The millenium prize problems

P versus NP

The question is whether, for all problems for which a computer can verify a given solution quickly (that is, in polynomial time), it can also findthat solution quickly. The former describes the class of problems termed NP, whilst the latter describes P. The question is whether or not all problems in NP are also in P. This is generally considered the most important open question in mathematics and theoretical computer science as it has far-reaching consequences in mathematics, biology, philosophy^{[citation needed]} and cryptography (see P versus NP problem proof consequences).

If the question of whether P=NP were to be answered affirmatively it would trivialise the rest of the Millennium Prize Problems (and indeed all but the unprovable propositions in mathematics) because they would all have direct solutions easily solvable by a formal system.

"If P = NP, then the world would be a profoundly different place than we usually assume it to be. There would be no special value in 'creative leaps,' no fundamental gap between solving a problem and recognizing the solution once it’s found. Everyone who could appreciate a symphony would be Mozart; everyone who could follow a step-by-step argument would be Gauss..."

— Scott Aaronson, MIT

Most mathematicians and computer scientists expect that P≠NP.

The official statement of the problem was given by Stephen Cook.

The Hodge conjecture

The Hodge conjecture is that for projective algebraic varieties, Hodge cycles are rational linear combinations of algebraic cycles.

The official statement of the problem was given by Pierre Deligne.

The Poincaré conjecture (proven)

In topology, a sphere with a two-dimensional surface is essentially characterized by the fact that it is simply connected. It is also true that every two-dimensional surface which is both compact and simply connected is topologically a sphere. The Poincaré conjecture is that this is also true for spheres with three-dimensional surfaces. The question had long been solved for all dimensions above three. Solving it for three is central to the problem of classifying 3-manifolds.

The official statement of the problem was given by John Milnor.

A proof of this conjecture was given by Grigori Perelman in 2003; its review was completed in August 2006, and Perelman was selected to receive the Fields Medal for his solution. Perelman declined that award.^[1] Perelman was officially awarded the Millennium prize on March 18, 2010.^[2] On July 1, 2010, it was reported that Perelman declined the award and associated prize money from the Clay Mathematics Institute.^[3] In rejecting the Millennium Prize, Perelman stated that he believed the decisions by the organized mathematics community to be unjust and that his contribution to solving the Poincaré conjecture was no greater than that of Columbia University mathematician Richard Hamilton (who first suggested a program for the solution).^[4] In June 2011, after it was announced that the Million-dollar Clay Millennium Prizewould be divided between Richard Hamilton and Perelman, it is assumed that Perelman will accept half the prize.^[5]

]The Riemann hypothesis

The Riemann hypothesis is that all nontrivial zeros of the analytical continuation of the Riemann zeta function have a real part of ¹/₂. A proof or disproof of this would have far-reaching implications in number theory, especially for the distribution of prime numbers. This was Hilbert's eighth problem, and is still considered an important open problem a century later.

The official statement of the problem was given by Enrico Bombieri.

Yang–Mills existence and mass gap

In physics, classical Yang–Mills theory is a generalization of the Maxwell theory of electromagnetism where the chromo-electromagnetic field itself carries charges. As a classical field theory it has solutions which travel at the speed of light so that its quantum version should describe massless particles (gluons). However, the postulated phenomenon of color confinement permits only bound states of gluons, forming massive particles. This is the mass gap. Another aspect of confinement is asymptotic freedom which makes it conceivable that quantum Yang-Mills theory exists without restriction to low energy scales. The problem is to establish rigorously the existence of the quantum Yang-Mills theory and a mass gap.

The official statement of the problem was given by Arthur Jaffe and Edward Witten.

Navier–Stokes existence and smoothness

The Navier–Stokes equations describe the motion of fluids. Although they were found in the 19th century, they still are not well understood. The problem is to make progress toward a mathematical theory that will give insight into these equations.

The official statement of the problem was given by Charles Fefferman.

The Birch and Swinnerton-Dyer conjecture

The Birch and Swinnerton-Dyer conjecture deals with a certain type of equation, those defining elliptic curves over the rational numbers. The conjecture is that there is a simple way to tell whether such equations have a finite or infinite number of rational solutions. Hilbert's tenth problem dealt with a more general type of equation, and in that case it was proven that there is no way to decide whether a given equation even has any solutions.

The official statement of the problem was given by Andrew Wiles.

