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At a appointment in Paris in 1900, the German mathematician David Hilbert presented a account of baffling problems in mathematics. He ultimately put alternating 23 problems that to some admeasurement set the analysis calendar for mathematics in the 20th century. In the 120 years back Hilbert’s talk, some of his problems, about referred to by number, accept been apparent and some are still open, but best important, they accept spurred addition and generalization. The Clay Mathematics Institute’s Millennium Prizes are a 21st-century adaptation of Hilbert’s aboriginal proposal.



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1. CONTINUUM HYPOTHESIS. To mathematicians, all infinities are not the same. The aloft of the counting numbers — 1, 2, 3, … — is abate than the aloft of all the absolute numbers. And there are architecture of still greater infinities aloft the reals. Hilbert’s aboriginal problem, additionally accepted as the continuum hypothesis, is the account that there is no aloft in amid the aloft of the counting numbers and the aloft of the absolute numbers. In 1940, Kurt Gödel showed that the continuum antecedent cannot be accepted application the accepted axioms of mathematics. In 1963, Paul Cohen showed it cannot be disproved, authoritative the continuum antecedent absolute of the axioms of mathematics.



? 2.THE COMPATIBILITY OF THE STANDARD AXIOMS OF ARITHMETIC. Hilbert’s additional botheration was to prove that addition is consistent, that is, that no contradictions appear from the basal assumptions he had put alternating in one of his papers. This botheration has been partially bound in the negative: Kurt Gödel showed with his blemish theorems in 1931 that it is absurd to prove the bendability of a arrangement alleged Peano addition application alone the axioms of Peano arithmetic. Mathematicians agitation whether Gödel’s assignment is a acceptable resolution to the problem.

3. EQUIDECOMPOSABILITY. Any polygon can be cut into a bound cardinal of polygonal pieces and reassembled into the appearance of any added polygon with the aforementioned area. Hilbert’s third botheration — the aboriginal to be bound — is whether the aforementioned holds for three-dimensional polyhedra. Hilbert’s apprentice Max Dehn answered the catechism in the negative, assuming that a cube cannot be cut into a bound cardinal of polyhedral pieces and reassembled into a tetrahedron of the aforementioned volume.



4. THE STRAIGHT LINE AS THE SHORTEST DISTANCE BETWEEN POINTS. Hilbert’s fourth botheration is about what happens back you relax the rules of Euclidean geometry. Specifically, what geometries can abide in which a beeline band is the beeline ambit amid two credibility but in which some axioms of Euclidean geometry are abandoned? Some mathematicians accede the botheration too ambiguous to accept a absolute resolution, but there are solutions for some interpretations of the question.

5. UNDERSTANDING LIE GROUPS. Hilbert’s fifth botheration apropos Lie groups, which are algebraic altar that call connected transformations. Hilbert’s catechism is whether Lie’s aboriginal framework, which assumes that assertive functions are differentiable, works after the acceptance of differentiability. In 1952, Andrew Gleason, Deane Montgomery and Leo Zippin answered the question, assuming that the aforementioned approach arises whether differentiability is affected or not. Some mathematicians accept interpreted the catechism abnormally and appropriately accept altered answers.

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? 6.THE AXIOMATIZATION OF PHYSICS. One of Hilbert’s primary apropos was to accept the foundations of mathematics and, if none existed, to advance accurate foundations by abbreviation a arrangement to its basal truths, or axioms. Hilbert’s sixth botheration is to extend that axiomatization to branches of physics that are awful mathematical. Some advance has been fabricated in agreement some fields of physics on absolute foundations, but because there is no ‘theory of everything’ in physics yet, a accepted axiomatization has not occurred.

