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56-578: The Clay Mathematics Institute officially designated the title Millennium Problem for the seven unsolved mathematical problems, the Birch and Swinnerton-Dyer conjecture , Hodge conjecture , Navier–Stokes existence and smoothness , P versus NP problem , Riemann hypothesis , Yang–Mills existence and mass gap , and the Poincaré conjecture at the Millennium Meeting held on May 24, 2000. Thus, on

112-598: A group , called the homeomorphism group of X , often denoted Homeo ( X ) . {\textstyle {\text{Homeo}}(X).} This group can be given a topology, such as the compact-open topology , which under certain assumptions makes it a topological group . In some contexts, there are homeomorphic objects that cannot be continuously deformed from one to the other. Homotopy and isotopy are equivalence relations that have been introduced for dealing with such situations. Similarly, as usual in category theory, given two spaces that are homeomorphic,

168-417: A certain amount of practice to apply correctly—it may not be obvious from the description above that deforming a line segment to a point is impermissible, for instance. It is thus important to realize that it is the formal definition given above that counts. In this case, for example, the line segment possesses infinitely many points, and therefore cannot be put into a bijection with a set containing only

224-412: A continuous inverse function . Homeomorphisms are the isomorphisms in the category of topological spaces —that is, they are the mappings that preserve all the topological properties of a given space. Two spaces with a homeomorphism between them are called homeomorphic , and from a topological viewpoint they are the same. Very roughly speaking, a topological space is a geometric object, and

280-434: A curve E with rank r obeys an asymptotic law where C is a constant. Initially, this was based on somewhat tenuous trends in graphical plots; this induced a measure of skepticism in J. W. S. Cassels (Birch's Ph.D. advisor). Over time the numerical evidence stacked up. This in turn led them to make a general conjecture about the behavior of a curve's L-function L ( E ,  s ) at s = 1, namely that it would have

336-495: A finite number of points, including a single point. This characterization of a homeomorphism often leads to a confusion with the concept of homotopy , which is actually defined as a continuous deformation, but from one function to another, rather than one space to another. In the case of a homeomorphism, envisioning a continuous deformation is a mental tool for keeping track of which points on space X correspond to which points on Y —one just follows them as X deforms. In

392-576: A finite number of possibilities to check. However, for large primes it is computationally intensive. In the early 1960s Peter Swinnerton-Dyer used the EDSAC-2 computer at the University of Cambridge Computer Laboratory to calculate the number of points modulo p (denoted by N p ) for a large number of primes p on elliptic curves whose rank was known. From these numerical results Birch & Swinnerton-Dyer (1965) conjectured that N p for

448-470: A function exists, X {\displaystyle X} and Y {\displaystyle Y} are homeomorphic . A self-homeomorphism is a homeomorphism from a topological space onto itself. Being "homeomorphic" is an equivalence relation on topological spaces. Its equivalence classes are called homeomorphism classes . The third requirement, that f − 1 {\textstyle f^{-1}} be continuous ,

504-672: A given solution quickly (that is, in polynomial time ), an algorithm can also find that solution quickly. Since the former describes the class of problems termed NP, while the latter describes P, the question is equivalent to asking whether all problems in NP are also in P. This is generally considered one of the most important open questions in mathematics and theoretical computer science as it has far-reaching consequences to other problems in mathematics , to biology , philosophy and to cryptography (see P versus NP problem proof consequences ). A common example of an NP problem not known to be in P

560-417: A homeomorphism results from a continuous deformation of the object into a new shape. Thus, a square and a circle are homeomorphic to each other, but a sphere and a torus are not. However, this description can be misleading. Some continuous deformations do not result into homeomorphisms, such as the deformation of a line into a point. Some homeomorphisms do not result from continuous deformations, such as

616-461: A more general type of equation, and in that case it was proven that there is no algorithmic way to decide whether a given equation even has any solutions. The official statement of the problem was given by Andrew Wiles . The Hodge conjecture is that for projective algebraic varieties , Hodge cycles are rational linear combinations of algebraic cycles . We call this the group of Hodge classes of degree 2 k on X . The modern statement of

