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The Seven Most Difficult Math Problems of Our Time

Many scientific disciplines use mathematics. Chemists, physicists, statisticians, computer scientists, and other studies all rely on some sort of mathematical reasoning or computations. The usage of math principles allows scientists to gain a deeper understanding of their respective studies as well as preforming calculations for practical use. However, the amount that they are able to do so is often limited because of the many unsolved problems in theoretical math. Seven unsolved problems that have stood out in terms of difficulty and importance are the P vs. NP problem, the Birch Swinnerton-Dyer Conjecture, the Riemann Hypothesis, the Poincare Conjecture, Yang-Mills existence and mass gap, and the Navier existence and smoothness problem. These were all together coined as the “Millennium Prize Problems” by the Clay Mathematics Institute (C.M.I.) of Cambridge, Massachusetts in 2000. The complexity and depth of these conundrums leaves mathematicians all around the world in confusion.

The P vs. NP question is one which has roots primarily in the field of computer science, but has also left many questions in mathematics as well. The debate has been between whether problems that can be easily verified by a computer can be solved with similar ease, or if the opposite holds true. P represents polynomial time, defined as the time it takes for a computer program to successfully execute a given algorithm. NP, on the other hand, represents non-deterministic polynomial time. The term “non-deterministic” is applied to NP because the execution time is not explicitly defined, or, in other terms, when the computer attempts potential solution until it finds one that is successful, meaning that the time that it takes for an algorithm to find a correct solution is, therefore, unknown. The controversy lies in the fact that non-deterministic polynomial time may equal polynomial time, and that the reason that P may not equal NP many times is because of an insufficient algorithm, rather than from an impossibility. For example, If one is told to find a solution to a given equation or a given task, they, after attempting a certain trial, can easily see whether the solution that they tried was successful or not. Thus, the solution was easy to verify. To actually find a valid solution, however, appears to be a much harder task at hand than to see whether a solution is right or wrong. If P equals NP, then a solution to the equation can be found as easily as it is to verify a particular solution, whereas if P does not equal NP, then the solution is much more difficult to find than to verify. Mathematicians have been trying to answer this question for a half a century using numerical methods, but so far, there are no conclusive proofs to prove either side right or wrong.

The Riemann Hypothesis another critical problem, which has, currently, been unsolved for about 150 years. It was first theorized Gregorg Riemann, the famous Russian mathematician who made groundbreaking contributions in integral calculus. First published in 1869, the Riemann Hypothesis is revolved around the obscure properties of the Riemann-Zeta Function, a three-dimensional function which has become an important part of physics and number theory. The Riemann-Hypothesis is one of the reasons why the properties are unknown. Riemann hypothesized certain properties on a complex plane of a Riemann-Zeta function. He hypothesized that in a Riemann-Zeta complex plane, that there are non-trivial zeros of the function at the critical value x=1/2. The Riemann Hypothesis, when solved, will open doors to and discoveries to mathematicians in the field of number theory.

The Birch Swinnerton-Dyer conjecture concerns the properties of the Riemann-Zeta function as well. The conjecture states that groups of numbers that have the communicative property of over an elliptic curve have similar properties a Lemma function with respected properties. This thesis had originated from the proof of Mendel’s theorem in 1922 that stated that there are a finite number of rational points in a given subset in an elliptic curve. The amount of subsets is called the rank of a curve. Therefore, a curve that has a rank of zero has finite rational points, and a curve with a rank that is greater than zero has infinite rational points. While Mordell’s theorem is proven, there is currently no equation that can calculate the rank of all elliptic curves. There are only ones that work for some curves and other equations that work for others, such as the L-functions, which define elliptic points with an Euler product. Two mathematicians – Bryan Birch and Peter Swinnerton-Dyer – attempted to make a more generalized equation in the early 1960’s by using the EDSAC computing system. They took elliptic curves to which they knew the rank was and formulated a generalized equation. Currently, though there are proof that the Birch and Swinnerton-Dyer conjecture holds true in certain cases, there is currently no proof that it is true for all elliptic curves.

The Yang-Mils existence and mass gap is another unresolved issue in mathematics and physics. It originated from the roots of the complex and abstract theory of quantum mechanics, which was from the physicist Max Planck in 1900. He conducted his well-known “The Black Body Experiment” in order to explain the properties of light. After he had gotten unexpected results, he concluded that particles in light were not infinite–as was thought in classical physics theory–but only existed in definite amounts, or quanta, of energy. Scientists had also observed sporadic behavior of electrons after the Black Body Experiment, which also lead to the theory of Quantum Mechanics. Electrons, at some moments, act as regular particles with a definite position, while at other times; they have similar properties of waves. Classical theories of physics could not explain the phenomena, which lead to the divergence of scientists, such as Albert Einstein, from classical theory, and to turn their beliefs towards Quantum physics instead.