7. IRRATIONALITY AND TRANSCENDENCE OF CERTAIN NUMBERS.A cardinal is alleged algebraic if it can be the aught of a polynomial with rational coefficients. For example, 2 is a aught of the polynomial x − 2, and √2 is a aught of the polynomial x2 − 2. Algebraic numbers can be either rational or irrational; abstruse numbers like π are aberrant numbers that are not algebraic. Hilbert’s seventh botheration apropos admiral of algebraic numbers. Accede the announcement ab, area a is an algebraic cardinal added than 0 or 1 and b is an aberrant algebraic number. Charge ab be transcendental? In 1934, Aleksandr Gelfond showed that the acknowledgment is yes.

? 8.PROBLEMS OF PRIME NUMBERS. Hilbert’s eighth botheration includes the acclaimed Riemann hypothesis, forth with some added questions about prime numbers.

? 9.RECIPROCITY LAWS AND ALGEBRAIC NUMBER FIELDS. Hilbert’s ninth botheration is on algebraic cardinal fields, extensions of the rational numbers to include, say, √2 or assertive circuitous numbers. Hilbert asked for the best accepted anatomy of a advantage law in any algebraic cardinal field, that is, the altitude that actuate which polynomials can be apparent aural the cardinal field. Partial solutions by Emil Artin, Teiji Takagi and Helmut Hasse accept pushed the acreage further, although the catechism has not been answered in full. The carefully accompanying 12th problem, which deals with added extensions of the rational numbers, is unresolved.

10. SOLVABILITY OF A DIOPHANTINE EQUATION. Polynomial equations in a bound cardinal of variables with accumulation coefficients are accepted as Diophantine equations. Equations like x2 − y3 = 7 and x2 y2 = z2 are examples. For centuries, mathematicians accept wondered whether assertive Diophantine equations accept accumulation solutions. Hilbert’s 10th botheration asks whether there is an algorithm to actuate whether a accustomed Diophantine blueprint has accumulation solutions or not. In 1970, Yuri Matiyasevich completed a affidavit that no such algorithm exists.

? 11.ARBITRARY QUADRATIC FORMS. Hilbert’s 11th botheration additionally apropos algebraic cardinal fields. A boxlike anatomy is an expression, like x2 2xy y2, with accumulation coefficients in which anniversary appellation has unknowns aloft to a absolute amount of 2. The cardinal 9 can be represented application integers in the aloft boxlike anatomy — set x according to 1 and y according to 2 — but the cardinal 8 cannot be represented by integers in that boxlike form. Some altered boxlike forms can represent the aforementioned sets of accomplished numbers. Hilbert asked for a way to allocate boxlike forms to actuate whether two forms represent the aforementioned set of numbers. Some advance has been made, but the catechism is unresolved.

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? 12.EXTENSION OF KRONEKER’S THEOREM ON ABELIAN FIELDS TO ANY ALGEBRAIC REALM OF RATIONALITY. With his 12th problem, Hilbert approved to generalize a assumption about the anatomy of assertive extensions of the rational numbers to added cardinal fields. It is currently unresolved.

13. SEVENTH-DEGREE POLYNOMIALS. Hilbert’s 13th botheration is about equations of the anatomy x7 ax3 bx2 cx 1 = 0. He asked whether solutions to these functions can be accounting as the agreement of finitely abounding two-variable functions. (Hilbert believed they could not be.) In 1957, Andrey Kolmogorov and Vladimir Arnold accepted that anniversary connected action of n variables — including the case in which n = 7 — can be accounting as a agreement of connected functions of two variables. However, if stricter altitude than bald chain are imposed on the functions, the catechism is still open.

14. FINITENESS OF CERTAIN SYSTEMS OF FUNCTIONS. The action for Hilbert’s 14th botheration came from antecedent assignment he had done assuming that algebraic structures alleged rings arising in a accurate way from beyond structures charge be finitely generated; that is, they could be declared application alone a bound cardinal of architecture blocks. Hilbert asked whether the aforementioned was accurate for a broader chic of rings. In 1958 Masayoshi Nagata bound the catechism by award a counterexample.

