Of course I am interested in having an exact sharp criterion from topologists about the obstructions to embeddability to because these are the topological restrictions on the physical universe itself and so this question is exceedingly important, and it is a purely topological question. I am very proud of my discovery that the actual universe is a round four sphere partly because we can ask this question and take the mathematical answer seriously as question about Nature. The fact that a great topologist like Michael Freedman is thinking about the issue is also encouraging. So that the topological restriction from the pure embeddability point of view. On the other hand one could imagine that the topology of the three manifold would not be very different from the equator by soft reasoning that well, there’s not really much matter in the universe compared to its size so how complicated can the topology be? I think this is deceptive. My intuition tells me that there is some topological complexity that arises from the complications in the microscopic scales. I am being vague about this still but a different way of seeing the point is that we are not dealing with a simple mean curvature flow type situation but a NavierStokes equation in the macroscopic scale and we don’t have much clue about topological complexity of the singularities of a NavierStokes equation. One of the reasons I sort of dance around the edges on these issues is because I don’t have a good sense of which point of view might show more of the the fundamental geometry of the universe as an evolving three manifold. Of course I am frustrated that the best minds of our time are wasting their time on string theory and quantum field theory rather than focusing on the real issues of what’s at stake. But this is probably unfair and I don’t have the energy or interest of trying to convince people that this is where they should be focused on: the problem of three dimensional topology of the universe is the same as these two problems: which three manifolds can embed into and what topological complexity can arise from starting at the equator and following the NavierStokes nonlinear parabolic equation and understand the topological changes in the resulting hypersurfaces. Sure, Floer Homology type theories might do the job. I obtained a copy of Mrowka and Kronheimer’s Monopoles and Three Manifolds to get a better sense of where this can lead. I am reading a paper of Giga to get a better sense of how to solve the NavierStokes equation even though I have an explicit scheme for solving it. I don’t think I will get the right way to see this without some perspective. It’s exciting but frustrating to have the greatest opportunity to be staring at the heart of Nature and not have the right tools handy. I was thinking before of doing something simple like modeling the hypersurface as fluctuating Brownian motion but now that I am looking at this a bit clearer, it’s better not to waste my time on artificial models and get the right model at first shot. I’m not getting any younger.
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1:44 AM (0 minutes ago)



These are universal constants c^{mn}_j which appear in the nonlinear mixing terms of the solution of the NavierStokes equations in the eigenbasis of S4. These are scale invariant in the sense that scaling the eigenvalues by k should produce:
c^{mn}_p = c^{(km)(kn)}_{kp}
This does not seem like a hard problem since scaling the sphere by (1/k) should do this. I will actually spend some time to work this out at some point — it just looks a bit messy and besides 2D spherical harmonics the 4D case is not used so it’s still research territory but this would be a breakthrough in our understanding of Nature in terms of turbulence phenomena. When Andrei Kolmogorov in 1941 came up with his 5/3 scaling law it was a breakthrough in our understanding of turbulence. That underneath the hood the source of the scaling might be the geometry of the entire universe and thus scale invariance in all the quadratic coupling problems would instantly demystify stochasticity, turbulence, and source of fractal geometry of Nature instantly.
In other words, I claim that it is the CLEBSCHGORDON COEFFICIENTS in S4 spherical HARMONICS AND THEIR SCALE INVARIANCE that is the geometric explanation of all the fractal phenomena that we observe in Nature empirically. Note that no such explanation is natural to the euclidean geometry. You need compactness of space to have any expansion at all. Therefore SPHERICAL geometry of space is crucial to explain scale invariance phenomena. Thus my S4 model radically alters human understanding of the universe in which we live by clarifying the mathematical structure behind scale invariance and fractal geometry of Nature. Benoit Mandelbrot had gone far into modeling of fractal phenomena with a long history going back to Cantor but no one before me has completely solved this mystery of Nature with a concrete mathematical solution that can connect the classical atom, the scattering amplitude computations and the turbulence in hydrodynamics in one shot. This is indeed a joyous day for humanity.
