            This homepage provides solutions to the following Millennium problems

- the Riemann Hypothesis
- well-posed 3D-nonlinear, non-stationary Navier-Stokes equations
- the mass gap problem of the Yang-Mills equations.

A common distributional Hilbert space framework provides an answer to Derbyshine's question ((DeJ) p. 295): „ What on earth does the distribution of prime numbers have to do with the behavior of subatomic particles?". It also enables a quantum gravity theory based on an only (energy related) Hamiltonian formalism, as the corresponding (force related) Lagrange formalism is no longer defined due to the reduced regularity assumptions to the domains of the concerned pseudo differential operators.

The key ingredients of the Zeta function theory are the Mellin transforms of the Gaussian function and the fractional part function. To the author´s humble opinion the main handicap to prove the RH is the not-vanishing constant Fourier term of both functions. The Hilbert transform of any function has a vanishing constant Fourier term. Replacing the Gaussian function and the fractional part function by their corresponding Hilbert transforms enables an alternative Zeta function theory based on two specific Kummer functions and the cotangens function. The imaginary part of the zeros of one of the Kummer functions play a key role defining alternatively proposed arithmetic functions to solve the binary Goldbach conjecture.

All attempts failed so far to prove well-posed 3D-nonlinear, non-stationary Navier-Stokes equations analogue to the 2D-NSE case due to not appropriate Sobolev (energy) norm estimates. The underlying mathematical handicap is called the Serrin gap. The reduced regularity assumption for the domain of the corresponding momentum operator enables an energy norm estimate "closing" the Serrin gap, while at the same point in time overcomes current "blow-up" effect handicaps.

The classical Yang-Mills theory is the generalization of the Maxwell theory of electromagnetism where chromo-electromagnetic field itself carries charges. As a classical field theory it has solutions which travel at the speed of light so that its quantum version should describe massless particles (gluons). However, the postulated phenomenon of color confinement permits only bound states of gluons, forming massive particles. This is the Yang-Mills mass gap. The variational representation of the Maxwell equations in the proposed "quantum state" Hilbert space framework builds on truly fermions (with mass) & bosons (w/o mass) quantum states / energies, i.e. a Yang-Mills equations model extention is no longer required.

The „mass generation process“ is modelled as a „selfadjoint property“ break down by the orthogonal projection from the momentum domain H(1/2) onto the Hilbert (sub-) space H(1), whereby the complementary closed subspace H(1/2)-H(1) provides the model for the (remaining) ground state (vacuum) energy; this extention of the standard energy Hilbert space H(1) is and can be neglected in all („less granular“) Lagrange formalism based physical models.

A common mathematical model of unified quantum and gravity theories requires a truly infinitesimal geometry framework. The Hilbert space based framework in quantum theory is certainly the more suitable geometrical framework compared to Weyl’s manifold based ones. At the same point in time both theories need to leave something out as they are not compatible. In quantum theory already in the simple quantum harmonic oscillator model the eigenvalues converge equidistant to infinity, i.e. the total energy is represented as an infinite series. A similar situation is given by the concept of „wave packages“ requiring other (less regular) domains as the domain H(1) for standard Fourier waves. A related concept is a about wavelets leading to the extended Hilbert space H(1/2). The standard quantum theory Hilbert space L(2)=H(0) is (just) chosen in order to enable statistical analysis in the context of statistical thermodynamics.

In the proposed model the (standard) „calculus in the small“ meets the „calculus in the large“ in a Hamiltonian formalism for classes of non-linear equations, where the kinetic (matter, Lagrange formalism) energy part is based on a Krein space setting/ decomposition modelling repulsive and attractive mass elementary particles. The Hilbert subspaces (H(0),H(1)) are compactly embedded into the Hilbert spaces (H(-1/2),H(1/2). This is about the same cardinality relationships as for the embededness of the set of the (physical observation relevant) rational numbers into the fields of real or hyper-real numbers.

