Not everyone likes math but it is that one thing that gives hope that every problem can be solved. If somehow, you are really into maths and very talented in the field, then there are some problems that can make you rich if you can find their solution. The Millennium problems are the seven most difficult problems and if you can solve any one of these, you can earn $1 Million as a reward.
These problems were first laid out by Clay Mathematics Institute (CMI). The Institute explained the reason behind the attractive prize on these problems saying, “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.”
Following are the seven Millennium Problems:
- Yang-Mills and Mass Gap
- Riemann Hypothesis
- P vs NP Problem
- Navier-Stokes Equation
- Hodge Conjecture
- Poincare Conjecture
- Birch and Swinnerton-Dyer Conjecture
One Russian mathematician Grigori Perelman managed to solve the Poincare Conjecture problem in 2003. The solution was approved three years later. However, he turned down the million dollar prize and the Fields Medal as well. According to him, the award was unfair and his contributions were not greater than Hamilton who discovered Ricci Flow. The solution Perelman gave of Poincare Conjecture used Ricci Flow.
The officials thought of using the refused prize money for the benefit of Mathematics. There are still 6 unsolved problems and you can give a try to solve them.
Yang-Mills and Mass Gap
Quantum mechanics is an important theory that enables us to understand the behavior of matter and energy on the subatomic particle level. Yang and Mills provided a framework to describe these particles using mathematical structures. The theory has been verified by numerous experiments but its foundation is still unclear.
According to the theory, the quantum particles have a positive mass which is called mass gap. The particles cannot be of zero masses even when they are analogous to massless photons. The mass gap is difficult to explain since nuclear forces are extremely strong and short range as compared to electromagnetism and gravity. The property was already discovered by physicists with computer simulations. The problem is about establishing a theory to explain the mass gap.
Prime numbers have immense importance in mathematics and there is a lot of interest in knowing that how these numbers are distributed on the number line. In the 19th century, a mathematician discovered the Prime Theorem that gives an idea of the average distance between the numbers. However, it is still unknown that how close the true distribution is. The Riemann Hypothesis suggests that the frequency of the prime numbers is related to the behavior of Riemann Zeta function.
The hypothesis states that an input value that makes the result zero in the function will fall on the same line. This has been checked for the first 10 trillion solutions. But it still needs a solid proof and therefore it also lies in the Millennium problems’ list.
P vs NP Problem
P is for easy to find, known as polynomial time and NP is for easy to check, known as nondeterministic polynomial time. It is an unsolved problem in theoretical computer science. The problem basically asks that if it is easy to check that the solution to a problem is correct, is it also easy to solve the problem? These are the problems that are solved and checked by a computer.
If you have a list of all the possible factors, they can be multiplied together and you can check if the original number is back or not. But finding factors of a very large number is not very easy. There is no way yet to prove if the proof of this problem exists or not. This problem was formulated by Stephen Cook and Leonid Levin in 1971.
Navier-Stokes equation governs the fluid dynamics at most. It helps in understanding the changes in the fluid flow during internal or external forces like pressure, velocity, and gravity. The equation has been used to model weather, ocean currents, air flow around a plane wing and also explains how stars move in a galaxy. Most mathematical tools do not prove useful to accurately predict the flow behavior. Since fluids behave differently in different cases. It might also be possible that N-S equation cannot be solved for all cases. The Millennium problem is to solve the equations for all cases or give an example of the case where it cannot be solved.
This is the hardest problem to explain. It asks if the complex mathematical shapes can be built from the simple ones or not. It is like building objects from LEGOs. The basic idea is to ask that to what extent the shape of a given object is changed by combining other simple geometric building blocks of larger dimensions.
The technique became popular and was generalized in many ways. But, the generalization was ignoring the geometric origins and it was important to add objects with no geometric interpretation. The Hodge Conjecture says that the pieces are called Hodge cycles and are a combination of geometric pieces called algebraic cycles.
Birch and Swinnerton-Dyer Conjecture
This describes the rational solutions to define elliptic curve and is recognized as the most challenging unsolved problem. The conjecture is that the elliptic curve has many rational solutions. So solving the equation will come down to a number to tell whether there are finitely or infinitely solutions. The solution is related to the behavior of an associated Zeta function. It is supported by experimental pieces of evidence but a solid proof is still lacking.