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672-453: A rank greater than 1. There is extensive numerical evidence for the truth of the conjecture. Much like the Riemann hypothesis , this conjecture has multiple consequences, including the following two: There is a version of this conjecture for general abelian varieties over number fields. A version for abelian varieties over Q {\displaystyle \mathbb {Q} } is

728-513: A zero of order r at this point. This was a far-sighted conjecture for the time, given that the analytic continuation of L ( E ,  s ) was only established for curves with complex multiplication, which were also the main source of numerical examples. (NB that the reciprocal of the L-function is from some points of view a more natural object of study; on occasion, this means that one should consider poles rather than zeroes.) The conjecture

784-689: Is bijective and continuous, but not a homeomorphism ( S 1 {\textstyle S^{1}} is compact but [ 0 , 2 π ) {\textstyle [0,2\pi )} is not). The function f − 1 {\textstyle f^{-1}} is not continuous at the point ( 1 , 0 ) , {\textstyle (1,0),} because although f − 1 {\textstyle f^{-1}} maps ( 1 , 0 ) {\textstyle (1,0)} to 0 , {\textstyle 0,} any neighbourhood of this point also includes points that

840-503: Is called the Navier–Stokes existence and smoothness problem. The problem, restricted to the case of an incompressible flow , is to prove either that smooth, globally defined solutions exist that meet certain conditions, or that they do not always exist and the equations break down. The official statement of the problem was given by Charles Fefferman . The question is whether or not, for all problems for which an algorithm can verify

896-523: Is essential. Consider for instance the function f : [ 0 , 2 π ) → S 1 {\textstyle f:[0,2\pi )\to S^{1}} (the unit circle in ⁠ R 2 {\displaystyle \mathbb {R} ^{2}} ⁠ ) defined by f ( φ ) = ( cos ⁡ φ , sin ⁡ φ ) . {\textstyle f(\varphi )=(\cos \varphi ,\sin \varphi ).} This function

952-460: Is the Boolean satisfiability problem . Most mathematicians and computer scientists expect that P ≠ NP; however, it remains unproven. The official statement of the problem was given by Stephen Cook . The Riemann zeta function ζ(s) is a function whose arguments may be any complex number other than 1, and whose values are also complex. Its analytical continuation has zeros at

1008-485: Is the regulator of E which is defined via the canonical heights of a basis of rational points, c p {\displaystyle c_{p}} is the Tamagawa number of E at a prime p dividing the conductor N of E . It can be found by Tate's algorithm . At the time of the inception of the conjecture little was known, not even the well-definedness of the left side (referred to as analytic) or

1064-512: Is the current grounding for the majority of theoretical applications of thought to the reality and potential realities of elementary particle physics . The theory is a generalization of the Maxwell theory of electromagnetism where the chromo -electromagnetic field itself carries charge. 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,

1120-480: Is the order of the torsion group , # S h a ( E ) = {\displaystyle \#\mathrm {Sha} (E)=} #Ш(E) is the order of the Tate–Shafarevich group , Ω E {\displaystyle \Omega _{E}} is the real period of E multiplied by the number of connected components of E , R E {\displaystyle R_{E}}

1176-528: The Poincaré conjecture in the 1990s, released his proof in 2002 and 2003. His refusal of the Clay Institute's monetary prize in 2010 was widely covered in the media. The other six Millennium Prize Problems remain unsolved, despite a large number of unsatisfactory proofs by both amateur and professional mathematicians. Andrew Wiles , as part of the Clay Institute's scientific advisory board, hoped that

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1232-484: The dual abelian variety A ^ {\displaystyle {\hat {A}}} . Elliptic curves as 1-dimensional abelian varieties are their own duals, i.e. E ^ = E {\displaystyle {\hat {E}}=E} , which simplifies the statement of the BSD conjecture. The regulator R A {\displaystyle R_{A}} needs to be understood for

