Millennium Prize Problems

The Millennium Problems are a set of seven unsolved mathematical problems designated by the Clay Mathematics Institute in 2000. These challenges cover a wide range of mathematical disciplines, from number theory to Topology, and solving any one of them would earn the solver a $1 million prize. They represent some of the most profound and perplexing questions in mathematics, captivating both professional mathematicians and enthusiasts alike. The Millennium Problems are 7 big mathematical problems that if answered, could bring change all over math and science education, and may even affect technology.

Whoever solves any one of the Millennium Problems could win $1,000,000, as well as potentially other prizes for it, like the Fields Medal, or even the Nobel Prize.

Some of the problems have a relationship to the de Fermat conjecture called Fermat's Last Theorem or FLT.

As described by the Clay Institute of Massachusetts, the Millennium Problems are:

Riemann hypothesis

A 200-year old question that is one of the most famous mathematical problems ever. Solving it will make mathematicians understand a lot more about prime numbers. It has applications in cryptography, number theory, and might even be useful in physics.

Mathematicians want to know when a certain function, called the Riemann Zeta function, written ζ(s), equals zero. There are many well known values of s where ζ(s) is zero, composing the negative even integers. The Riemann Hypothesis says that apart from these, ζ(s) equals zero only when s is a complex number with real part 1/2 or a negative whole number. As to whether this hypothetical is one of pure or applied math, a particular characterization of the machine to measure earthquakes gives real-part-1/2 spikes.

Clay description

Some numbers have the special property that they cannot be expressed as the product of two smaller numbers, e.g., 2, 3, 5, 7, etc. Such numbers are called prime numbers, and they play an important role, both in pure mathematics and its applications. The distribution of such prime numbers among all natural numbers does not follow any regular pattern. However, the German mathematician G.F.B. Riemann (1826 - 1866) observed that the frequency of prime numbers is very closely related to the behavior of an elaborate function

   ζ(s) = 1 + 1/2^s + 1/3^s + 1/4^s + ...

called the Riemann Zeta function. The Riemann hypothesis asserts that all interesting solutions of the equation

   ζ(s) = 0

lie on a certain vertical straight line.

This has been checked for the first 10,000,000,000,000 solutions. A proof that it is true for every interesting solution would shed light on many of the mysteries surrounding the distribution of prime numbers.

Yang-Mills equations

The laws of quantum physics stand to the world of elementary particles in the way that Newton's laws of classical mechanics stand to the macroscopic world. Almost half a century ago, Yang and Mills introduced a remarkable new framework to describe elementary particles using structures that also occur in geometry. Quantum Yang-Mills theory is now the foundation of most of elementary particle theory, and its predictions have been tested at many experimental laboratories, but its mathematical foundation is still unclear. The successful use of Yang-Mills theory to describe the strong interactions of elementary particles depends on a subtle quantum mechanical property called the "mass gap": the quantum particles have positive masses, even though the classical waves travel at the speed of light. This property has been discovered by physicists from experiment and confirmed by computer simulations, but it still has not been understood from a theoretical point of view. Progress in establishing the existence of the Yang-Mills theory and a mass gap will require the introduction of fundamental new ideas both in physics and in mathematics.

P vs. NP problem

This is the most famous one in computer science. It would be solved "if" someone discovered whether a computer can always find a solution to a problem given, as fast as check it. An ideal solution would contain an accessible how, including machine code. The metaphor can easily be asked of a pocket calculator: can it always find the solution of a typed equation as easily as it could check multiplication by division, for instance? The heart of the problem is in one-way operators' speed factor.

Clay description

Suppose that you are organizing housing accommodations for a group of four hundred university students. Space is limited and only one hundred of the students will receive places in the dormitory. To complicate matters, the Dean has provided you with a list of pairs of incompatible students, and requested that no pair from this list appear in your final choice. This is an example of what computer scientists call an NP-problem, since it is easy to check if a given choice of one hundred students proposed by a coworker is satisfactory (i.e., no pair taken from your coworker's list also appears on the list from the Dean's office), however the task of generating such a list from scratch seems to be so hard as to be completely impractical. Indeed, the total number of ways of choosing one hundred students from the four hundred applicants is greater than the number of atoms in the known universe! Thus no future civilization could ever hope to build a supercomputer capable of solving the problem by brute force; that is, by checking every possible combination of 100 students. However, this apparent difficulty may only reflect the lack of ingenuity of your programmer. In fact, one of the outstanding problems in computer science is determining whether questions exist whose answer can be quickly checked, but which require an impossibly long time to solve by any direct procedure. Problems like the one listed above certainly seem to be of this kind, but so far no one has managed to prove that any of them really are so hard as they appear, i.e., that there really is no feasible way to generate an answer with the help of a computer. Stephen Cook and Leonid Levin formulated the P (i.e., easy to find) versus NP (i.e., easy to check) problem independently in 1971.

