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The Poincaré conjecture has some prerequisite assertions, like that there is a "surface" (yes, in surface physics there is the mystery, but the main point like in Quantum Biology is not the surface itself, but the bonding and folding (in Biology: hydrogen bridge bonding) or in physics lattice physics and Penrose patterns, or better stated, the dimensional transition as a Movement (energy, scaling, work) and bonding or topological ("layer") folding (dimensional transiting (what happens at the "horizons" of this 3D Sphere -> 4D sphere? Nothing, it is about the transition fractal 0-string to N-string (oscillation, qparticles, virtuals, finally hadrons, gluons swirling): the phase transition within the space topology AT the N-Sphere Horizons: The singularity would by a mathematical state, which can never exist (Riemann Zeta) it is about the infinity mediation at the horizons (topology):

Riemann zeta to Ramanujan infinity conjecture: the assertion of an invariance (Eichfelder) is then a mathematical tautology (the rotation of a 2D world sheet in space) - but you need the maths for the application, in this case, you should dissolve the circle number Pi - Pi needs to be a fractal N-Pi that describes how two tangents meet at infinity at the N-Body Horizons within N-shaped bodies (fractal space topology at Prime-steps (stability constraints for quarks in view to conditions (entropy or "Boltzmann" (->thermodynamics) and dimensional quantum vacuum (Dark Matter Field for the photonic "antimatter" gate to A SuperFluid BH density).

The transitions of fractal N-Body Diameters (Schwarzschild Horizons to Surface Areas for the "real" world modeling) are the main "parameters".

Computational Complexity and Black Hole Horizons

Leonard Susskind

(Submitted on 23 Feb 2014 (v1), last revised 25 Feb 2014 (this version, v2))

Computational complexity is essential to understanding the properties of black hole horizons. The problem of Alice creating a firewall behind the horizon of Bob's black hole is a problem of computational complexity. In general we find that while creating firewalls is possible, it is extremely difficult and probably impossible for black holes that form in sudden collapse, and then evaporate. On the other hand if the radiation is bottled up then after an exponentially long period of time firewalls may be common. It is possible that gravity will provide tools to study problems of complexity; especially the range of complexity between scrambling and exponential complexity.

https://arxiv.org/abs/1402.5674

Reply to:

by Ulla Mattfolk

https://www.facebook.com/groups/thinkallowed/permalink/1720831234806000/

Perelman proved the conjecture by deforming the manifold using the Ricci flow (which behaves similarly to the heat equation that describes the diffusion of heat through an object). The Ricci flow usually deforms the manifold towards a rounder shape, except for some cases where it stretches the manifold apart from itself towards what are known as singularities. Perelman and Hamilton then chop the manifold at the singularities (a process called "surgery") causing the separate pieces to form into ball-like shapes. Major steps in the proof involve showing how manifolds behave when they are deformed by the Ricci flow, examining what sort of singularities develop, determining whether this surgery process can be completed and establishing that the surgery need not be repeated infinitely many times.

The first step is to deform the manifold using the Ricci flow. The Ricci flow was defined by Richard S. Hamilton as a way to deform manifolds. The formula for the Ricci flow is an imitation of the heat equation which describes the way heat flows in a solid. Like the heat flow, Ricci flow tends towards uniform behavior. Unlike the heat flow, the Ricci flow could run into singularities and stop functioning. A singularity in a manifold is a place where it is not differentiable: like a corner or a cusp or a pinching. The Ricci flow was only defined for smooth differentiable manifolds. Hamilton used the Ricci flow to prove that some compact manifolds were diffeomorphic to spheres and he hoped to apply it to prove the Poincaré Conjecture. He needed to understand the singularities.

Hamilton created a list of possible singularities that could form but he was concerned that some singularities might lead to difficulties. He wanted to cut the manifold at the singularities and paste in caps, and then run the Ricci flow again, so he needed to understand the singularities and show that certain kinds of singularities do not occur. Perelman discovered the singularities were all very simple: essentially three-dimensional cylinders made out of spheres stretched out along a line. An ordinary cylinder is made by taking circles stretched along a line. Perelman proved this using something called the "Reduced Volume" which is closely related to an eigenvalue of a certain elliptic equation.

Sometimes an otherwise complicated operation reduces to multiplication by a scalar (a number). Such numbers are called eigenvalues of that operation. Eigenvalues are closely related to vibration frequencies and are used in analyzing a famous problem: can you hear the shape of a drum?. Essentially an eigenvalue is like a note being played by the manifold. Perelman proved this note goes up as the manifold is deformed by the Ricci flow. This helped him eliminate some of the more troublesome singularities that had concerned Hamilton, particularly the cigar soliton solution, which looked like a strand sticking out of a manifold with nothing on the other side. In essence Perelman showed that all the strands that form can be cut and capped and none stick out on one side only.

Completing the proof, Perelman takes any compact, simply connected, three-dimensional manifold without boundary and starts to run the Ricci flow. This deforms the manifold into round pieces with strands running between them. He cuts the strands and continues deforming the manifold until eventually he is left with a collection of round three-dimensional spheres. Then he rebuilds the original manifold by connecting the spheres together with three-dimensional cylinders, morphs them into a round shape and sees that, despite all the initial confusion, the manifold was in fact homeomorphic to a sphere.

One immediate question was how can one be sure there aren't infinitely many cuts necessary? Otherwise the cutting might progress forever. Perelman proved this can't happen by using minimal surfaces on the manifold. A minimal surface is essentially a soap film. Hamilton had shown that the area of a minimal surface decreases as the manifold undergoes Ricci flow. Perelman verified what happened to the area of the minimal surface when the manifold was sliced. He proved that eventually the area is so small that any cut after the area is that small can only be chopping off three-dimensional spheres and not more complicated pieces. This is described as a battle with a Hydra by Sormani in Szpiro's book cited below. This last part of the proof appeared in Perelman's third and final paper on the subject.

Friedmann-Lemaitre-Robertson-Walker universe corresponds to a time evolving radius of a S3 space. It argues that if this universe is modified in M3, at the end the acceleration it may produce a phase transition changing M3 to a space of constant curvature which corresponds precisely to a de Sitter phase associated with S3.

William Thurston's elliptization conjecture states that a closed 3-manifold with finite fundamental group is spherical, i.e. has a Riemannian metric of constant positive sectional curvature. A 3-manifold with such a metric is covered by the 3-sphere, moreover the group of covering transformations are isometries of the 3-sphere. Note that this means that if the original 3-manifold had in fact a trivial fundamental group, then it is homeomorphic to the 3-sphere (via the covering map). Thus, proving the elliptization conjecture would prove the Poincaré conjecture as a corollary. In fact, the elliptization conjecture is logically equivalent to two simpler conjectures: the Poincaré conjecture and the spherical space form conjecture.

The elliptization conjecture is a special case of Thurston's geometrization conjecture.

https://en.wikipedia.org/wiki/Poincar%C3%A9_conjecture#Ricci_flow_with_surgery

https://en.wikipedia.org/wiki/Thurston_elliptization_conjecture

https://en.wikipedia.org/wiki/Geometrization_conjecture

Image Courtesy: Von Salix alba in der Wikipedia auf Englisch, CC BY 2.5, https://commons.wikimedia.org/w/index.php?curid=1972330