For anyone confused, he says that a 4-element set can be split into two pairs in exactly 3 ways, e.g. { { 1, 2 }, { 3, 4} }, { { 1, 3 }, { 2, 4 } }, { { 1, 4 }, { 2, 3 } }. Any permutation of the 4-element set naturally induces a permutation on these 3 partitions: if f : { 1, 2, 3, 4 } -> { 1, 2, 3, 4 } is a permutation, then the induced permutation maps { { a, b }, { c, d } } to { { f(a), f(b) }, { f(c), f(d) } }. This assignment is a non-trivial homomorphism from permutations on 4 elements to permutations on 3 elements, and the kernel of that homomorphism is a witness that S_4 is not a simple group. Somehow this further proves that SO(4) is not a simple Lie algebra, but I didn't follow that part.
@scottychen2397
Күн бұрын
Ok, but this is geometry. So this group theory, …. Its not necessarily relevant if it isnt appreciated as an action on fibres. Clearly, youre considering real matrix groups. Its not necessarily intuitively extendable to take this stuff to complex analytical moduli spaces, so , for me, there truly is a different kind of meaning associated to this stuff. This stuff: is not pure group theory…. So your inappropriate vocabulary could truly be considered more confused than a geometer’s irrelevant musings on this topic in way that may indeed be technically confused’. Alas, such a one would not have made the mistake of not truly grasping what groups are doing here to these manifolds and these vector bundles. I’m not sure youre in a position to be claiming anybody is confused until this can be translated into a vocabulary of group actions on fibre bundles.
@scottychen2397
Күн бұрын
Obviously, this is exactly what he was talking about. Maybe to add to this, His lesson is that: Since this matrix group is not simple, there exists an ‘explosive’ beginning to the theory of yang mills connections.
@adayah2933
22 сағат бұрын
@@scottychen2397 What are you even talking about? Firstly, I explained the group theoretical part. For that purpose it would be nonsensical to use the "vocabulary" of group actions on fibre bundles, since they are not relevant to the picture. Secondly, I don't claim that anyone is actually confused, but if they happen to be, I offer an explanation. If you're a mathematician, you should easily understand that the form "for anyone confused, ..." is a conditional. Just like when the statement (∀x)( P(x) => Q(x) ) is true, it does not follow that there is x such that P(x) is true. The rest of your message is too incomprehensible for me to even try to decipher.
@tesset8828
3 жыл бұрын
At 0:04 is that Gromov, Atiyah, Bill Thurston, and Cedric Villani? I'm probs wrong, but if that's right this is crazy.
@nvstvsi
3 жыл бұрын
You're right. Steven smale is in the front row too.
@jameson44k
Жыл бұрын
Louis Nirenberg is in the front row too, and Robert Bryant next to Atiyah.
@samueldeandrade8535
2 ай бұрын
It seems you would lose your mind if you saw Socrates and Plato together. Oh ... wait a second ...
@sandeepthilakan7728
27 күн бұрын
First row - Gabai, Thurston, Nirenberg, Smale Second row - Atiyah, Bryant That's all I could identify. I am sure others can add.
@bigfrankalbigguy789
24 күн бұрын
Why would it be crazy that in a tiny community the most esteemed members would be invited to the esteemed persons conference?
@scottychen2397
Ай бұрын
This is only just begining: I can’t immediately tell if this is C*C = (C) * (C) or R^4 = (R^3) * (R) Neither of which has to do with what this is probably talking about, which is the technically 4 dimensional TM = 2 + 2 in a way that’s over the real as it were but is sincerely different than what can be topologically deduced is the natural implication of the previous demonstrations of explicit geometry. An exact comparison must be given to this kind of thing
@bigfrankalbigguy789
24 күн бұрын
He's talking about the fact that the alternating group of degree n >= 3 is non-simple if and only if n = 4. en.wikipedia.org/wiki/Alternating_group
@scottychen2397
Күн бұрын
Oh - that’s very different: clearly youre actually watching the video. I’m presently concentrating on the concept of a tangent bundle, which exhibits a very suspicious 4=2+2, which is why I chose to comment. This is not something he is ignorant toward as a geometer, although its true I didn’t really watch the video for what it actually is.
@scottychen2397
Күн бұрын
@bigfrankalbigguy789 I don’t know about this….. this is donaldson theory: whatever it is, Im sure its referring to the tangent bundle at the end of the day. Please don’t comment this if you end being incorrect in your implication that I dont know what Im talking about. TM is the most important 2n =n+n , no matter what kind of principal class cohomology group exists acting on whatever fibres…. Any result of 4=2+2 should - in some way- be related to the TM’s dimension. Spatial dimension has nothing to do with the dimension of the cotangent space nor tangent space….. This has to be throughly demonstrated as consistent with other things. Please dont imply I don’t know what im talking about. This is geometry.
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