PREFACE
OVERVIEW
RIEMANN HYPOTHESIS
NAVIER-STOKES EQUATIONS
YANG-MILLS EQUATIONS
WHO I AM
LITERATURE


Albert Einstein, "we can't solve problems by using the same kind of thinking we used when we created them",

Wolfgang E. Pauli, "all things reach the one who knows how to wait".

This homepage addresses the following three Millenium problems (resp. links to corresponding homepages):

A. The Riemann Hypothesis (RH)
B. The 3D-Navier-Stokes equations (NSE)
C. The Yang-Mills equations (YME)

A. The Riemann Hypothesis

All nontrivial zeros of the analytical continuation of the Riemann zeta function have a real part of 1/2. The Hilbert-Polya conjecture states that the imaginary parts of the zeros of the Zeta function corresponds to eigenvalues of an unbounded self adjoint operator.

We provide a solution for the RH building on a new Kummer function based Zeta function theory, alternatively to the current Gauss-Weierstrass function based Zeta function theory. This primarily enables a proof of the Hilbert-Polya conjecture (but also of other RH criteria like the Bagchi formulation of the Nyman-Beurling criterion or Polya criteria), whereby the imaginary parts of the zeros of the corresponding alternative Zeta function definition corresponds to eigenvalues of a bounded, self adjoint operator with (newly) distributional Hilbert space domain.

The proposed framework also provides an answer to Derbyshine's question, ("Prime Obsession")

... “The non-trivial zeros of Riemann's zeta function arise from inquiries into the distribution of prime numbers. The eigenvalues of a random Hermitian matrix arise from inquiries into the behavior of systems of subatomic particles under the laws of quantum mechanics. What on earth does the distribution of prime numbers have to do with the behavior of subatomic particles?"

The answer, in a nutshell:

"identifiying "fluids" or "sub-atomic particles" not with real numbers (scalar field, I. Newton), but with hyper-real numbers (G. W. Leibniz) enables a truly infinitesimal (geometric) distributional Hilbert space framework (H. Weyl) which corresponds to the Teichmüller theory, the Bounded Mean Oscillation (BMO) and the Harmonic Analysis theory. The distributional Hilbert scale framework enables the full power of spectral theory, while still keeping the standard L(2)=H(0)-Hilbert space as test space to "measure" particles' locations. At the same time, the Ritz-Galerkin (energy or operator norm minimization) method and its counterpart, the methods of Trefftz/Noble to solve PDE by complementary variational principles (A. M. Arthurs, K. Friedrichs, L. B. Rall, P. D. Robinson, W. Velte) w/o anticipating boundary values) enables an alternative "quantization" method of PDE models (P. Ehrenfest), e.g. being applied to the Wheeler-de-Witt operator.

Regarding the proposed alternative quantization approach we also refer to the Berry-Keating conjecture. This is about an unknown quantization H of the classical Hamiltonian H=xp, that the Riemann zeros coincide with the spectrum of the operator 1/2+iH. This is in contrast to canonical quantization, which leads to the Heisenberg uncertainty principle and the natural numbers as spectrum of the harmonic quantum oscillator. The Hamiltonian needs to be self-adjoint so that the quantization can be a realization of the Hilbert-Polya conjecture.


B. The Navier-Stokes Equations

The Navier-Stokes equations describe the motion of fluids. The Navier–Stokes existence and smoothness problem for the three-dimensional NSE, given some initial conditions, is to prove that smooth solutions always exist, or that if they do exist, they have bounded energy per unit mass.

We provide a global unique (weak, generalized Hopf) H(-1/2)-solution of the generalized 3D Navier-Stokes initial value problem. The global boundedness of a generalized energy inequality with respect to the energy Hilbert space H(1/2) is a consequence of the Sobolevskii estimate of the non-linear term (1959):

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

The "standard" weak Hopf solution is not well posed (which is therefore also the case for the corresponding classication solution(s) due to not appropriately defined domains of the underlying velocity and pressure operators.

The proposed solution also overcomes the "Serin gap" issue, as a consequence of the bounded non-linear term wih respect to the appropriate energy norm.


C. The Yang-Mills Equations

The YME are concerned with quantum field theory. One of ist challenge is about an appropriate mathematical model to govern the "mass gap" (i.e. to end up with finite energy norms), which is the difference in energy between the vacuum and the next lowest energy field.

We propose to apply the same solution concept to solve the "mass gap" issue of the YME. This provides a truly infinitesimal geometry (H. Weyl), enabling the concept of Riemann that force is a pseudo force only, which results from distortions of the geometrical structure. The baseline is a common Hilbert space framework (for all (nearby action) differential equations)

- providing the mathematical concept of a geometrical structure (while Riemann's manifold concept provides only a metric space and related affine connections)

- replacing "force type" specific gauge fields and its combination model(s) for the electromagnetic, the strong and the weak nuclear power "forces"

- building an integrated (no longer "force" dependent dynamical matter-field interaction laws) universal field model (including the gravity "force")

As a consequence there is no "mass" and therefore no (YME-) "mass gap" anymore, but there is an appropriate vacuum (Hilbert) energy space, which is governed by the Heisenberg uncertainty principle.

                     http://www.quantum-gravitation.de/