? 15.RIGOROUS FOUNDATION OF SCHUBERT’S ENUMERATIVE CALCULUS. Hilbert’s 15th botheration is addition catechism of rigor. He alleged for mathematicians to put Schubert’s enumerative calculus, a annex of mathematics ambidextrous with counting problems in geometry, on a accurate footing. Mathematicians accept appear a continued way on this, admitting the botheration is not absolutely resolved.

? 16.TOPOLOGY OF ALGEBRAIC CURVES AND SURFACES. Hilbert’s 16th botheration is an amplification of brand academy graphing questions. An blueprint of the anatomy ax by = c is a line; an blueprint with boxlike agreement is a cone-shaped area of some anatomy — parabola, ambit or hyperbola. Hilbert approved a added accepted approach of the shapes that higher-degree polynomials could have. So far the catechism is unresolved, alike for polynomials with the almost baby amount of 8.

17. EXPRESSION OF DEFINITE FORMS BY SQUARES. Some polynomials with inputs in the absolute numbers consistently booty non-negative values; an accessible archetype is x2 y2. Hilbert’s 17th botheration asks whether such a polynomial can consistently be accounting as the sum of squares of rational functions (a rational action is the caliber of two polynomials). In 1927, Emil Artin apparent the catechism in the affirmative.

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18. BUILDING UP OF SPACE FROM CONGRUENT POLYHEDRA. Hilbert’s 18th botheration is a accumulating of several questions in Euclidean geometry. First, for anniversary n, does Euclidean amplitude of ambit n accept alone a bound cardinal of fundamentally audible translation-invariant symmetries? In 1910, Ludwig Bieberbach answered this allotment of the catechism in the affirmative. Second, in a tiling of the even by squares, any aboveboard can be mapped to any added square. Such a tiling, in any dimension, is alleged isohedral. This allotment of the botheration apropos the actuality of non-isohedral tilings in three-dimensional space. In 1928, Karl Reinhardt begin such a tiling. (Later, Heinrich Heesch begin a tiling in two-dimensional space; admitting Hilbert did not say why he did not ask the aforementioned catechism about two-dimensional space; abounding bodies accept it is because he did not apprehend such a tiling could abide there.) Finally, what is the densest way to backpack spheres? In 1998, Thomas Hales presented a computer-aided affidavit assuming that the archetypal agreement in aftermath stands is absolutely optimal.

19 and 20. ARE THE SOLUTIONS OF REGULAR PROBLEMS IN THE CALCULUS OF VARIATIONS ALWAYS NECESSARILY ANALYTIC? Do solutions in accepted exist? The calculus of variations is a acreage anxious with optimizing assertive types of functions alleged functionals. In his 19th and 20th problems, Hilbert asked whether assertive classes of problems in the calculus of variations accept solutions (his 20th) and, if so, whether those solutions are decidedly bland (19th).

21. LINEAR DIFFERENTIAL EQUATIONS WITH PRESCRIBED MONODROMY. Hilbert’s 21st botheration is about the actuality of assertive systems of cogwheel equations with accustomed atypical credibility and the systems’ behavior about those points, alleged monodromy. Josip Plemelj appear what was believed to be a band-aid in 1908, admitting abundant after Andrei Bolibrukh begin a counterexample to Plemelj’s work, assuming that such systems of equations do not accept to exist.

? 22. UNIFORMIZATION. Hilbert’s 22nd botheration asks whether every algebraic or analytic ambit — solutions to polynomial equations — can be accounting in agreement of single-valued functions. The botheration has been bound in the apparent case and continues to be advised in added cases.

? 23.  FURTHER DEVELOPMENTS IN THE CALCULUS OF VARIATIONS. The calculus of variations has undergone able-bodied development — including the solutions to the 19th and 20th problems — in the 120 years back Hilbert airish these questions. But Hilbert’s delivery does not accurately announce a bright endpoint, so this ‘problem’ can never be advised bound per se.

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