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