Math is quintessential to solve real-world problems, starting from the big bang to black holes. Needless to say, without mathematics, many scientific phenomena would have been left unanswered. However, despite using enough math, there are certain phenomena that mathematicians are yet to decipher.
Millennium Prize Problems
Going into the 21st century, the quest for finding solutions for such unproven theories thickened. Clay Mathematics Institute came up with a solution to encourage the participation of mathematicians. In 2000, the institute declared that they will award $1 Million for anyone who solves the most important math problems. There are 6 problems that require the most attention from the mathematicians. Proving these problems would be detrimental in the world of science.
1. Navier-Stokes Equations
Newton’s laws of motion shaped the world of physics and gave us a deterministic approach on an object’s movement. However, what Newton couldn’t explain was the flow of fluids. Now, the Navier-Stokes equation is in search of it. The essence of the Navier-Stokes equation is to describe how the flow of fluid evolves under various conditions. To put it into perspective, Newton explained how an object’s velocity is influenced by force, through his second law. Similarly, the Navier-Stokes equation will tell us how a fluid flow is influenced by various inherent factors such as.: Pressure and Viscosity. Moreover, solving this equation will also tell us how gravity influences the flow of fluid.
Mathematically, the Navier-stokes equation is a representation of Newton’s law in differential equations. Different equations will help in understanding the change in a particular entity over time. In mathematics, differential equations hold key for various phenomena such as vibrations, the flow of heat etc. Unfortunately, the existing methods to solve differential equations are not enough to solve Navier-Stokes equations. Because the equation itself is very complex. For instance, it is not feasible to track how smoke coming from a factory would change over time. Because it is chaotic and turbulent. However, if you are fortunate to find a solution for this, not only will you earn $1 million but also a respectable position in the world of science.
2. Yang-Mills Existence and Mass Gap
Many physicists are avidly searching for the reason as to why nuclear forces are extremely strong as opposed to gravitational and electromagnetic. However, Yang-Mills theory suggests that “mass gap” is the reason behind it. Yang-Mills theory describes the quantum behaviour of electromagnetism. It is often regarded as a foundation for Quantum mechanics. In addition to that, the theory also explains nuclear forces in terms of mathematical structures that arise in studying geometric symmetries.
Coming back to the mass gap, it is a theory by virtue of which physicists believe that certain subatomic particles, analogous to massless photons, have a positive mass. Even though there are a lot of physical experiments that prove this, there is not a single mathematical formulation that can support it. Anyone with a mathematical explanation behind the mass gap will get the millennium prize.
3. Riemann Hypothesis
The Riemann hypothesis is one of the oldest math problems on this list. The gist of this problem is simple. But to understand its formulation would require graduate-level mathematics courses in complex analysis. Concretely, the essence of this hypothesis is to find the distribution of prime numbers within the integers. Many Greek mathematicians have said that there are infinitely many prime numbers. However, mathematicians were more interested in finding the exact number of primes between two limits. This led them to “prime pi”, which finds the number of primes less than a given number.
“Prime pi” wasn’t that significant. It was reformulated by German mathematician Bernhard Riemann. He coined the term, “Riemann Zeta Function”. Using this, the hypothesis establishes bounds on how far the distribution of primes can stray from the average. There is evidence that this hypothesis is true, but proof for the same is still elusive. In 2018, Professor Michael Atiyah claimed to have found a proof for this, but his work is yet to be reviewed.
4. P versus NP
Not just for mathematics, P vs NP is extremely important in the field of computation. Solving this problem will have a huge impact on modern computing. However, with a price tag of $1 Million, it is obvious that it won’t be easy. In order to give you a clear cut definition as to what this problem pertains to, let us break this down into.: P and NP problems.
P problems are those that are easy to solve in a matter of seconds. For instance, 2+5 or sorting 3, 1, 5, 2 in ascending order. Concretely, basic arithmetic operations, sorting a list, searching through a data table are all P problems. These problems can be solved in polynomial time. Hence, the name “P”
On the other hand, NP problems are different. For such problems, it is easy to check whether the solution is correct, but it is hard to find an efficient way to find the solution. For instance, it is easy to check whether a given set of numbers are a prime factor of a large number, you just need to multiply them together. However, to find a list of possible print factors for an arbitrarily large number is a time-intensive task. Generally, the problems whose solution can be checked quickly are said to be solvable in “nondeterministic polynomial time”. Hence the name, “NP”
Mathematically, any problem in P is in NP. Because, if you are able to solve the question quickly, then it is easy for you to quickly check whether the solution is correct. For instance, 2+5=7 is in P. We can check by seeing if 7-5 yields 2. Concretely, you can check the solution by solving the actual problem in hand. Now, the P versus NP problem is to find whether the reverse is true. That is if you can check whether the solution is correct, is there an efficient way to actually find those solutions.
Mathematicians and Computer scientists believe that the reverse is not true. But they haven’t developed a proof for the same.
5. Hodge Conjecture
Hodge Conjecture has profound importance in algebraic geometry. It helps in understanding hyper-surfaces. But before we get there, it is important to know what algebraic geometry does. Simply put, it ties algebra and geometry. For instance, for y=x, you will get a straight line when you plot the points on the graph. However, things are not that easy in mathematics. There are higher-dimensional analogues of this curve. Analogues that demand multiple equations or even the use of the complex plane.
But mathematicians didn’t give up. The 20th century saw a lot of advancements in algebraic geometry. Hard-to-imagine hypershapes were made more tractable with the help of computational tools. Along with these was Hodge conjecture. It suggests that certain geometric structures having a useful algebraic counterpart can be used to study and classify these hypershapes.
6. Birch and Swinnerton-Dyer Conjecture
Mathematicians are obsessed with something called “Diophantine equations”. These are polynomial equations with whole-number solutions. A perfect example of this is the Pythagorean triplet. However, recently mathematicians found a certain set of elliptical curves that are defined using these diophantine equations. These curves are at utmost importance in number theory and cryptography. The Birch and Swinnerton-Dyer Conjecture will come in handy for finding solutions to these curves. Once solved, there would be an unprecedented growth in the world of cryptography!