unsolved mysteries in maths

There are many unsolved problems in mathematics. Some prominent outstanding unsolved problems (as well as some which are not necessarily so well known) include 1. The Goldbach conjecture. 2. The Riemann hypothesis. 3. The conjecture that there exists a Hadamard matrix for every positive multiple of 4. 4. The twin prime conjecture (i.e., the conjecture that there are an infinite number of twin primes). 5. Determination of whether NP-problems are actually P-problems. 6. The Collatz problem. 7. Proof that the 196-algorithm does not terminate when applied to the number 196. 8. Proof that 10 is a solitary number. 9. Finding a formula for the probability that two elements chosen at random generate the symmetric group . 10. Solving the happy end problem for arbitrary . 11. Finding an Euler brick whose space diagonal is also an integer. 12. Proving which numbers can be represented as a sum of three or four (positive or negative) cubic numbers. 13. Lehmer's Mahler measure problem and Lehmer's totient problem on the existence of composite numbers such that , where is the totient function. 14. Determining if the Euler-Mascheroni constant is irrational. 15. Deriving an analytic form for the square site percolation threshold. 16. Determining if any odd perfect numbers exist. Other still-unsolved problems Additive number theory Goldbach's conjecture and its weak version The values of g(k) and G(k) in Waring's problem Collatz conjecture (3n + 1 conjecture) Gilbreath's conjecture Erdős conjecture on arithmetic progressions Erdős–Turán conjecture on additive bases Pollock octahedral numbers conjecture Number theory: prime numbers Catalan's Mersenne conjecture Twin prime conjecture Are there infinitely many prime quadruplets? Are there infinitely many Mersenne primes (Lenstra–Pomerance–Wagstaff conjecture); equivalently, infinitely many even perfect numbers? Are there infinitely many Sophie Germain primes? Are there infinitely many regular primes, and if so is their relative density e − 1 / 2? Are there infinitely many Cullen primes? Are there infinitely many palindromic primes in base 10? Are there infinitely many Fibonacci primes? Are there infinitely many Wilson primes? Are there any Wall–Sun–Sun primes? Is every Fermat number 22n + 1 composite for n > 4? Is 78,557 the lowest Sierpinski number? Is 509,203 the lowest Riesel number? Fortune's conjecture (that no Fortunate number is composite) Polignac's conjecture Landau's problems Does every prime number appear in the Euclid–Mullin sequence? General number theory abc conjecture Do any odd perfect numbers exist? Do quasiperfect numbers exist? Do any odd weird numbers exist? Do any Lychrel numbers exist? Is 10 a solitary number? Do any Taxicab(5, 2, n) exist for n>1? Brocard's problem: existence of integers, n,m, such that n!+1=m2 other than n=4,5,7 Distribution and upper bound of mimic numbers Littlewood conjecture Algebraic number theory Are there infinitely many real quadratic number fields with unique factorization? Discrete geometry Solving the Happy Ending problem for arbitrary n Finding matching upper and lower bounds for K-sets and halving lines The Hadwiger conjecture on covering n-dimensional convex bodies with at most 2n smaller copies Ramsey theory The values of the Ramsey numbers, particularly R(5,5) The values of the Van der Waerden numbers General algebra Hilbert's sixteenth problem Hadamard conjecture Existence of perfect cuboids Combinatorics Number of Magic squares  Finding a formula for the probability that two elements chosen at random generate the symmetric group Sn Frankl's union-closed sets conjecture: for any family of sets closed under sums there exists an element (of the underlying space) belonging to half or more of the sets The Lonely runner conjecture: if k + 1 runners with pairwise distinct speeds run round a track of unit length, will every runner be "lonely" (that is, be more than a distance 1 / (k + 1) from each other runner) at some time? Singmaster's conjecture: is there a finite upper bound on the multiplicities of the entries greater than 1 in Pascal's triangle? The 1/3–2/3 conjecture: does every finite partially ordered set contain two elements x and y such that the probability that x appears beforey in a random linear extension is between 1/3 and 2/3? Conway's thrackle conjecture Graph theory Barnette's conjecture that every cubic bipartite three-connected planar graph has a Hamiltonian cycle The Erdős–Gyárfás conjecture on cycles with power-of-two lengths in cubic graphs The Hadwiger conjecture relating coloring to clique minors The Erdős–Faber–Lovász conjecture on coloring unions of cliques The total coloring conjecture The list coloring conjecture The Ringel–Kotzig conjecture on graceful labeling of trees The Hadwiger–Nelson problem on the chromatic number of unit distance graphs Deriving a closed-form expression for the percolation threshold values, especially pc (square site) Tutte's conjectures that every bridgeless graph has a nowhere-zero 5-flow and every bridgeless graph without the Petersen graph as a minor has a nowhere-zero 4-flow The Reconstruction conjecture and New digraph reconstruction conjecture concerning whether or not a graph is recognizable by the vertex deleted subgraphs. The cycle double cover conjecture that every bridgeless graph has a family of cycles that includes each edge twice. Does a Moore graph with girth 5 and degree 57 exist? Analysis the Jacobian conjecture Schanuel's conjecture Lehmer's conjecture Pompeiu problem Is γ (the Euler–Mascheroni constant) irrational? the Khabibullin’s conjecture on integral inequalities Dynamics Fürstenberg conjecture – Is every invariant and ergodic measure for the  action on the circle either Lebesgue or atomic? Margulis conjecture — Measure classification for diagonalizable actions in higher-rank groups Partial differential equations Regularity of solutions of Vlasov–Maxwell equations Regularity of solutions of Euler equations Group theory Is every finitely presented periodic group finite? The inverse Galois problem For which positive integers m, n is the free Burnside group B(m,n) finite? In particular, is B(2, 5) finite? Set theory The problem of finding the ultimate core model, one that contains all large cardinals. If ℵω is a strong limit cardinal, then 2ℵω < ℵω1. The best bound, ℵω4, was obtained by Shelah using his pcf theory. Woodin's Ω-hypothesis. Does the consistency of the existence of a strongly compact cardinal imply the consistent existence of a supercompact cardinal? (Woodin) Does the Generalized Continuum Hypothesis below a strongly compact cardinal imply the Generalized Continuum Hypothesiseverywhere? Does there exist a Jonsson algebra on ℵω? Without assuming the axiom of choice, can a nontrivial elementary embedding V→V exist? Is it consistent that ? Does the Generalized Continuum Hypothesis entail  for every singular cardinal λ?