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First of all it is a trivial observation that in the S4 model the physical universe must be a hypersurface topologically into \mathbf{R}^4 just by inverse stereographic projection from any point not in the physical hypersurface \sigma: \mathbf{R}^4\rightarrow S^4 and \sigma^{-1}(M) is diffeomorphic to M.  I just don’t know very much about three-manifold topology so I have to defer to the master of topology such as Michael Freedman on this issue.
algol-freedman

Of course I am interested in having an exact sharp criterion from topologists about the obstructions to embeddability to \mathbf{R}^4 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 S^3 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 Navier-Stokes equation in the macroscopic scale and we don’t have much clue about topological complexity of the singularities of a Navier-Stokes 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 \mathbf{R} and what topological complexity can arise from starting at the equator and following the Navier-Stokes 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 Navier-Stokes 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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I return with the gigantic accomplishment of finally seeing that ANDREI KOLMOGOROV’S 1941 5/3 scaling invariance for high Reynold’s number turbulence in Navier-Stokes equation is finally demystified in the S4 model for the first time in human history by the link to scale invariance of the Clebsch-Gordon coefficients under scaling of the spheres.  Is this explanation simple and stupid?  Yes most certainly.  It is a formal explanation whose power resides in understanding the source of the scale invariance once again to the geometry of the universe.  In contrast, understanding the scale invariance in the context of Navier-Stokes equations on \mathbf{R}^3 is a very hard problem and the results of Sheffer for the bounds on the Hausdorff dimension of the singular set by Cafarelli, Kohn and Nirenberg were celebrated.  Thus it may have seemed that the fundamental source of the scale invariance would be extremely complicated hard problem but of course as with scattering amplitude computations in QED it is trivial because the universe is compact and computations of this type in physics reduces to eigenbasis computations that are well known and well studied and appear in Courant’s book on Mathematical Methods in Physics and elsewhere.  I am now going to take a break from pushing against the broken physics of this age especially at the high point after a decade of obsession with the S4 physics model now that I have resolved partly, at least to my own satisfaction the issue of smoothness, stochasticity, chaos, and determinism in Nature.  Chaos, stochasticity, smoothness, non-locality, all these features can be made concrete and mathematical and explicit in S4 model of physics.  There are many people who can work out the details (which includes myself but I want to push forward with the liberty of not being chained and gagged by academic disciplines — Kissinger said of Harvard academics that they wage wars against each other for minutiae — and I don’t know if he is right or not but I am not interested in such things because I have better ideas to spend my genius).  Now let me consider the issue of existence of magnetic monopoles in experiments in spin ice.  The details are very interesting but since I have my own personal obsessions and I am not beholden to the academic institutions, I will go back to my own thoughts from 2008 which began the S4 physics project which I will declare a resounding success since it concretely culls expansion as well as explain other things.  Now the S4 model posits four spatial dimensional universe and here I know that I am taking a bold speculative leap but this is the sort of bold speculative leap that has really done wonders for me so I will continue.  In my last years at Princeton, I took a course on Aestheticists with Professor Stanley Corngold who did not have much of a good opinion of my understanding of the issues involved with Kafka and Nietzsche at the time but I regard his Kafka criticism to be extremely valuable regardless and I had always thought of Nietzsche as a great genius in the same caliber as Einstein and Hilbert and Dirac.  Now Nietzsche had a theory of Nature which is of course not well known today and certainly not privileged among the scientists.  But he was a very great genius so I take him seriously.  His theory was that the universe was composed of atoms of will to power.  This is an internal worldview that produced such a theory.  Now with close to absolute confidence that there exists a vast four dimensional universe and there exist magnetic monopoles I wonder if magnetic monopoles are ubiquitous and constitute the material through which we experience things that we consider psychological.  Of course the thought of such a link will be considered highly unscientific in this age but it is something that I think is actually of scientific interest.  Nietzsche predicted that the most central science of humanity would be psychology, that psychology would become the queen of the sciences one day.  This prediction of course is underestimating the depth of the sciences and mathematics but regardless I hope that Nietzsche has been rescued from the bad politics of the Nazi era.  I have tremendous sympathy for him for personal reasons as well.  He was totally destroyed by his sister and her Nazism.  Oh yes, magnetic monopoles.  I was wondering for a while if magnetic monopoles are abundant in the four dimensional spherical universe of radius 3246 Mpc and somehow because of the way in which we experience the world we are only aware of such things as ‘internal’ experiences.  In 2008 I was much more confident than I am today that in dreams and other states we are continuously experiencing magnetic monopoles and other non-physical purely electromagnetic phenomena.  It was not an accident that I wrote down my matter and electromagnetic equations on S4 model as occurring on spinor fields on the global four-sphere rather than taking pains to consider the complement of the physical hypersurface of S4 as non-physical.  This is not because I have a penchant for believing in supernatural creatures but rather because we don’t know what a very large part of existence looks like and what exists in a vast extra dimension and the only way we can explore such a giant frontier that I have opened up for humanity is by ensuring that our physics is simple and perfect.  If we have a wobbly broken physics that don’t fit together and speculate about events 13.7 billion years ago with a ballooning cosmos like the Church Cosmology from Nicolas Copernicus era of course such a vast new frontier will have to be hidden with U(1) invariance of electromagnetism and what have you.  There is no question at all in my mind that Bohm-Ahranov effect implies existence of a large EM phase dimension and this EM phase dimension allows us to consider the possibility of pushing rigorous mathematical physics to bear on these issues of what is there to be found.  We are now locked in a box with special relativity.   We are now stuck with the idea that our reach out into the cosmos is strictly limited by the speed of light.  I want to disabuse humanity from this idea because special relativity is not right.  There are vast worlds in the S4 universe whose exploration is held up by a medieval scientific establishment whose optimism for progress is superficial and too focused on needs of governments and industry.  Nietzsche’s theory of the universe formed from atoms of will to power is interesting because it seems to me that Nietzsche — who was consciously interested in critique of science from an artistic metaphysics from his first effort on The Birth of Tragedy — was in a sense a maverick theorist in the old tradition of Ancient Greek philosophers.  I believe that he was theorizing based on internal experiences — well more than believe; he wrote as much — and that this will not be explained by biochemistry of the brain. I think our ideas of thinking that the mind is the brain and that chemical analysis of the complexities of the brain will reveal the complexities of the mind are silly.  There is much that can be learned by empirical methods but they are primitive and simple things and we will not be able to reach by this way — especially in a four spatial dimensional universe where there is no possibility whatsoever that we do not experience all four space dimensions although we today are unable to have sharp notions that can map for us how we experience them.  Empiricism is extremely powerful; but empiricist science will suffer greatly now especially when the Christian right has taken control of the most powerful hegemonic empire that this planet has ever seen in its entire history.

