Each of the Millennium 7 Math Problems

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I can’t find an article or Youtube video that dumbs them down enough. Anyway, the 7 problems are:

– Yang Mills and Mass Gap

– Riemann Hypothesis

– P vs NP Problem

– Navier – Stokes Equation

– Hodge Conjecture

– Poincaré Conjecture (the only one solved)

– Birch and Swinnerton – Dyer Conjecture

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Yang Mills and Mass gap:

Describe strong force in physics (which hold quarks together to form proton and neutron, among others matter), and show that it’s impossible to quark to be isolated on its own.

Riemann hypothesis:

To count number of prime numbers up to some upper bound x, we have a prime number theorem that give a very good approximation. The question is about the error of that approximation for large x. We know that the error is at least x^1/2 ln(x) (up to a constant factor). The conjecture is that the error is as small as possible, that is it’s actually this.

P vs NP:

If a problem have an easy to check if a solution is correct, is it also easy to solve the problem in the first place?

Navier-Stokes:

There is an equation that describe how fluid should move according to Newtonian physics, in 3 dimensions. Given fluid that start at a certain state and change over time according to the equation, is it possible for it to reach a point in time when energy concentrate too much in a too small region? In a more poetic term, can air on a gentle day suddenly explode?

Hodge conjecture:

Given a system of equation of with complex numbers such that all terms have the same degree, this system form a set of solutions. You can study what kind of holes it has, by figuring out which shape that are subset of this set enclose something missing. Certain such subset come from set of solutions to system of equations by adding more equations to the previous system, or combination of these subsets. These subset satisfy some obvious property. The conjecture is the converse: if a subset satisfy these obvious property, must it be obtained using the above method?

Poincare conjecture:

Given a 3-dimensional space has no boundary, every path on it must approach at least one point an infinite number of times, and every loop on it can be shrink to a point. If we remove one point from it, must it be the case that we can deform it into usual 3-dimensional Euclidean space?

BSD:

Given an equation y^2 =x^3 +ax+b where a and b are integer constants, we want to know points with rational coordinate on the curve drawn by this equation. There are potentially infinitely many points, but we can construct new point from old point using a tangent-and-chord construction: draw a chord through 2 points (or tangent line of the curve if both point is the same), and look at the intersection of that line with the curve. That way, we don’t have to worry about all possible points, but just a few points that can be used to generate all other points. There are a few exceptional points that cannot generate an infinite number of new points. We want to know the minimum number of points such that, given these points and the exceptional points, we can generate every points. This minimum number is called the rank. The conjecture said that it can be found as follow. For each prime p, find the number of integer almost-solutions between 0 and p-1, where almost solution means the equation is off by a multiple of p. Then take that number of almost-solution and divide by p. Do that for every prime p, and multiply the result up to some upper bound x. This number should be approximately ln(x) to the some power, up to a constant factor, for large x. This power is conjectured to be the rank.

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