Thanks,
Zulf
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10/1/16



where is the energy dissipation rate and is the energy injection rate and is the ‘energy flow rate’. I am not interested in the NavierStokes equation with a forcing term so I ignore the energy injection rate but Terry defines the three types of regimes of fluid flow in terms of the size of the terms on the right: when dominates, ‘injection regime’, when dominates, ‘dissipation regime’; then dominates ‘energy flow regime’. I will explain in a moment how we can use the same concepts and solve the NavierStokes equations exactly in the S4 model which is not very easy to do on $latex\mathbf{R}^3$ and I am stealing these concepts from Terry’s analysis because I am interested in not inventing anything new that I don’t have to. In my case I will be expanding things in an eigenspinor basis rather than a dyadic basis and the analogy of the mixing terms being the ‘energy flow’ term and the diffusion term being the dissipation regime is perfectly sane and sensible. We want to use this description to understand what Kolmogorov was after in turbulence and then we want to make an identification between this problem and the classical atom problem and note that technically the issues are almost identical and therefore claim that in fact the source of the turbulence is at the source of the Dirac equation for electromagnetism rather than in the NavierStokes macroscopic model. This link is the most important thing that I can say that Kolmogorov was not looking for.
Terry’s heuristic is using the form of the nonlinear term innerproducted with the solution for the energy while the dissipation term is . Therefore Terry expects dissipation regime — and I won’t bother using the precise symbol for dealing with the constants and just use greater or smaller with the understanding that constants don’t matter for order of magnitude estimates and Terry has a precise blog about this — for and he expects energy flow regime when .
This is beautiful because we won’t change a word of this analysis which is extremely valuable to me. The ideas here go back to Andrei Kolmogorov and that is what is interesting. The important issue is Kolomogorov’s 5/3 law which Terry gets to using the power law asymptotic . Read his blog because he does a fabulous explanation for analytic argument for the Kolmogorov’s argument. I will just use the whole setup as a program and try to see whether in the exact solution all of the pieces Terry is describing can be proven. I certainly have the sort of simplicity where I can try to prove these.
The structural constants $latec c_{N,N_1,N_2}$ in Terry’s setup will be related to the ClebschGordan coefficients of expanding products of spherical harmonics in terms of spherical harmonics of .
Are these constants scaleinvariant? In other words are scale invariant if they are the ClebschGordan coefficients? The first idea that pops to my mind is that they should be because the sphere of radius 1 is being compared with the sphere of a radius . I like to check these simple intuitions because I hate getting and mixed up. So if you consider and is an integer then that is the th Fourier mode of a circle. Let’s say high frequency. Then you can shrink the circle to radius and get it as the first Fourier mode. Right so the ClebschGordon coefficients of will be scale invariant in this way although there may be complications for proving a theorem that I am evading but at the moment we are more interested in understanding the source of the scale invariance for the NavierStokes in the S4 model where it should be coming from the ClebschGordon coefficients rather than being pedantic. So in the S4 model we would be explaining the SCALE INVARIANCE phenomena in NavierStokes in Nature directly from the ClebschGordon scale invariance which is mathematically a relatively trivial observation say compared to Fermat’s Last Theorem or something hard. So this is mathematically not so interesting in difficulty but it is tremendously insightful about why we can be assured of scale invariance in turbulence in Nature from the S4 model. In other words Kolmogorov was not interested, I don’t think, in turbulence because he wanted to be an applied mathematician but rather because he was attempting to get to the heart of Nature through the turbulence problem. And here Nature is bared before us in a transparent manner — scale invariance of turbulence can be described through a beautiful geometry of S4.
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4:52 PM (12 minutes ago)



The technical solution is sufficiently easy in this case that the important issue is to note that the NavierStokes equation — so long as it describes a fluid composed of atoms interacting by LennardJones potentials governed by my S4 matter and electromagnetic equations (which I have not established but trust that the equilibrium statistical mechanics theory provides justification for the link since it seems to have a universal character to such interacting systems) implies that NavierStokes equation governs the hypersurface evolution.
This is an extremely important reiteration of ANDREI KOLMOGOROV’S obsession with the question of stochasticity in the NavierStokes equation. In fact this is the dynamics governing gravitational evolution in our universe and is the source of fractal geometry of Nature literally for this is the macroscopic description of microscopic interacting systems in equilibrium and scale invariance is encoded in the equation. This is providing the link between Richard Hamilton’s Ricci flow type geometric evolution and Benoit Mandelbrot’s view of Nature’s geometry as fractal. And in the S4 model this can be solved exactly.
Note that without the S4 model this unity between perfect symmetry of the foursphere and the stochasticity that we observe in Nature cannot be understood with a concrete realization of the solution in the eigenbases where these fundamental equations lose their analytic sophistication and can be solved easily in an explicit basis.