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zulfikar.ahmed@gmail.com <zulfikar.ahmed@gmail.com>

Attachments1:44 AM (0 minutes ago)

to harrington, jharris, jhp, jhricko_4, jianjunp, jinha, jjbrehm, jlind, jlondon, jlw, jmateo, jmerseth, jmetcalf, jmg, jmogel, joel, joelms, john.aldrich, john.beatty, johncrawford53, jose.oliveira, josesoto, josue, jpadgett, jpbalz
Ladies and Gentlemen,
Around October of 2016 when I was working on stochastic volatility models (which were quite successful empirically) I had the idea of using Navier-Stokes equations to model FINANCIAL MARKET VOLATILITY because at the time it seemed that turbulence in the financial markets might have the non-linearity of the Navier-Stokes equation.  In the intervening time my views have certainly developed as I had been focused on establishing S4 physics for a Scientific Revolution where the world can see through new eyes the universe a static cosmos eternal and unchanging with galaxies relatively still governed by the single geometry of S4 and macroscopically the single force of electromagnetism that I pronounced on July 4 2018.  In the process I have been developing my ideas about how to replace Einstein’s gravitational field equations properly and I stumbled upon the realization that the Navier-Stokes equation is the dynamic system for gravity.  This is not new per se, since Navier-Stokes equations are the standard macroscopic fluid model for classical equilibrium statistical mechanics.  However this time around revisiting the Navier-Stokes equations I realized that the scale invariance of Navier-Stokes solved in the spherical harmonics basis is dictated by the Clebsch-Gordan coefficients of spherical harmonics on S4 (i.e. phi_m phi_n = sum_{m,n} c^{mn}_j phi_j) where the phi_j are spherical harmonics of S4.

