Millennium Prize Problems solutions

Elementary solutions to the Millennium Prize Problems.

1. P=NP

By definition, polynomila algorithms admit decomposition in chains of smaller polynomial algorithms. Consequently, polynomial time algorithms do not solve problems where blocks, whoose order is the same as the underlying problem, require simultaneous resolution. Thus, in fact P≠NP

2. Hodge conjecture

Assuming that if a compact Kähler mainfold is complex-analytically rigid, the area-minimizing subvarieties approach complex analytic subvarieties. The set of singularities of an area-minimizng flux is zero in measure. The rest it left to the reader as an easy routine excersize.

3. Riemann hypothesis

This is a simple experimental fact. 1013 roots of the Riemann hypothesis have been already tested and it suffices for all practical applications. In fact, one state a suitable statistical hypothesis and check it on the sample of, say, 105 roots.

4. Yang–Mills existence and mass gap

Well, discrete infinite bosonic energy-mass spectrum of gauge bosons under Gelfand nuclear triples admits non-perturbative quantization of Yang-Mills fields whence the gauge-invariant quantum spectrum is bounded below. A particular consequence is the existence of the mass gap.

5. Navier–Stokes existence and smoothness

 haven't worked this one in such detail, but observing that

∥L(u,v)∥2=∑n≥25u22nv22n+1/n2≤C∥(un/n√)∥24∥(vn/n√)∥24≤C∥(un/n√)∥22∥(vn/n√)∥22=C(∑u2n/n)(∑v2n/n)

one can easily find at least one closed-form solution applying the bubble integral. In the equation, L is a bilinear operator.

6. Birch and Swinnerton-Dyer conjecture

The problem with former attempts has been in the way elliptic curves have been dealt with. But this really admits a proof with a computer by checking the (finitely many) categories of curves.