Quantum Mechanics theories have their own sub-theories as well. One of them is gauge theory, which is what the Yang-Mills theory falls under, which asserts that the geometric structure of a particle is the cause of the unpredictable behavior of electrons. The Yang Mills theory attempts to build a gap between the classical theory – that particles are massless – and quantum theory, that particles, in fact, possess mass. New mathematical techniques will be able to explain the gap between “existence” of particle mass and theories of “mass”. Up to this moment, however, there has not been devised a successful mathematical method to explain such. The solution of this problem will not only demystify the behavior of particles, but it will also explain the workings of electronics and semiconductors.

The mathematical properties of the Navier-Stokes equations are not understood, which poses limits to physicists’ understanding of fluid flow. These equations were an attempt to understand the basic laws of incompressible fluid mechanics, such as waves in water. They are broken down to three categories: conservation of energy, momentum, and mass. The properties and solutions of the equations, however, are not completely understood, as to their relation of occurrence to the physical world. In particular, the Navier-Stokes equation for turbulence in a liquid has been a puzzlement to mathematicians. There exists an equation that represents the three-dimensional flow of liquids, and it is continuous. The more that the properties are understood, the more that mathematicians and physicists will know about the properties of turbulence, and would gain a deeper understanding in the properties of water flow and currents.

The Hodge Conjecture is a mathematical dilemma in algebraic geometry. The conjecture originated from the former Cambridge professor William Vallance Douglas Hall in the 1940’s, and had presented it at the International Conference of Mathematicians of in 1950. The roots of the Hodge Conjecture come from the most basic studies of geometry: topology. Topology concerns the external properties rather than any measurements, as in Euclidean geometry. Mathematicians are able to study the properties of an object by describing its topological features; that is, by their dimensions, number of holes. Hodge stated that for any topological object, in order to characterize it, one could analyze the smaller parts, called the algebraic cycles. Again, mathematicians were and are still unable to prove his conjecture.

The Poincare conjecture was another unsolved mathematical problem in topology. To this day, it remains the only millennium problem that has been solved. The conjecture was made by Henri Poincare, a prominent French mathematician and professor in the mid-1800. Poincare was actually the first one who originated the field of topology when he studied the Konigsberg problem, in which tour guides wanted to find a way so that, when they went on touring guides, they wanted to cross each of the seven bridges only once so the tourists would not get bored. Mathematicians, initially, thought that this was a non-mathematical question, until Poincare suggested that the study of geometry does not have to be limited to measurements, but can also include the characteristics of a geometrical shape. That led to the birth of his book on algebraic geometry, Analysis situs, published in 1895, which would become “the Bible of Topology”, and would inspire other mathematicians to work in the newly developing area as well. 

Topology, though, would not become an emerging field without problems. Poincare's attempts to even out the rough patches, ironically, led him to ask a question that is considered one of the most difficult of all mathematical questions in history. Poincare had developed a system in which to characterize topological objects. One way was by dimensions. Another was by an object's Betti number, which represents the number of times a topological object can be cut before its surface is cut into two different areas. Both of these classifications, however, proved to be unsuccessful in determining topological invariants when Poincare had discovered the homology sphere – which was a three-dimensional object that had the equivalent torsion coefficients and Betti numbers as a regular sphere – but could not be morphed into a regular sphere, thus proving them to actually be topologically different. After his findings, Poincare sought out to find a different classification for topological figures, which would later become groups. Poincare asked whether a homology sphere that is twisted into one point can still be made into a sphere, and thought that the question could be easily answered. However, the conjecture would puzzle mathematicians for almost a century. Some mathematicians have spent all of their lives on solving this problem and have failed. It was only until 2006 that it was proven. Grigori Perelman, a Russian mathematician in Russia, published the proof online in three separate documents. Perelman was offered The Fields Metal, prize money from C.M.I., and was invited to the International Conference of Mathematicians (ICM) in Madrid, Spain. He rejected both of the prizes and the invitation to ICM.

All except one of the Millennium Prize problems are currently left unanswered. For years, mathematicians all around the world – including even the most prominent figures – have unsuccessfully tried to find answered. Some have wasted their entire lives in an attempt to find a solution. The necessity of solutions for these age-old problems is dire. Unraveling these questions allow scientific and mathematical knowledge to reach new heights. CMI will award the solver to any of these problems – along with a valid written proof – a one million dollar cash prize. 

Grigori Perelman, the Solver of the Poincare Conjecture

Grigori Perelman