These are universal constants c^{mn}_j which appear in the nonlinear mixing terms of the solution of the Navier-Stokes 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 CLEBSCH-GORDON 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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Attachments10/1/16

to renkel, jhansen, jharrington, jharris, jhp, jhricko_4, jhs, jhutasoi, jianjunp, jinha, jjbrehm, jlind, jlondon, jlw, jmateo, jmerseth, jmetcalf, jmg, jmogel, joel, joelms, john.aldrich, john.beatty, johncrawford53, jose.oliveira
 Ladies and Gentlemen,
Intuition suggests that if we replace the hydrodynamic Navier-Stokes equation in dimension d>=3,
d/dt u – nu*Laplacian(u) = (u, grad(u))
u(0,x) = u0(x)
to the situation I claim in financial volatility, the Laplacian will be replaced by a SYMMETRIC FELLER GENERATOR, an integro-differential operator.  The Navier-Stokes theory tells us that smoothness of solutions is a hard problem.  Weak solutions always exist and satisfy an equality like
d/dt ||u|| + \int_t || grad(u)||_2^2 ds = || u(0)||_2^2
and it’s a famous open problem to determine if there can be smooth solutions or singularities develop in finite time.  If they do develop, singularities must have one-dimensional Hausdorff-meansure zero (Cafarelli-Kohn-Nirenberg).  Since now I have produced statistical evidence that this (u,grad(u)) type nonlinearity exists for volatility a natural question is whether we can cheaply push through the Navier-Stokes theory to our volatility models.  As luck would have it, Giga’s work (for long time existence for Navier-Stokes with small data initial conditions etc.) is nicely phrased in terms of a more general operator than Laplacian.  His basic setup is
d/dt u + Au = Fu
where A say a self-adjoint operator that satisfies || exp(-tA) f||_p <= C || f ||_s / t^sigma for some constant sigma.  Now even though the types of Feller generators A we are interested in such as a stable process etc. satisfy this type of bound because there are heat kernel bounds of the form p(t,x,y) <= C t^{-d/beta} ( 1 + d(x,y)/t^{1/beta})^{-(d+beta)}.  So we can use Giga’s theory to claim some parts of the Navier-Stokes theory to our volatility setting such as small initial data existence — although this is more of a schematic proposal.  If we use a more general operator A instead of Laplacian, we have scaling different so the Hausdorff dimension of the singular set should change slightly.  Otherwise, we should have all the results of turbulent solutions of Navier-Stokes hold for volatility as well.  So this is a slightly more precise step toward making VOLATILITY STORMS a significant rigorous concept that will put the link between turbulent hydrodynamics and financial volatility in firm footing.

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Terry Tao have a very good blog on the Navier-Stokes equation from which I am learning some of the basic issues about how to think about them.  What he does to think about these is use a dyadic decomposition of frequencies.  He analyses the energy E(t) = \int u(t.x)^2 dx of the solution of the Navier-Stokes equation by a dyadic decomposition and considering the pieces of energy in the dyadic basis:

\partial_t E_ = \Pi_{N,N_1,N_2} - D_N + F_N

where D_N is the energy dissipation rate and F_N is the energy injection rate and \Pi_{N,N_1,N_2} is the ‘energy flow rate’.  I am not interested in the Navier-Stokes 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 F_N dominates, ‘injection regime’, when D_N dominates, ‘dissipation regime’; then \Pi_{N,N_1,N_2} dominates ‘energy flow regime’.  I will explain in a moment how we can use the same concepts and solve the Navier-Stokes 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 Navier-Stokes macroscopic model.  This link is the most important thing that I can say that Kolmogorov was not looking for.

Terry’s heuristic is \Pi_{N,N_1,N_2} = O( NE_N^{3/2}) using the form of the nonlinear term inner-producted with the solution for the energy N while the dissipation term is D_N = \nu N^2E_N.  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 N \ge \nu^{-1} E_N^{1/2} and he expects energy flow regime when 1 \le N \le \nu^{-1} E_N^{1/2}.

