Although the Millennium Prize Problems carry a total of 7 million dollars in prize money, very few people know about them. In my opinion, this should be propagated more instead of the latest viral videos on YouTube, or the latest celebrity gossip. This was exactly the thinking of Landon Clay, who founded the Clay Mathematics Institute in 1998. Clay, a Harvard alumnus and successful businessman, is a firm believer and supporter of math and science as beneficial to all mankind, and he has set up this institute as well as the prize money.
I will give a very brief summary of each problem except the Poincaré Conjecture, which has been solved by Grigori Perelman. Keep in mind that these descriptions are not exhaustive as the areas of study of these fields are highly specialized. It cannot be possible to explain in a paragraph what mathematicians have been studying for decades or even centuries, so these descriptions will be very loose.
P vs NP
This is a problem in theoretical computer science. It involves two characters, P and NP. P represents problems that are easily solved by a computer. NP represents problems that are not necessarily easily solved by a computer but whose solutions can easily be checked if they are provided. By default, every P problems is an NP problem because if the computer can solve it, then it can check it. However, there are many examples of NP problems that aren’t known to be P (problems for which a computer can easily check a solution but for which we don’t yet know a fast algorithm for a computer to solve). Currently, many people believe that the evidence points to P not being equal to NP. Solving P vs NP would allow us to solve problems involving trillions of combinations without trying each one, and would greatly propel computing.
When Descartes married algebra and geometry by drawing a graph represented by a function, this revolutionized mathematics and it was never the same again. This enabled us to visualize and solve problems both geometrically or algebraically. The Hodge Conjecture implies a similar relationship exists between topology and algebra. Mathematicians soon found ways to describe more complicated shapes that were hard to imagine and only accessible via complicated equations. These shapes are known as “manifolds”. Depending on their properties, these manifolds have different “homology classes”. One example of manifolds with different homology classes is the sphere and the torus (AKA donut). In the case of the sphere, there is only one homology class: all shapes that are drawn on a sphere are homologically equivalent. In the case of a donut, there are multiple distinct homology classes. The Hodge Conjecture basically says that if you drew a random shape on a manifold, there is a rule you can apply to guarantee it can be described algebraically. Easy to describe in words but tricky to describe mathematically.
The Riemann Hypothesis has profound implications in number theory and tries to tackle perhaps the longest standing question in mathematics: Where are all the prime numbers, and how are they distributed? The hypothesis involves the “trivial” and “non-trivial” zeroes of the complex Zeta Function. A complex function is a function that takes in complex numbers of the form a + bi, and spits out a complex number. Trivial zeroes of the Zeta function occur at negative, even integers (-2,-4,-6,…). The conjecture is that all other zeroes (“non-trivial) have the form ½+bi. Although these zeroes have been computed to the trillion digits and held true, this does not constitute a proof and a general proof is being sought. The distribution of prime numbers is seemingly random and sporadic with no telling when or where the next one will pop up in the number line. This also leads to a deeper philosophical question. How can something as structured and ordered like mathematics have something as chaotic and random as prime numbers as one of its foundations?
Birch and Swinnerton-Dyer Conjecture
A Diophantine equation is a polynomial equation for which mathematicians are searching for integer or rational solutions. The study these equations, is known as arithmetic geometry. A well-known example of a Diophantine equation is the Pythagorean theorem. These equations are named in honor of the Greek mathematician, Diophantus, who studied these types of equations. An elliptic curve is a graphic representation of a Diophantine Equation of the form y2 = x3 + ax + b. This conjecture states that for an elliptic curve, E, the algebraic rank and geometric rank are the same. In other words, you can find the algebraic rank by finding the geometric rank and vice versa. The rank is essentially the number of rational solutions with a 0 rank meaning a finite number of solutions, and a rank greater than or equal to 1 having infinite solutions.
The Navier-Stokes equations of fluid flow are partial differential equations that physicists use to model ocean currents, weather patterns, and other phenomena. These equations, named after Claude-Louis Navier and George Gabriel Stokes, have been stumping mathematicians for about 150 years. The problem lies in that these equations are so complex, that one cannot tell whether these equations will suddenly have a spike or blow up as time goes on. It’s similar to the story of the cat in Dr. Seuss’s “The Cat in the Hat Comes Back”, where the cat makes a stain he cannot clean up. He calls on the help of a smaller version of himself called Little Cat A, who then calls on an even smaller cat called Little Cat B and so on until the microscopic Little Cat Z unleashes a VOOM on the stain and it disappears. The Navier-Stokes problem is asking whether we can predict where the VOOMs are. Fluids can be both viscous liquids and gases and solving these equations will impact areas such as meteorology and fluid dynamics.
This theory is the basis of elementary particle theory, but relies on a very weird concept, the concept of infinitely small numbers to describe the weight of these “massless” particles. This is a contradiction since these particles travel at the speed of light and anything travelling that fast must have infinite mass. New foundations and approaches to physics is required to solve this theory. Solving the mass-gap problem would mean the existence of a rigorous Quantum Field Theory. The techniques used could also be applied to other results in Quantum Field Theory.
P vs NP