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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For Riemann I can recommend this: https://m.youtube.com/watch?v=zlm1aajH6gY
P=NP asks whether any problem for which a solution can be verified in polynomial time, that solution can also be computed in polynomial time. The current consensus is “probably not” but it hasn’t been proven either way.
The others I don’t know enough about.
None of these are particularly easy to explain, but I’ll try and give an overview of the ones I’m familiar with:
> Navier – Stokes Equation
There is a very important mathematical concept called a “differential equation”. This is where we have an indirect description of how a quantity changes in time and/or space, which involves rates of change in the quantity, and we want to turn that into a direct description where we can just plug in the time and/or position and get its value. This is called “solving” the differential equation.
A simple example is the motion of a projectile affected by air resistance. We know that the amount of air resistance is dependent on the speed of the object. We also know from Newton’s second law that the acceleration of an object is equal to the overall force acting on it, divided by its mass. This gives us a differential equation that relates the acceleration of the projectile to its speed. To find out where the projectile will actually be at a given time, we need to solve this differential equation.
The simplest differential equations can be solved exactly, but more often we need to use numerical methods to find approximate solutions. There is a broad class of differential equations, called “linear” differential equations, for which this is straightforward – you can even prove that your approximate solution is within a certain distance of the real solution. There are many processes in science and engineering that can be understood in terms of linear differential equations, but some require nonlinear equations. The most well-known example of this is the motion of fluids, which is described by the Navier-Stokes equations. People have come up with various numerical methods for the Navier-Stokes equations which *seem* to give good approximate solutions, but nobody has been able to prove that they actually work. In fact, nobody even knows, outside some simple special cases, how to prove that the Navier-Stokes equations even *have* solutions to approximate. That’s what this Millennium Prize problem is about: proving that the Navier-Stokes equations have solutions with certain basic properties. That would be the first step towards proving that these numerical methods actually work, and it would presumably revolutionise our understanding of non-linear differential equations more generally. By the way: the fact that fluids do weird stuff like turbulence is closely related to the fact that they are described by nonlinear equations.
> P vs NP Problem
This is from an area of maths called computational complexity theory, which studies how many steps are required to solve problems. For example, suppose we have a list of numbers (positive and negative), and we want to know if there is some subset of those numbers which sum to zero. We could, of course, write down every possible subset of the numbers and check whether any of them sum to zero, but is there a faster way?
Much of the work in this field has involved classifying problems into “complexity classes”, where the number of necessary steps grows in a certain way as the size of the problem (in our case, the length of the list of numbers) grows.
Two of the most important and well known complexity classes are P and NP. P is the class of problems for which the minimum number of steps required to solve the problem is “polynomial” in the size of the problem. That is, the number of steps is less than s^n, where s is the size of the problem and n is some fixed number. NP is slightly more complicated to define, but roughly speaking its the class of problems for which an existing solution can be verified in a polynomial number of steps.
The thing is, it’s not actually known whether P and NP are the same class. There are many problems – and the “subset-sum” problem I described is one of them – for which we know it’s in NP, but we don’t know if it’s in P (though it’s trivial to show that any problem in P is also in NP). It’s strongly suspected that many of these problems are *not* in P, and that P and NP are two distinct classes. Proving that would presumably help towards understanding how various other complexity classes fit together too. If someone proved that P and NP *are* in fact the same (which would definitely be the more surprising outcome), then this would presumably lead to much faster methods for solving all kinds of problems.
> Riemann Hypothesis
This one is much more abstract. It’s about something called the Riemann zeta function, which is an example of a complex function, i.e. one that takes a complex number (a real number + an imaginary number) and gives you another complex number. There are certain points at which the value of the Riemann zeta function is zero, and it seems that a certain subset of these points all lie along a certain line. The Riemann hypothesis says that all of this subset of “zeros” lie on that line.
The reason why this is so significant is because people have developed many conditional proofs which show that if the Riemann hypothesis (or a variant of it) is true, then some other interesting conjecture is also true.
I can give short ELI5 for some of them; the others I don’t understand enough (or not at all).
>- Riemann Hypothesis
As we go to higher numbers, prime numbers get rarer. Do they do so predictably, or more erratically? (Answer: As far as we know, predictably, but there’s no strict mathematical proof). At least that’s the original motivation for the problem; formally it’s about the solutions to a certain equation which are needed to calculate the exact value of how many prime numbers there are smaller than a certain number.
>- P vs NP Problem
Are some math problems genuinely hard, or might there be an easy solution that we just haven’t found yet? (Answer: No easy solution to these problems has been found yet, and most modern cryptography is based on the assumption that such an easy solution does not exist. But there’s no strict mathematical proof that such a solution *cannot* exist).
>- Poincaré Conjecture (the only one solved)
In our three-dimensional world, there are spheres and there are donuts. In four dimensions, is there a hyper-shape that combines the properties of a sphere and a donut? (Answer: no).
Navier-Stokes
Describes fluid flow by tracking the energy types and directions in the flow. It can be solved using simplified assumptions and approximations but mathematically it’s not known if there is a general or continuous solution to it for all possible inputs.
P-NP
Some computational problems seem to be unable to solve with an algorithm quicker than it takes to brute force the problem. The problems is mathematically proving that there is no algorithm for any given comuptation and it’s not that we haven’t found it.
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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