The Million Dollar Question
After Andrew Wiles, an English mathematician, proved the notorious Fermat’s last theorem in 1993, mathematicians struggled to find new problems to solve. As a result, in May 2000 the Clay Mathematics Institute announced seven sets of mind-boggling questions known as the millennium problems. Proving one of these questions will not only bring fame to the one who solves it but also a prize of a million dollars. While announcing the million-dollar prize, British mathematician Michael Atiyah andAmerican mathematician John Tate said that these problems were “the seven most difficult open problems at the time.”
One of the most well-known millennium problems is the Riemann hypothesis. The hypothesis states that any nontrivial zeros of the Riemann Zeta function lie on the complex plane with a real value of ½. The Riemann Zeta function is the sum of all reciprocals of n with a power of s where n goes from one to infinity and s is a complex number. In simple terms, mathematicians need to prove that a particular function called the Zeta function has its zeros when ½ is entered as an input.
While the Riemann hypothesis may seem arbitrary, the implications of proving it are groundbreaking. Through the Riemann hypothesis, mathematicians can find patterns exhibited by prime numbers. Prime numbers are a set of numbers where its factors include only itself and one. However, these numbers are very difficult to confirm and are seemingly located at random positions. Interestingly, the Riemann hypothesis states that the solutions to the Zeta function are constructed of prime numbers. Also, branches of physics like quantum mechanics are based on the assumption that the Riemann hypothesis is true. Consequently, proving the Riemann hypothesis would help scientists to understand the structure of math and physics.
Another millennium problem is the Birch and Swinnerton-Dyer Conjecture. This conjecture is one of the most challenging problems in number theory. The conjecture describes the set of rational solutions to an elliptic curve.
Elliptic curves are a special kind of relation that can be expressed as y2=ax3-bx+c. The conjecture states that elliptic curve solutions can be modeled by a function known as the L function, which incorporates the Zeta function in it. The significance of proving this conjecture lies in the fact that it gives mathematicians a better understanding of elliptic curves. With the recent developments of quantum computers, hacking computer algorithms became fairly easy. As a result, scientists started to make algorithms like elliptic curve cryptography to ensure safety and productivity. Through the Birch and Swinnerton-Dyer Conjecture, our computer devices and government information can become safer.
On a similar note, P vs. NP is another millennium problem that helps computer scientists to understand algorithms. P vs. NP asks whether algorithms can find a solution to a problem if it can quickly check the solution to a problem. The implications of the P vs. NP problem are very useful in modern days because computers and algorithms are ubiquitous. Complicated studies like protein folding in biology and identification of prime numbers in mathematics are related to the P vs. NP problem. If mathematicians are able to prove this problem, humanity will be able to quickly solve problems that were previously known to take time to solve.
Unlike the Pvs. NP problem, the Navier Stokes Equation is one of the millennium problems that deals with physics. Specifically, the Navier Stokes equation describes the motion of all fluids in the universe.
For example, the equation explains properties of water, human blood, the atmosphere, and other fluids on earth. The equation itself can be intuitively sound. The elements of the equation is actually an extension of Newton’s second law, which states that force is equal to mass times acceleration. However, mathematicians are not able to explain this equation mathematically.
The Yang’s Mill theory is another Millennium problem that deals with physics. The theory asks whether or not mathematics can have theories that agree with quantum physics experiments. By proving this theory, scientists can get a better understanding of subatomic particles. Currently, the Standard Model, which is the mathematical model of all elementary particles in the universe, is based on Yang’s Mill theory. As a result, proving this Millenium problem will give physicists a better grasp of how particles behave mathematically.
One of the abstract millennium problems is the Hodge conjecture. This conjecture states that any topological object can be described using algebraic equations. In Geometry, theorems like the Pythagorean theorem help mathematicians understand and calculate geometric shapes easily. Unlike geometry, however, topology is not yet connected to algebra. As a result, calculating and understanding topological shapes can be difficult without proving the Hodge conjecture. Since Topology is a significant study related to many different subjects, proving it would be extremely beneficial.
Finally, the Poincare conjecture is one of the Millennium problems that has been proven. The conjecture asks whether or not objects in any dimensions with no holes are topologically spheres. In 2003, a Russian mathematician called Grigori Perelman proved this conjecture. He proved that any object without any holes in any dimension is equivalent to a sphere. Interestingly, this proof implies that the universe is actually a sphere that is infinitely expanding.
The Millenium problems were not selected just because of their difficulty. Each problem relates to a grand implication that either helps scientists understand more about the universe or develop human technology. Anyone who proves one of the Millenium problems will be leading humanity to greater success.