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 E_N \sim AN^{-\alpha}.  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.

terry-heuristic

The structural constants $latec c_{N,N_1,N_2}$ in Terry’s setup will be related to the Clebsch-Gordan coefficients of expanding products of spherical harmonics in terms of spherical harmonics of S^4.

Are these constants scale-invariant?  In other words are c_{\lambda N, \lambda N_1, \lambda N_2} scale invariant if they are the Clebsch-Gordan 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 1/\lambda>0.  I like to check these simple intuitions because I hate getting 1/\lambda and \lambda mixed up.  So if you consider e^{i \lambda \theta} and \lambda is an integer then that is the \lambda-th Fourier mode of a circle.  Let’s say \lambda >> 1 high frequency.  Then you can shrink the circle to radius 1/\lambda and get it as the first Fourier mode.  Right so the Clebsch-Gordon coefficients of S^4 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 Navier-Stokes in the S4 model where it should be coming from the Clebsch-Gordon coefficients rather than being pedantic.  So in the S4 model we would be explaining the SCALE INVARIANCE phenomena in Navier-Stokes in Nature directly from the Clebsch-Gordon 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)

to harrington, jharris, jhp, jhricko_4, jianjunp, jinha, jjbrehm, jlind, jlondon, jlw, jmateo, jmerseth, jmetcalf, jmg, jmogel, joel, joelms, john.aldrich, john.beatty, johncrawford53, jose.oliveira, josesoto, josue, jpadgett, jpbalz
Ladies and Gentlemen,
I am glad I took a step back from considering a direct N=10^{82} atom model on the equator S3 of S4 of radius 3246 Mpc.  Since JOSIAH WILLARD GIBBS established foundations of equilibrium statistical mechanics 1902 it seems that there is strong feeling that thermodynamic models for large atoms in such a situation should be macroscopically described by the NAVIER-STOKES equation, which I had began thinking about several years ago in the context of models for finance but now I can see that they are models for fundamental stochasticity in Nature in S4 physics. Ebin and Marsden proved short time existence for these for closed manifolds in the late 1960s.  These equations are nonlinear parabolic equations and on riemannian manifolds with Levi-Civita connection D, they have the form
dv/dt – nu*Laplacian(v) + D_v(v) =  -grad(p)
div(v) = 0
v=v(x,t)
v(x,0)=v0(x)
Consider the fluid restricted to a tubular neighbordhood of a hypersurface and solve this equation by extending the v0(x) to all of S4 and using spherical harmonics decomposition after translating the equation to the differential form duals.  The nonlinear term D_v(v) in local coordinates is quadratic with Christoffel symbol coefficients in v = v^i d/dx^i so the translation to 1-forms produces a quadratic nonlinearity  Q(v^#) as well where v^# denotes the metric-dual 1-form of v.  Then use the spherical harmonics basis where we get a system of ordinary differential equations with eigenspaces being mixed in a quadratic manner but these can be solved by the SAME strategy that I employed for the classical atom problem and I will have to make this a bit more rigorous but essentially nu will be an analog to the coupling constant for EM interactions.  Formally the solution is almost identical and we get transition in the S4 model smooth bounded solutions to turbulence when the nu parameter (reynold’s constant is sufficiently high).

The technical solution is sufficiently easy in this case that the important issue is to note that the Navier-Stokes equation — so long as it describes a fluid composed of atoms interacting by Lennard-Jones 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 Navier-Stokes equation governs the hypersurface evolution.

This is an extremely important re-iteration of ANDREI KOLMOGOROV’S obsession with the question of stochasticity in the Navier-Stokes 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 four-sphere 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,

Zulf

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This is a test run not an attempt at any final model.  Consider an interaction Hamiltonian in classical mechanics on $latex\mathbf{R}^4$.  This \mathbf{R}^4 will represent the tangent bundle of S^4 of radius 3246 Mpc.  We want to see what to do on the flat 4-space following this Hamiltonian and then consider the geometry later.  We would like to take many iterations to examine different aspects of this problem until we get a model that is right for S4 physics.

H = \sum_{i=1}^N p_i^2/2m + U(r_1,\dots,r_N)

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 Navier-Stokes 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 S^4.  We’ll stick in the diffusion term to make it Navier-Stokes.

euler-equation

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