The Clay Mathematics Institute is a non-profit foundation in Denver, CO containing a wide variety of math-based programs; its stated goal is to further the increase and spread of mathematical knowledge. In 2000, the Institute selected seven as-of-then unsolved math problems and designated them as the Millennium Prize Problems
. Each Problem carries a prize of one million dollars for the mathematician(s) who prove or disprove it. All of these problems are exceedingly difficult and much time and effort had already been given to them prior to their selection by the Institute, but the cash prize provides an added incentive to find their solutions, if they even exist.
To date, only one of the Millennium Prize Problems, the Poincaré conjecture, has been solved.
NOTE: All of these deal with complex, PhD-level maths, and some are very abstract. To keep people's heads from exploding, they will be explained in layman's terms as much as is possible.
Solved
The Poincaré conjecture deals with a specific application of topology and states that if you have a shape in 4-dimensional space that doesn't have any holes in it and isn't separated into multiple pieces, then that shape is analogous to a sphere. This might seem like a no-brainer, as (at least in three dimensions) anyone can take a lump of clay and mold it into a sphere with their hands, but actually proving the 4-dimension case mathematically was a major challenge. In the early '00s, the Russian mathematician Grigori Perelman proved the conjecture in a set of three papers. Building on prior work by Richard Hamilton, who had tried to use a technique called Ricci flow to solve the conjecture, Perelman was able to fix the gaps in Hamilton's work and thus prove that the conjecture was true. Interestingly, Perelman refused both the $1M prize and other accolades, believing that to honor him for the achievement but not also honor Hamilton was unfair.
Unsolved
The Birch and Swinnerton-Dyer conjecture
The Birch and Swinnerton-Dyer conjecture deals with equations that define elliptic curves over the rational numbersnote . Specifically, it states that an elliptic curve will either have an infinite or finite number of solutions depending on if a zeta function connected to the curve is or isn't equal to 0, respectively. Several special cases of the conjecture have been proved, but the general case remains unproved.
The Hodge conjecture is the most abstract of the Millennium Problems. It deals with a connection between topology and algebra, and states, in essence, that a type of space called a projective algebraic variety, which usually has a very complicated shape, can be broken down into pieces called algebraic cycles.
Navier-Stokes existence and smoothness
This problem revolves around the Navier-Stokes equations, a set of equations that are used to describe the motion of fluids. First developed in the mid-1800s, they're an invaluable tool in many different industries and sciences. The unresolved issue is that it hasn't been proven that, in a 3-dimensional space, you can always get a solution that is infinitely differentiable, or smooth. This has been proven in 2-dimensional space, and certain applications and variants of the equations have been proven, but the general 3-dimensional case remains unproven. The Millennium Problem is, then, to either come up with such a proof or find a counterexample.
Let P be the set of all problems for which an algorithm can quickly find a solution, and let NP be the set of all problems for which an algorithm can quickly check whether or not a given solution is correct. The question then becomes, if a problem exists in NP, does it also exist in P? To put it another way, if you can quickly check a problem's solution for correctness, does that then mean that the problem itself can be quickly solved?
In this case, "quickly" means in polynomial time i.e. a time function that is expressed as a polynomial (say, 2x - 1) as opposed to something like an exponential (say, 2x - 1), which increases much more rapidly. The designations for the two sets of problems rise from this notion; "P" means "can be solved in polynomial time" and "NP" means "can be checked in polynomial time". The general thought among mathematicians considering this problem is that P ≠ NP; i.e. there are problems which are impossible to both quickly check and quickly solve (the traveling salesman problem being generally considered to be such a problem), but no conclusive proof one way or the other has been found.
The Riemann hypothesis is tied to the Riemann zeta function, a function defined as the infinite series 1/ns, where n starts at 1 and goes to infinity through the positive integersnote and s = a + bi, where a and b are real numbers and i is defined as the square root of -1note , making it imaginary. The German mathematician Bernard Riemann developed this function while studying ways to determine the distribution of prime numbers. As he worked with the function, he realized that when it was subjected to analytic continuation, it produced zeroes when s was a negative even integer (-2, -4, -6...); these values became defined as the trivial zeroesnote of the function. He also determined that the function's nontrivial zeroes would lie in the area between 0 and 1 on the complex planenote , and hypothesized that these nontrivial zeroes would only occur if, in the formula for s, a = 1/2. To date, astronomical numbers of nontrivial zeroes where 1/2 is the real part of s have been verified, and there are several mathematical concepts which are built around an assumption that the Riemann hypothesis is true, but an actual proof hasn't been found.
Yang-Mills existence and mass gap
This problem is almost as abstract as the Hodge conjecture, and is based in quantum mechanics. What it requires is for someone to develop a particular kind of quantum field theorynote called a Yang-Mills theory that has the same strength as current models (defined by three specific papers), with the condition that the mass of the smallest particle defined by this field theory is always > 0.
