Math might not teach us to add love or subtract hate, but it does give us all hope that every problem has a solution. And, if you happen to be really good in solving math problems, like, extremely talented in the field of Mathematics, there are even problems that can make you rich if you manage to solve them.
First laid out by Clay Mathematics Institute (CMI) in 2000, The Millennium Problems are seven most difficult math problems, and solving each has a reward worth $1 Million. The institute explains that there’s a reason to keep such attractive prize on these problems: “The Prizes were conceived to record some of the most difficult problems with which mathematicians were grappling at the turn of the second millennium; to elevate in the consciousness of the general public the fact that in mathematics, the frontier is still open and abounds in important unsolved problems; to emphasize the importance of working towards a solution of the deepest, most difficult problems; and to recognize achievement in mathematics of historical magnitude.”
Here are the seven Millennium Problems:
Yang–Mills and Mass Gap
P vs NP Problem
Birch and Swinnerton-Dyer Conjecture
Russian mathematician Grigori Perelman managed to solve the Poincaré Conjecture problem in 2003, which was approved three years later. The mathematician, however, turned down the million dollar prize and also the Fields Medal. He said that the award was unfair and his contribution was no greater than that of Hamilton, the mathematician who discovered Ricci Flow, which actually led to the solution of Poincaré Conjecture problem.
While the officials thought of using the refused prize money for the benefit of Mathematics, there are still 6 problems that remain unsolved, and you can certainly try to solve them.
Let’s look in detail each of the remaining six Millenium Problems.
Yang–Mills and Mass Gap
Quantum mechanics is one of the most successful theories in the history, enabling us to understand the behaviour of matter and energy at atomic and subatomic particle levels. Yang and Mills provided an important framework to describe these elementary particles using mathematical structures, and the theory today plays an important role in the elementary particles theory.
The YM theory has been verified already by numerous experiments, but its mathematical foundation still remains unclear. The theory suggests that the quantum particles have positive masses defined by “mass gap” to describe the interactions of elementary particles. In other words, particles cannot be of zero masses even when they are analogous to massless photons. The mass gap is a critical part to explain why nuclear forces are extremely strong and short in range compared to electromagnetism and gravity. This property has already been discovered by physicists through experiments and validated with computer simulations. The Millennium Problem is then about establishing a general mathematical and physics theory to explain the mass gap.
Prime numbers have always been one of the important areas of interest for mathematicians. These numbers that are only divisible by themselves and 1, actually build the whole numbers. With their immense importance in mathematics and applications, there is a great amount of interest in knowing how these prime numbers are distributed along the number line. While it was believed that prime numbers do not follow a particular pattern compared to other natural numbers, in 19th century mathematicians discovered the Prime Theorem that gives an approximate idea on the average distance between the prime numbers. But, it remains unknown how close the true distribution remains to that average. The Riemann Hypothesis, however, limits this possibility by suggesting that the frequency of prime numbers is closely related to the behavior of an elaborate function, known as Riemann Zeta function. The hypothesis states that any input value in the equation that makes the result zero (except the negative integers) fall on the exact same line. While this has already been checked for the first 10 trillion solutions, it still needs a rigorous proof for every interesting solution, making it one of the unsolved Millennium Problems.
P vs NP Problem
The P (easy to find) Vs NP (easy to check) is an unsolved problem in the world of theoretical computer science. In simple terms, the problem basically asks this: if it is easy to check that the solution to a problem is correct, is it also easy to solve the problem? “P” here stands for polynomial time, i.e. problems that are easier to solve by the computer and “NP” stands for nondeterministic polynomial time, i.e. problems that not easy for computers to solve, but are easy to check. One of the examples is finding the prime factors of a large number. If you have all the list of possible factors, you can easily multiply them together and check if you can get back the original number. However, there is no possible way to find the factors of that large number. Mathematicians as such believe that no such possible proof exists, but proving the same in itself is a daunting task and as such remains one of the unsolved Millennium Problems. The problem was formulated by Stephen Cook and Leonid Levin in 1971.
Majority of fluid dynamics is governed by Navier-Stokes equations that explain the fluid’s motion. It essentially helps in understanding how the speed of the fluid flow will change under the internal and external forces like pressure, velocity and gravity respectively. Scientists and engineers use the Navier-Stokes equations to mathematically model weather, ocean currents, air-flow around an aircraft wing and even to understand how stars move inside a galaxy. But, our understanding of these equations is still minimal as most mathematical tools do not prove useful to accurately predict the flow behavior. This is because fluids behave differently in different cases. For example, smoke coming out from a cigarette or a candlestick shows smooth signs of flow initially, but turns abruptly into vortices that are unpredictable with the differential equations. While it might be possible that N-S equations cannot be solved exactly in all cases, it is also possible that an ideal mathematical fluid can be developed which follows the equations. The Millennium Problem is then about solving these equations in all cases or show an example where it cannot be solved.
The Hodge Conjecture is one of the hardest to explain. To make it simple though, the problem asks whether complex mathematical shapes can be built from simple ones. More or less the question is similar to building objects from Lego blocks. The fundamental idea is to ask that to what extent the shape of a given object can be approximated by sticking together simple geometric building blocks of increasing dimensions. The technique became popular and got generalized in many ways, enabling mathematicians to progress in studying variety of objects in their investigations. However, the generalization ignored the geometrical origins and it became important to add pieces having no geometric interpretation. The Hodge Conjecture as such says that these pieces called Hodge cycles are actually nothing but a combination of geometric pieces called algebraic cycles.
Birch and Swinnerton-Dyer Conjecture
The Birch and Swinnerton-Dyer Conjecture describes rational solutions in order to define elliptic curve. It is also recognized as one of the most challenging mathematical problems that is still unsolved. The conjecture is that the elliptic curve has infinitely many rational solutions. Thus, solving the equation as such will boil down to a single number to tell you whether there are finitely or infinitely many solutions. This solution is related to the behavior of an associated Zeta function with the size of the group of rational points on the curve. The conjecture is already supported by experimental pieces of evidence, but the correct proof is still left to be provided. The conjecture as such was chosen as one of the Millennium Prize Problems.