1288-476: 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. More specifically, the Millennium Prize version of the conjecture is that, if the elliptic curve E has rank r , then the L -function L ( E , s ) associated with it vanishes to order r at s = 1 . Hilbert's tenth problem dealt with

1344-500: The "worst manifestations of present-day mass culture", and thought that there are more meaningful ways to invest in public appreciation of mathematics. He viewed the superficial media treatments of Perelman and his work, with disproportionate attention being placed on the prize value itself, as unsurprising. By contrast, Vershik praised the Clay Institute's direct funding of research conferences and young researchers. Vershik's comments were later echoed by Fields medalist Shing-Tung Yau , who

1400-543: The Hodge conjecture is: The official statement of the problem was given by Pierre Deligne . The Navier–Stokes equations describe the motion of fluids , and are one of the pillars of fluid mechanics . However, theoretical understanding of their solutions is incomplete, despite its importance in science and engineering. For the three-dimensional system of equations, and given some initial conditions , mathematicians have not yet proven that smooth solutions always exist. This

1456-469: The Poincaré conjecture, the precise formulation of which states: Any three-dimensional topological manifold which is closed and simply-connected must be homeomorphic to the 3-sphere . Although the conjecture is usually stated in this form, it is equivalent (as was discovered in the 1950s) to pose it in the context of smooth manifolds and diffeomorphisms . A proof of this conjecture, together with

1512-485: 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 problem has been well-known ever since it was originally posed by Bernhard Riemann in 1860. The Clay Institute's exposition of

1568-548: The behaviour of the Hasse–Weil L -function L ( E ,  s ) of E at s  = 1. More specifically, it is conjectured that the rank of the abelian group E ( K ) of points of E is the order of the zero of L ( E ,  s ) at s = 1. The first non-zero coefficient in the Taylor expansion of L ( E ,  s ) at s = 1 is given by more refined arithmetic data attached to E over K ( Wiles 2006 ). The conjecture

1624-419: The case of homotopy, the continuous deformation from one map to the other is of the essence, and it is also less restrictive, since none of the maps involved need to be one-to-one or onto. Homotopy does lead to a relation on spaces: homotopy equivalence . There is a name for the kind of deformation involved in visualizing a homeomorphism. It is (except when cutting and regluing are required) an isotopy between

1680-566: The choice of US$ 1 million prize money would popularize, among general audiences, both the selected problems as well as the "excitement of mathematical endeavor". Another board member, Fields medalist Alain Connes , hoped that the publicity around the unsolved problems would help to combat the "wrong idea" among the public that mathematics would be "overtaken by computers". Some mathematicians have been more critical. Anatoly Vershik characterized their monetary prize as "show business" representing

1736-540: The curve is greater than 0, then the curve has an infinite number of rational points. Although Mordell's theorem shows that the rank of an elliptic curve is always finite, it does not give an effective method for calculating the rank of every curve. The rank of certain elliptic curves can be calculated using numerical methods but (in the current state of knowledge) it is unknown if these methods handle all curves. An L -function L ( E ,  s ) can be defined for an elliptic curve E by constructing an Euler product from

Millennium Prize Problems - Misplaced Pages Continue

1792-592: The field of Riemannian geometry . For his contributions to the theory of Ricci flow, Perelman was awarded the Fields Medal in 2006. However, he declined to accept the prize. For his proof of the Poincaré conjecture, Perelman was awarded the Millennium Prize on March 18, 2010. However, he declined the award and the associated prize money, stating that Hamilton's contribution was no less than his own. The Birch and Swinnerton-Dyer conjecture deals with certain types of equations: those defining elliptic curves over

1848-439: The following: All of the terms have the same meaning as for elliptic curves, except that the square of the order of the torsion needs to be replaced by the product # A ( Q ) tors ⋅ # A ^ ( Q ) tors {\displaystyle \#A(\mathbb {Q} )_{\text{tors}}\cdot \#{\hat {A}}(\mathbb {Q} )_{\text{tors}}} involving

