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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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.
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