References in media
- Perplex City: One of the cards from the first season was "Riemann", which required a solution for the Riemann hypothesis to win the point.
- Psychic Squad: In one chapter, Minato uses the Riemann hypothesis as Psychic Static, concentrating on the complicated mathematical formula as a psychic defense against a mind-reading middle schooler.
- Gifted is about a child prodigy and the custody battle for her between her uncle, Frank, and her maternal grandmother, Evelyn. It's revealed that her mother found a solution to the Navier-Stokes problem but stipulated in her will that her work would not be made public until Evelyn's death.
- Traveling Salesman revolves around a team of mathematicians hired by the US Department of Defense to try to solve the P vs. NP problem, and who have to deal with the moral and ethical ramifications of making their results known.
- The Humans: Aliens take over a math professor who proved the Riemann hypothesis (which would be the last step to godhood for mankind) and want to wipe out the fact thoroughly...including the relatives of the professor, to be safe.
- The Mathematicians Shiva is about a mathematician who purportedly solved the Navier-Stokes problem but who died with her solution unrevealed, and the efforts by a group of mathematicians to uncover her work.
- Angel: "Supersymmetry" contains references to many mathematical concepts and theories, including the P vs. NP problem.
- Elementary: In "Solve for X", Holmes and Watson try to solve the murders of two mathematicians who had come very close to cracking the P vs. NP problem, with the eventual reveal that another mathematician who'd been working on the problem had used applications resulting from it to set up the murders.
- NUMB3RS:
- Charlie has a bad tendency to get buried in mathematical work when he's troubled, with the P vs. NP problem being particularly important because it's what he immersed himself in after his mother died.
- In "Prime Suspect", four criminals kidnap a mathematician's five-year-old daughter because they believe he solved the Riemann hypothesis, which can be used as a master key for virtually any internet encryption, and they're using her to get the solution as ransom. When Charlie finds out that his solution will not work, Don comes up with the idea of giving the criminals a fake solution that will open an electronic door that the FBI would set up for them. This allows them to track down their location, allowing the FBI to raid it, arrest the criminals, and save his daughter.
- Leopoldstadt: One of the main characters, Ludwig, is a professor of mathematics at the University of Vienna, and is obsessed with number theory, particularly solving the Riemann hypothesis.Ludwig: If I went to sleep for a hundred years, the first thing I'd ask when I woke up is, "Has Riemann been proved?"
Hermann: Why?
Ludwig: Because if it has, I can state with certainty how may prime numbers exist below a given number however high; and if it hasn't, I can't. Not without certainty.
Hermann: That is a very annoying answer.
- The Sexy Brutale: The Riemann hypothesis is referenced in a rant by the ghost in the library.The DRIVEL in this library! I ask you! I mean, that shelf, there? Handwritten, unpublished notes from some tiresome chap..."Da Vinci", I think... That whole section over here? Some boring TIT banging on about a "Riemann Hypothesis" they've cracked! Yawn! That bookshelf at the back? Pull the Balzac out and the shelf next to it MOVES! Health hazard, eh?! Where is all the SMUT, sir? Where are the illustrated geographical "studies" of disrobed tribespeople? Where are the "art" books with the bosoms and the gentlemen statues? I find the lack of good, old fashioned obscenity to be completely indecent! ...I think I need a moment to myself.
- Nine Hours, Nine Persons, Nine Doors: A punny reading of a book titled "Riemann Hypothesis" leads to a crude exchange between Clover and Snake.Clover: It says "Riemann Hypothesis". What is there to hypothesize about with a reamin'? Isn't it pretty straightforward?
Snake: Heavens no. There are many factors — Length, girth, lubrication or lack of... It's an exciting and rapidly growing field.
Clover: Whoa!
- Cr1TiKaL makes a passing reference to the Hodge conjecture during his commentary on Among the Sleep.
- Another Gaming Comic: 'Joe Chaos' apparently has the definition of the Riemann hypothesis memorized, but doesn't understand it.
- Irregular Webcomic!: In strip #1960, Adam and Jamie test the omniscience granted by the waters of the Mnemosyne by asking about whether or not the Riemann hypothesis has been proved.
- American Dad!: In "National Treasure 4: Baby Franny: She's Doing Well: The Hole Story", Steve convinces Francine to prove her worth by solving one of the Millennium Prize Problems in mathematics. After an intense mathematics montage, Francine goes up to a professor at a college lecture and announces that she solved the Yang-Mills existence and mass gap problem. Her solution? The number 6. The professor informs her that the answer wouldn't be a number, but rather an entirely new concept in mathematics, and she frowns and walks away.
- The Simpsons: In "Treehouse of Horror VI", when Homer jumps into the weird space behind the bookcase, "P = NP" is one of the maths equations floating around in the space.