The physical concepts of „time“ and „change“ are different sides of the same coin, i.e. there is no „time“ w/o „change“ and there is no „change“ w/o „time“. In other words, the concepts of „time“ and „change“ are and need to be in scope of the „matter/kinetic“ energy model H(1), while its complementary ground state (vacuum) energy model H(1/2)-H(1) is per definition independent from the thermodynamical concept of „time“.

The Friedrichs extension of the Laplacian operator is a selfadjoint, bounded operator B with domain H(1). Thus, the operator B induces a decomposition of H(1) into the direct sum of two subspaces, enabling the definition of a potential and a corresponding „grad“ potential operator. Then a potential criterion defines a manifold, which represents a hyperboloid in the Hilbert space H(1) with corresponding hyperbolic and conical regions ((VaM) 11.2). The direct sum of the corresponding two subspaces of  are proposed as a model to define a decomposition of H(1) into subspaces modelling repulsive resp. attractive elementary mass particles. We note that dark matter is only subject to gravity;  the energy/matter type distribution in the universe is estimated with "dark energy ~74%", "dark matter 22%" and "atoms 4%", whereby about 99.9% of the atoms are hydrogen (90%) and helium (10%).

(KaD1): „It is most amazing that the dynamics of the universe as determined by the equations of GR can be derived from purely Newtononian considerations. … The only difference between the Newtonian and Einstein version of cosmology becomes apparent only by differentiating the Newton energy conservation equation taking into account the relation between and the pressure from local energy conservation“.

From a mathematical point of view the cosmological inflation model (which can be derived from both, the Newtonian and the Einstein version of cosmology (!)), is about a simple, classical ordinary differential equation, which is even not well-posed due to the missing initial value condition at the Planck time. As the unit of measure of "pressure" is identical to the unit of measure of the "energy density" this unbalanced (not well-posed (!)) conservation law is the model to estimate the required (to be built somehow) expansion energy during the inflation period. We note that the assumption of the above Einstein version of cosmology to derive the ODE is based on the cosmological principle assumption; it states that the universe should look like the same for all observers. It „tells us that the universe must be homogeneous and isotropic. This then tells us, which metric must be used, which is the Friedmann-Robertson-Walker (FRW) metric“ (KaD1). In other words, the physical assumptions about the state of the universe during the inflation period is the stable one, which we currently have; we further note, that the same model can be also derived from the classical Newtonian version of cosmology. This reminds to the origin of one of the titles of the books from R. Penrose: "The Emperor's New Clothes".

(ChF) 1.1: occurance of plasmas in nature, "It has often been said that 99% of the matter in the universe is in the plasma state; that is, in the form of an electrified gas with atoms dissociated into positive ions and negative electrons. This estimate may not be very accurate, but it is certainly a reasonable one in view of the fact that stellar interiors and atmospheres, gaseous nebulae, and much of the interstellar hydrogen are plasmas. In our own neighborhood, as soon one leaves the earth's atmosphere, one encounters the plasma comprising the Van Allen radiation belts and the solar wind. On th other hand, in our everyday lives encounters with plasmas are limited to a few examples: the flash of a lightning bolt, the soft glow of the Aurora Borealis, the conducting gas inside a fluorescent tube or neon sign, and the slight amount of ionization in a rocket exhaust. It would seem that we live in the 1% of the universe in which plasmas do not occur naturally."

(ChF) 7.5: the meaning of Landau damping, "The theoretical discovery of wave damping without energy dissipation by collisions is perhaps the most astonishing result of plasma physics research. That this is a real effect has been demonstrated in the laboratory. ... Landau damping (spontaneous stabilization of plasma; return to an equilibrium w/o increase of entropy)  is a characteristic of collisionsless plasmas, but it may also have application in other fields. For instance, in the kinetic treatment of galaxy formation, stars can be considered as atoms of a plasma interacting via gravitational rather than electromagnetic forces. Instability of the gas of stars can cause spiral arms to form, but his process is limited by the Landau damping."

