physicist’s proof”?

That is, even if Chern was wrong, it may be that someone is going to prove it rigorously, so it is basically correct (like a lot of QFT proofs).