History UsefulNotes / TheMillenniumPrizeProblems

This problem revolves around the [[https://en.wikipedia.org/wiki/Navier-Stokes_equations 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 3-dimensional case remains unproven. The Millennium Problem is, then, to either come up with such a solution or find a counterexample.

The Riemann hypothesis is tied to the [[https://en.wikipedia.org/wiki/Riemann_zeta_function Riemann zeta function]], a function defined as the infinite series 1/n[[superscript:s]], where n starts at 1 and goes to infinity through the positive integers[[note]]so 1/1[[superscript:s]]+1/2[[superscript:s]]+1/3[[superscript:s]]...[[/note]] and s = a + bi, where a and b are real numbers and i is defined as the square root of -1[[note]]yes, this actually works in a mathematical sense, counterintuitive as it might sound[[/note]], 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 it produced zeroes when s was a negative even integer (-2, -4, -6...); these values became defined as the trivial zeroes[[note]]"trivial" in the sense that it's relatively easy to see how the result is produced[[/note]] of the function. He also determined that the function's nontrivial zeroes would lie in the area between 0 and 1 on the complex plane[[note]]a 2D plane where the x axis is the real numbers and the y axis is positive and negative multiples of i[[/note]], 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.

* LetsPlay/{{Critikal}} makes a passing reference to the Hodge conjecture during his commentary on ''VideoGame/AmongTheSleep''.

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 it 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.

[[folder:Anime and Manga]]
* ''Manga/PsychicSquad'': In one chapter, Minato uses the Riemann hypothesis as PsychicStatic, concentrating on the complicated mathematical formula as a psychic defense against a mind-reading middle schooler.

[[folder:Tabletop Games]]
* ''TabletopGame/PerplexCity'': One of the cards from the first season was "Riemann", which required a solution for the Riemann hypothesis to win the point.
[[/folder]]

The Riemann hypothesis is tied to the [[https://en.wikipedia.org/wiki/Riemann_zeta_function Riemann zeta function]], a function defined as the infinite series 1/n[[superscript:s]], where n starts at 1 and goes to infinity through the positive integers[[note]]so 1/1[[superscript:s]]+1/2[[superscript:s]]+1/3[[superscript:s]]...[[/note]] and s = a + bi, where a is a real number and b is a multiple of the imaginary number i, defined as the square root of -1[[note]]yes, this actually works in a mathematical sense, counterintuitive as it might sound[[/note]]. 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 it produced zeroes when s was a negative even integer (-2, -4, -6...); these values became defined as the trivial zeroes[[note]]"trivial" in the sense that it's relatively easy to see how the result is produced[[/note]] of the function. He also determined that the function's nontrivial zeroes would lie in the area between 0 and 1 on the complex plane[[note]]a 2D plane where the x axis is the real numbers and the y axis is positive and negative multiples of i[[/note]], and hypothesized that these nontrivial zeroes would 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.

The Riemann hypothesis is tied to the [[https://en.wikipedia.org/wiki/Riemann_zeta_function Riemann zeta function]], a function defined as the infinite series 1/n[[superscript:s]], where n starts at 1 and goes to infinity through the positive integers and s = a + bi, where a is a real number and b is a multiple of the imaginary number i, defined as the square root of -1[[note]]yes, this actually works in a mathematical sense, counterintuitive as it might sound[[/note]]. 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 it produced zeroes when s was a negative even integer (-2, -4, -6...); these values became defined as the trivial zeroes[[note]]"trivial" in the sense that it's relatively easy to see how the result is produced[[/note]] of the function. He also determined that the function's nontrivial zeroes would lie in the area between 0 and 1 on the complex plane[[note]]a 2D plane where the x axis is the real numbers and the y axis is positive and negative multiples of i[[/note]], and hypothesized that these nontrivial zeroes would 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.

[[folder:Film]]

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 anyone can take a lump of clay and mold it into a sphere with their hands, but actually proving it 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.

* ''Manga/PsychicSquad'': In one chapter, Minato uses the Riemann hypothesis as PsychicStatic, concentrating on the complicated mathematical formula as a psychic defense against a mind reading middle schooler.