As H(1) is compactly embedded into H(1/2), and given an initial universe w/o any "thermodynamical time“ (i.e. H(1) is empty with an only existing ground state energy for the whole mathematical model system) the probability for „symmetry break down“ events to generate mass were and are zero; obviously those events happened and will go on to be happen. At the same point in time the generated and still being generated „matter world“ is governed by physical principles like the „least action principle“ and the principles of „statistical thermodynamic“, whereby the classical action variable of the system determines the „thermodynamical time“. The thermodynamical clock started to run with probability "100%" later than t=0 with the start of the inflation process.

DeR) p. 93: „In general the resonance between the wave phase velocity and the velocity of individuals electrons cannot be neglected. It involves coupling between single-particle and collective aspects of plasma behaviour, and give rise to an energy flow which is known as Landau damping. Before continuing, we should note that this topic is related to one of the main unsolved questions of physics. It has not yet been possible to resolve fully the contrast between the reversibility in time of microscopic phenomena – for example, the dynamics of a particle described by Newton’s laws of motion – and the irreversibility in time of macroscopic phenomena, as desribed by the second law of the thermodynamics. Any thermodynamic system is in fact constructed from a large number of particles, all of which obey Newton’s laws, so that this contrast is central to physics. A resolution of this contrast would be particulary helpful to a full understanding of Landau damping; this is because Landau damping involves a flow of energy between single particles on the one hand side, and collective excitations of the plasma on the other.“

We note that the Landau damping property is complementary to the properties of electro-magnetic "forces", which weaken themselves spontaneously over time w/o increase of entropy or friction.

The Landau damping phenomenon can be interpretated as the capability of stars to organize themselves in a stable arrangement. The proposed alternative inflation model with integrated quantum fluctuation initial "value" conditions is based on the non-linear Landau collision operator in a weak H(-1/2) representation. The current classical related Landau-Boltzmann PDE are interpreted as an approximation model to it and not the other way around. The Vlasov equation (a current alternative plasma dynamics model) is discarded, as this model overlooks the important physical phenomenon of electrons travelling with exactly the same material and wave speed.

The H(1/2) space as first cohomology is fundamental to explain the properties of period mapping on the universal Teichmüller space. We note that a vector space and any linear subspace are convex cones, i.e. the tool „convex analysis and general vector spaces“ can be applied. Morse’s calculus of variations in the large enables a calculus of variations in the large e.g. on varifolds ((MoM), (SeH), (AlF)).

In the context of varifolds we quote from (MiJ): "For a particle P, which moves among a surface M during a given time interval, the action of the particle during this time interval is defined to be a certain constant times the action integral "E". If no forces act on P (except for the constraining forces which hold it within M), then the principle of least action asserts that E will be minimized within the class of all paths joining w(0) and w(1), or at least that the first variation of E will be zero. Hence P must traverse a geodesic. But a quite different physical model is possible. Think a rubber band which is stretched between two points of a slippery curved surface. If the band is described parametrically by the equation x=w(t), then the potential energy arising from tension will be proportional to our integral E (at least to a first order of approximation). For an equilibrium position this energy must be minimized, and hence the rubber band will describe a geodesic."

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latest update: Oct. 15, 2019, p. 17

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The journey started in 2010 resulting in the paper

Braun K., The three Millennium problem solutions (RH, NSE, YME) and a Hilbert scale based quantum geometrodynamics, July 2019

The journey is still ongoing: in order to support an appropriate change traceability this paper is split into

PART I:
Braun K., A Kummer function based alternative Zeta function theory to solve the Riemann Hypothesis and the binary Goldbach conjecture

last update :  July 31, 2019

PART II:
Braun K., 3D-NSE and YME mass gap solutions in a distributional Hilbert scale frame enabling a quantum gravity theory

last update :  July 31, 2019

With respect to the NSE topic we also refer to

http://www.navier-stokes-equations.com/

With respect to the ground state energy and quantum gravity topic we also refer to

https://quantum-gravitation.de/        