Navier–Stokes equations

Perhaps the most important equations in fluid mechanics, which studies liquids and gases— to design better cars and jet airplanes, to learn how the atmosphere and water cycle work, among other things. They are shorthanded in engineering, much used in mathematics, and regarded in applied sciences.

Clay description

Waves follow our boat as we meander across the lake, and turbulent air currents follow our flight in a modern jet. Mathematicians and physicists believe that an explanation for and the prediction of both the breeze and the turbulence can be found through an understanding of solutions to the Navier-Stokes equations. Although these equations were written down in the 19th Century, our understanding of them remains minimal. The challenge is to make substantial progress toward a mathematical theory which will unlock the secrets hidden in the Navier-Stokes equations.

Hodge conjecture

In the twentieth century mathematicians discovered powerful ways to investigate the shapes of complicated objects. The basic idea is to ask to what extent we can approximate the shape of a given object by gluing together simple geometric building blocks of increasing dimension. This technique turned out to be so useful that it got generalized in many different ways, eventually leading to powerful tools that enabled mathematicians to make great progress in cataloging the variety of objects they encountered in their investigations. Unfortunately, the geometric origins of the procedure became obscured in this generalization. In some sense it was necessary to add pieces that did not have any geometric interpretation. The Hodge conjecture asserts that for particularly nice types of spaces called projective algebraic varieties, the pieces called Hodge cycles are actually (rational linear) combinations of geometric pieces called algebraic cycles.

Poincaré conjecture

The only Millennium Problem that has been conditionally solved, as of 2018. Mathematician Grigori Perelman found it true in certain 4d characterizations of the 3-sphere, but declined the $1M prize.

Because Perelman published his papers over the Internet rather than in a peer-reviewed journal, as required by the Clay Math Institute's rules, he was not initially offered CMI’s award, though representatives for the organization indicated they might relax their requirements in his case. Complicating their decision was uncertainty over whether Perelman would accept the money; he publicly stated that he would not decide until the award was offered to him. Finally, in 2010, CMI offered Perelman the reward for proving the Poincaré conjecture, and Perelman refused.

The Poincaré Conjecture states that the sphere is the only 3D object that can be shrunk to a single point, given certain conditions.

By most schools of natural philosophy, this conjecture would be seen immediately as either obviously correct or obviously flawed. For one thing, it is not irreversible. The 3-sphere characterization, like various Happy End boards/Hademard tilings, has higher dimensional characterization, so the most desired solution would be an equation for y per dimension x.

Clay description

If we stretch a rubber band around the surface of an apple, then we can shrink it down to a point by moving it slowly, without tearing it and without allowing it to leave the surface. On the other hand, if we imagine that the same rubber band has somehow been stretched in the appropriate direction around a doughnut, then there is no way of shrinking it to a point without breaking either the rubber band or the doughnut. We say the surface of the apple is "simply connected," but that the surface of the doughnut is not. Poincaré, almost a hundred years ago, knew that a two dimensional sphere is essentially characterized by this property of simple connectivity, and asked the corresponding question for the three dimensional sphere.

This question turned out to be extraordinarily difficult. Nearly a century passed between its formulation in 1904 by Henri Poincaré and its solution by Grigoriy Perelman, announced in preprints posted on ArXiv.org in 2002 and 2003. Perelman's solution was based on Richard Hamilton's theory of Ricci flow, and made use of results on spaces of metrics due to Cheeger, Gromov, and Perelman himself. In these papers Perelman also proved William Thurston's Geometrization Conjecture, a special case of which is the Poincaré conjecture. See the press release of March 18, 2010.

Birch and Swinnerton-Dyer conjecture

Mathematicians have always been fascinated by the problem of describing all solutions in whole numbers x,y,z to algebraic equations like

x² + y² = z²

Euclid gave the complete solution for that equation, but for more complicated equations this becomes extremely difficult. Indeed, in 1970 Yu. V. Matiyasevich showed that Hilbert's tenth problem is unsolvable, i.e., there is no general method for determining when such equations have a solution in whole numbers. But in special cases one can hope to say something. When the solutions are the points of an abelian variety, the Birch and Swinnerton-Dyer conjecture asserts that the size of the group of rational points is related to the behavior of an associated zeta function ζ(s) near the point s=1. In particular this amazing conjecture asserts that if ζ(1) is equal to 0, then there are an infinite number of rational points (solutions), and conversely, if ζ(1) is not equal to 0, then there is only a finite number of such points.