As I had promised, here is the stochasticity of Nature explained with a crutch — that crutch being the leap from a per atom model to the macroscopic fluid dynamics model. But my classical atom model can be repeated here almost exactly. The missing link in my own understanding then reduces to the issue of classical statistical mechanics of interacting particles. This is of course on top of other wonderful properties of this model like no need for renormalizability for scattering amplitude computations.
Thanks,
The aspect that is interesting right now is setting things up so that the distribution of directions of the particles is uniform. Ok so this is a problem of equilibrium statistical mechanics for inhomogeneous fluids. The established equation for fluid dynamics is the NavierStokes equations which is the macroscopic dynamical equation for the movement of the ensemble and Vladimir Arnold has produced a hydrodynamic theory for riemannian manifolds. This is TopologicalMethodsHydrodynamics.
What we need assuming for the moment that we make the leap from microdynamics to macrodynamics, is the fluid motion equation on . We’ll stick in the diffusion term to make it NavierStokes.
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12:56 PM (1 minute ago)



PHYSICAL INTUITION — and this is unique in the PLANET today until S4 is accepted physics because no one else can claim to have physical intuition about S4 hypersurfaces — the EinsteinHilbert action does NOT have critical points at these Clifford tori. This is physical intuition for S4 that is verified by simple computation of the first variation. But the conjecture is that NO critical point of EinsteinHilbert action on S4 should exist that is not the equator BECAUSE that should be empty space without matter.
vanishes. The space of solutions do not depend on or . This information constitutes the SeibergWitten invariants of . Let
where is the group of gauge transformations. Then a Theorem says is always compact. Fixing for generic the space is a smooth finite dimensional manifold with a smooth circle action. The dimension is where is the signature of . Here .
The SeibergWitten invariant is defined as follows on spinc structures over . First for topological reasons the dimension is even call it .
(a) If is even .
(b) If then is a finite set of points then .
(c) If then where represents the first Chern class of a bundle .
Consider the K3 surface . This fourmanifold has and . There is a unique spinc structure with . For this spinc structure and for all others it is zero. Freedman’s theorem tells us is homeomorphic to but SeibergWitten is nonzero for the first and zero for the second.
(These are quick notes from Clifford Taubes’ Park City Lectures to refresh my mind).
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We want to check whether these Clifford tori can pop up as critical points. This is to check for for these tori. The Ricci curvature varies . The mean curvature for these is zero. These have two distince principal curvatures for the 2sphere and for the circle. We have relations:
(a)
(b) C
Solving the system gets us and . Now just plug into This is not zero.
This is fabulous. This tells us that the Clifford tori cannot arise as a critical point of the EinsteinHilbert action for hypersurfaces of . Therefore we will have uniqueness of the critical point of EinsteinHilbert action for the range of squared length of the second fundamental form . Extending beyond approximately is a very hard problem according to this paper. We need to check what this squared length of the second fundamental form translates to in terms of the radius of the universe but this is quite respectable. We know that for totally umbilical hypersurfaces, i.e. the geodesic spheres in the first variation will vanish and now we know it will not vanish for these Clifford tori. It would be nice to prove that there are no other critical points besides these but I am willing to take it on faith for now.
This is actually a very nice geometric result in and of itself because although I basically used the hard work of Alencar and do Carmo, of Cheng and Yau and here QM Cheng and S. Ishikawa and did very little work thus far myself, I can now make the bolder conjecture that these are the only critical points for the restricted length squared of second fundamental form. It’s a little surprising result because if you look at the heavily worked field of minimal (zero mean curvature) and constant mean curvature and constant scalar curvature hypersurfaces it’s quite busy with even a nice Master’s Thesis on these things. I don’t have intrinsic interest in working on these problems but I think this is fascinating nonetheless because based on physical intuition in this case I obtained the correct answer despite all the results for constant curvature cases that Clifford tori should not be critical points of the EinsteinHilbert action which is generally not considered in the field. So I conjecture that empty space should only be the equator of the round foursphere according to the EinsteinHilbert action in S4 physics. In other words I am conjecturing that while it is more difficult to extend the minimal hypersurfaces without pathological examples that these are not going to be a problem for the EinsteinHilbert action. I am making this conjecture based on physical intuition rather than mathematical here and this is another theme that I am stealing from the string theorists who have been successful with their brand of physical intuition. I think this is the first time in history that anyone has made a conjecture based on physical intuition for the round four sphere model of physics, so I am quite proud of this moment.
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