1904-434: The function maps close to 2 π , {\textstyle 2\pi ,} but the points it maps to numbers in between lie outside the neighbourhood. Homeomorphisms are the isomorphisms in the category of topological spaces . As such, the composition of two homeomorphisms is again a homeomorphism, and the set of all self-homeomorphisms X → X {\textstyle X\to X} forms

1960-418: The homeomorphism between a trefoil knot and a circle. Homotopy and isotopy are precise definitions for the informal concept of continuous deformation . A function f : X → Y {\displaystyle f:X\to Y} between two topological spaces is a homeomorphism if it has the following properties: A homeomorphism is sometimes called a bicontinuous function. If such

2016-468: The left side is now known to be well-defined and the finiteness of Ш(E) is known when additionally the analytic rank is at most 1, i.e., if L ( E , s ) {\displaystyle L(E,s)} vanishes at most to order 1 at s = 1 {\displaystyle s=1} . Both parts remain open. The Birch and Swinnerton-Dyer conjecture has been proved only in special cases: There are currently no proofs involving curves with

2072-423: The more powerful geometrization conjecture , was given by Grigori Perelman in 2002 and 2003. Perelman's solution completed Richard Hamilton 's program for the solution of the geometrization conjecture, which he had developed over the course of the preceding twenty years. Hamilton and Perelman's work revolved around Hamilton's Ricci flow , which is a complicated system of partial differential equations defined in

2128-418: The most challenging mathematical problems. It is named after mathematicians Bryan John Birch and Peter Swinnerton-Dyer , who developed the conjecture during the first half of the 1960s with the help of machine computation. Only special cases of the conjecture have been proven. The modern formulation of the conjecture relates to arithmetic data associated with an elliptic curve E over a number field K to

2184-414: The negative even integers; that is, ζ(s) = 0 when s is one of −2, −4, −6, .... These are called its trivial zeros. However, the negative even integers are not the only values for which the zeta function is zero. The other ones are called nontrivial zeros. The Riemann hypothesis is concerned with the locations of these nontrivial zeros, and states that: The Riemann hypothesis is that all nontrivial zeros of

2240-538: The number of points on the curve modulo each prime p . This L -function is analogous to the Riemann zeta function and the Dirichlet L-series that is defined for a binary quadratic form . It is a special case of a Hasse–Weil L-function . The natural definition of L ( E ,  s ) only converges for values of s in the complex plane with Re( s ) > 3/2. Helmut Hasse conjectured that L ( E ,  s ) could be extended by analytic continuation to

2296-410: The number of rational points on a curve is infinite then some point in a finite basis must have infinite order . The number of independent basis points with infinite order is called the rank of the curve, and is an important invariant property of an elliptic curve. If the rank of an elliptic curve is 0, then the curve has only a finite number of rational points. On the other hand, if the rank of

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2352-604: The official website of the Clay Mathematics Institute, these seven problems are officially called the Millennium Problems . To date, the only Millennium Prize problem to have been solved is the Poincaré conjecture. The Clay Institute awarded the monetary prize to Russian mathematician Grigori Perelman in 2010. However, he declined the award as it was not also offered to Richard S. Hamilton , upon whose work Perelman built. The Clay Institute

2408-610: The pairing between a basis for the free parts of A ( Q ) {\displaystyle A(\mathbb {Q} )} and A ^ ( Q ) {\displaystyle {\hat {A}}(\mathbb {Q} )} relative to the Poincare bundle on the product A × A ^ {\displaystyle A\times {\hat {A}}} . The rank-one Birch-Swinnerton-Dyer conjecture for modular elliptic curves and modular abelian varieties of GL(2) -type over totally real number fields

2464-429: 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

2520-462: The problem was given by Arthur Jaffe and Edward Witten . Birch and Swinnerton-Dyer conjecture In mathematics , the Birch and Swinnerton-Dyer conjecture (often called the Birch–Swinnerton-Dyer conjecture ) describes the set of rational solutions to equations defining an elliptic curve . It is an open problem in the field of number theory and is widely recognized as one of

