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The Millennium Problems

Updated: Oct 18, 2022

Even if you wouldn't consider yourself a mathematician, you have probably heard of the infamous 'Millennium Problems'.

They are 7 famous mathematical problems that were selected by the Clay Institute in 2000, and their answers are worth 1 million dollars each. They are vast and complex questions, spanning the fields of algebraic geometry, arithmetic geometry, geometric topology, mathematical physics, number theory, partial differential equations, and theoretical computer science.

I thought it might be worthwhile to give you a quick rundown on what each of them concern, and maybe go into a bit more detail in a later post. Although they represent some of the largest and most noble of mathematical struggles, they do give an interesting insight into the explorations and adventures of real mathematicians.

  1. Poincare Conjecture

  2. P vs. NP

  3. Hodge Conjecture

  4. Riemann Hypothesis

  5. Yang-Mills Existence and Mass Gap

  6. Navier-Stokes Existence and Smoothness

  7. Birch-Swinnerton-Dyer Conjecture

1. Poincare Conjecture (or as I like to call it, the donut conjecture)

This is the only Millennium Problem that has been solved! It concerns the topology of manifolds.

A manifold X of dimension n is a geometric object with a weird shape. If you zoomed up on each point, you will notice that its local area looks like n-dimensional Euclidean space. Think of an ant on the surface of a football; to the ant it looks pretty flat (a sphere is a 22-manifold; the flat ground looks like 22-dimensional Euclidean space) Two manifolds are distinct if one cannot be continuously deformed into the other.

As you can imagine, mathematicians would like to know if two manifolds are the same by deformation or not. This is therefore known as the classification problem: is there a way to identify if two manifolds are equivalent, without explicitly writing down their maps and comparing them?

Want the answer? Post coming up!

2. P vs. NP

Next, we jump to theoretical computer science. The question revolves around the time it take to execute an algorithm. That may seem like a very simple concept at first, but it is considered the most important problem in the entire field.

P and NP are classes of problems. P refers to all problems which can be solved in 'polynomial time' (essentially, the number of 'things' that a P problem has to deal with, to a certain power, corresponds to the time it takes to solve it.). NP is are problems that can be verified in polynomial time. That is, the problem depends on a positive integer n, and there is an algorithm that takes the prospective solution as input and returns "yes" or "no" .

As a little taster of this one's importance: proving P=NP would essentially break the internet.

3. Hodge Conjecture

This one concerns geometric shapes and polynomial equations.

Imagine drawing special polynomial equations (over complex numbers), and using them to 'cut out' shapes. These, known as complex algebraic varieties. Group theory emerged as a good technique to study the structures of these varieties, especially ones known as cohomology groups. Essentially, cohomology groups 'behave' like varieties but are much easier to deal with on a computer.

When trying to find a cohomology group to study a variety, you can sometimes find it simply by looking at information about sub-varieties of that variety.

This is the one that I perhaps know the least about, so I will leave it up to MIT for this one The Hodge conjecture states that certain cohomology groups studied by Hodge over certain nice complex varieties are generated by the classes of sub-varieties. The cohomology groups in question are often called the groups of Hodge classes, and classes generated by sub-varieties are often called algebraic. So in these terms, the conjecture becomes Every Hodge class on a projective complex manifold is algebraic.

4. Riemann Hypothesis

PRIMES!! What would any list of important maths problems be without a moment of silence for the elusive structures that riddle every field and problem mathematicians encounter.

The Riemann zeta function is as follows

ζ (n)= 1/(2^n) +1/(3^n) +1/(4^n)....

This function can be 'analytically continued' to one that is differentiable (we can find where its roots are) everywhere in the complex plane (think Argand diagrams), except for at at n=1. On the negative real line, this extension function has what are known as 'trivial zeroes' n=-2,-4,-6 .... The location of the OTHER zeros is more mysterious.

The Riemann hypothesis states that the real coordinate of all non-trival zeroes of the RZ Functions is 1/2.

Proving this hypothesis would have huge implications concerning our knowledge of number theory and the distribution of primes, and of course is heavily connected to other areas of maths.

5. Yang-Mills Existence and Mass Gap

Let's get small.

This theorem concerns the strong and weak nuclear force, which are supposed to hold together an atomic nucleus. They are fundamental components of the 'Standard Model' under which modern quantum physics operates. There are many gaps in this model, and many wonder if it is internally consistent.

The "mass gap" is formally defined as the difference between the energy in a vacuum (zero) and the next lowest energy state. The YM theory is a generalisation of Maxwell's theory of electromagnetism which allows for the majority of modern applications of quantum physics.

A solution would require set of formal axioms, and a proof that it is internally logically consistent. It would also require some proof of a positive lower bound on the masses of particles predicted by the theory.

6. Navier-Stokes Existence and Smoothness

Still hanging on the more applied end, we know look at partial differential equations, whch are used to model the movement of fluids. I need not even BEGIN to describe the importance of these methods in modern engineering and physics.

The Navier-Stokes equations are controlled by forces including pressure p, viscous stress v, and an external force. It uses Newton's second law (the one with force and motion) to describe partial derivatives of the velocity v of the fluid as a function of position and time.

These equations though massively important, are incomplete, as scientists are yet to prove that that smooth solutions to these equations always exist. The Millenial problem seeks a proof that smooth, globally defined solutions either always exist or do not always exist in incompressible fluids.

7. Birch-Swinnerton-Dyer Conjecture

Elliptic curves are quirky curves, which are characterised by nonnegative integers called their 'genus'. They sometimes have rational points on them.

Let us concern ourselves with curves of genus 1, as curves of genus-0 can be easily described, and curves of genus greater than or equal to 2 have finitely many rational points on them.

Any elliptic curve can be described by a linear combination of n of its rational points and a torsion point. This number, n, is the RANK of the curve. Another weird function L takes an elliptic curve and parameter s. The 'order of vanishing' of a that curve at s=1 is known as its analytic rank. Part of the BSD Conjecture involves proving that the rank n of a curve is equal to its analytic rank.

The other part concerns the Taylor Series (a fancy expansion) of L, which links to finding an equation describing products and quotients of constants relating to the elliptic curves. While there is much computational evidence for the first half of this conjecture, no general proof has yet been found.

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