Millenium Problems (Clay Math org)

[Wolf DeVoon]

I have no dog in this hunt, because I'm a math dodo. Natural language predicate logic, sure. But not this stuff.

http://www.claymath.org/millennium-problems

Millennium Problems

Yang–Mills and Mass Gap

Experiment and computer simulations suggest the existence of a "mass gap" in the solution to the quantum versions of the Yang-Mills equations. But no proof of this property is known.

Riemann Hypothesis

The prime number theorem determines the average distribution of the primes. The Riemann hypothesis tells us about the deviation from the average. Formulated in Riemann's 1859 paper, it asserts that all the 'non-obvious' zeros of the zeta function are complex numbers with real part 1/2.

P vs NP Problem

If it is easy to check that a solution to a problem is correct, is it also easy to solve the problem? This is the essence of the P vs NP question. Typical of the NP problems is that of the Hamiltonian Path Problem: given N cities to visit, how can one do this without visiting a city twice? If you give me a solution, I can easily check that it is correct. But I cannot so easily find a solution.

Navier–Stokes Equation

This is the equation which governs the flow of fluids such as water and air. However, there is no proof for the most basic questions one can ask: do solutions exist, and are they unique? Why ask for a proof? Because a proof gives not only certitude, but also understanding.

Hodge Conjecture

The answer to this conjecture determines how much of the topology of the solution set of a system of algebraic equations can be defined in terms of further algebraic equations. The Hodge conjecture is known in certain special cases, e.g., when the solution set has dimension less than four. But in dimension four it is unknown.

Poincaré Conjecture

In 1904 the French mathematician Henri Poincaré asked if the three dimensional sphere is characterized as the unique simply connected three manifold. This question, the Poincaré conjecture, was a special case of Thurston's geometrization conjecture. Perelman's proof tells us that every three manifold is built from a set of standard pieces, each with one of eight well-understood geometries.

Birch and Swinnerton-Dyer Conjecture

Supported by much experimental evidence, this conjecture relates the number of points on an elliptic curve mod p to the rank of the group of rational points. Elliptic curves, defined by cubic equations in two variables, are fundamental mathematical objects that arise in many areas: Wiles' proof of the Fermat Conjecture, factorization of numbers into primes, and cryptography, to name three.

[Brant Gaede]

No idea why you posted this. My understanding of zero is it's not a number. It's a place-holder which revolutionized arithmetical computations.

--Brant

only Bob can help us now

[Wolf DeVoon]

No idea why you posted this. My understanding of zero is it's not a number. It's a place-holder which revolutionized arithmetical computations.

--Brant

only Bob can help us now

I always ask for help when I see something I don't understand.

[Brant Gaede]

Either that or shoot it.

--Joke alert!

[BaalChatzaf]

On 11/20/2015 at 10:47 PM, Brant Gaede said:

No idea why you posted this. My understanding of zero is it's not a number. It's a place-holder which revolutionized arithmetical computations.

--Brant

only Bob can help us now

Zero is the identity element for the additive subgroup of the rationals,  reals,  complex number 0 + x = x.  x - x = 0  zero is a genuine number but during the renaissance the mathematicians considered negative numbers  and zero  as fantastical elements,  not to be taken seriously.  The bankers were the first to catch on that zero  and negative balances were as real as rain.