2576-470: The problem was given by Enrico Bombieri . In quantum field theory , the mass gap is the difference in energy between the vacuum and the next lowest energy state . The energy of the vacuum is zero by definition, and assuming that all energy states can be thought of as particles in plane-waves, the mass gap is the mass of the lightest particle. For a given real field ϕ ( x ) {\displaystyle \phi (x)} , we can say that

2632-510: The problems selected by the Clay Institute were already renowned among professional mathematicians, with many actively working towards their resolution. The seven problems were officially announced by John Tate and Michael Atiyah during a ceremony held on May 24, 2000 (at the amphithéâtre Marguerite de Navarre ) in the Collège de France in Paris . Grigori Perelman , who had begun work on

2688-467: The right side (referred to as algebraic) of this equation. John Tate expressed this in 1974 in a famous quote. This remarkable conjecture relates the behavior of a function L {\displaystyle L} at a point where it is not at present known to be defined to the order of a group Ш which is not known to be finite! By the modularity theorem proved in 2001 for elliptic curves over Q {\displaystyle \mathbb {Q} }

2744-625: The space of homeomorphisms between them, Homeo ( X , Y ) , {\textstyle {\text{Homeo}}(X,Y),} is a torsor for the homeomorphism groups Homeo ( X ) {\textstyle {\text{Homeo}}(X)} and Homeo ( Y ) , {\textstyle {\text{Homeo}}(Y),} and, given a specific homeomorphism between X {\displaystyle X} and Y , {\displaystyle Y,} all three sets are identified. The intuitive criterion of stretching, bending, cutting and gluing back together takes

2800-458: The theory has a mass gap if the two-point function has the property with Δ 0 > 0 {\displaystyle \Delta _{0}>0} being the lowest energy value in the spectrum of the Hamiltonian and thus the mass gap. This quantity, easy to generalize to other fields, is what is generally measured in lattice computations. Quantum Yang–Mills theory

2856-447: The whole complex plane. This conjecture was first proved by Deuring (1941) for elliptic curves with complex multiplication . It was subsequently shown to be true for all elliptic curves over Q , as a consequence of the modularity theorem in 2001. Finding rational points on a general elliptic curve is a difficult problem. Finding the points on an elliptic curve modulo a given prime p is conceptually straightforward, as there are only

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2912-470: Was additionally critical of the idea of a foundation taking actions to "appropriate" fundamental mathematical questions and "attach its name to them". In the field of geometric topology , a two-dimensional sphere is characterized by the fact that it is the only closed and simply-connected two-dimensional surface. In 1904, Henri Poincaré posed the question of whether an analogous statement holds true for three-dimensional shapes. This came to be known as

2968-543: Was chosen as one of the seven Millennium Prize Problems listed by the Clay Mathematics Institute , which has offered a $ 1,000,000 (£771,200) prize for the first correct proof. Mordell (1922) proved Mordell's theorem : the group of rational points on an elliptic curve has a finite basis . This means that for any elliptic curve there is a finite subset of the rational points on the curve, from which all further rational points may be generated. If

3024-476: Was inspired by a set of twenty-three problems organized by the mathematician David Hilbert in 1900 which were highly influential in driving the progress of mathematics in the twentieth century. The seven selected problems span a number of mathematical fields, namely algebraic geometry , arithmetic geometry , geometric topology , mathematical physics , number theory , partial differential equations , and theoretical computer science . Unlike Hilbert's problems,

3080-484: Was proved by Shou-Wu Zhang in 2001. Another generalization is given by the Bloch-Kato conjecture . Homeomorphic In mathematics and more specifically in topology , a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré ), also called topological isomorphism , or bicontinuous function , is a bijective and continuous function between topological spaces that has

3136-410: Was subsequently extended to include the prediction of the precise leading Taylor coefficient of the L -function at s  = 1. It is conjecturally given by where the quantities on the right-hand side are invariants of the curve, studied by Cassels, Tate , Shafarevich and others ( Wiles 2006 ): # E t o r {\displaystyle \#E_{\mathrm {tor} }}

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