Hilbert's problems

The hilbert between problems are a list of 23 problems of mathematics. They were introduced by the German mathematician David Hilbert on August 8, 1900 International Congress of Mathematicians in Paris and were unresolved at this time.

2.1 Hilbert's first problem
2.2 Hilbert's second problem
2.3 Hilbert's third problem
2.4 Hilbert's fourth problem
2.5 Hilbert's fifth problem
2.6 Hilbert's sixth problem
2.7 Hilbert's seventh problem
2.8 Hilbert's eighth problem
2.9 Hilbert's ninth problem
2:10 Hilbert's tenth problem
2:11 Hilbert's eleventh problem
2:12 Hilbert's twelfth problem
2:13 Hilbert's thirteenth problem
2:14 Hilbert's fourteenth problem
2:15 Hilbert's fifteenth problem
2:16 Hilbert's sixteenth problem
2:17 Hilbert's seventeenth problem
2:18 Hilbert's eighteenth problem
2:19 Hilbert's nineteenth problem
2:20 Hilbert's twentieth problem
2:21 Hilbert's twenty-first problem
2:22 Hilbert twenty-second problem
2:23 Hilbert twenty-third problem
2:24 " Hilbert's twenty-fourth problem "

History

History and background

Hilbert was invited to give a lecture for the Second International Congress of Mathematicians in August 1900 in Paris. He decided to keep no " lecture " in which he would speak to date and worthy achievements in mathematics, but his lecture was to some extent provide a programmatic outlook on the future mathematics in the coming century. This objective is in his introductory remarks to the expression:

" Who among us would not want to lift the veil behind which the future lies hidden in order to take a look at the next advances of our science and at the secrets of its development during future centuries! What particular goals will there be toward which the leading mathematical spirits of coming generations? What new methods and new facts will the new century - in the wide and rich field of mathematical thought "

He took the Congress therefore an opportunity to put together a thematically wide-ranging list of unsolved mathematical problems. In December 1899, he began to think about the subject. At the beginning of the new year he asked his close friend Hermann Minkowski and Adolf Hurwitz then which areas must cover a corresponding lecture. Written down finally be taken Hilbert 's list, however, only in the short term before the Congress. In the official congress program it is therefore not recover.

Hilbert's presentation was part of the convention events to " Bibliography and History" instead. Due to time constraints he placed first in front of only ten problems. Those present were given a French summary of the list, which appeared a little later in the Swiss journal L' Enseignement Mathématique. The full German original article appeared a short time later on the news of the Royal Society of Sciences in Göttingen and in 1901, with some additions in the archives of mathematics and physics.

In 2000, the German historian Rüdiger Thiele discovered in the original notes a Hilbert 24th problem, which, however, was missing in the final version of the list and can be attributed to the taking of evidence theory.

Problems of Work

The mathematics of the century little was consolidated. The trend is to replace words with symbols and vague concepts through rigorous axiomatic theory was not yet very pronounced and should only allow the generation following mathematicians to formalize their subject more. Hilbert could not rely on the Zermelo -Fraenkel set theory, concepts such as the topological space and the Lebesgue integral or the Church - Turing thesis. The functional analysis, which was founded by, among others, Hilbert himself with the introduction of named after him Hilbert space had not yet separated as a mathematical field of calculus of variations.

Many of the problems in Hilbert's list are - partly for this reason - not as accurate and restricted formulated so that they could be clearly resolved by the publication of a proof. Some problems are less concrete issues rather than requests to conduct research in specific areas; other issues, the questions are too vague made to measure exactly what Hilbert had viewed as a solution.

An error Hilbert, however, does however not affect the formulation of the problems is to be found in the introduction to the article. He brings his belief that every problem has to be solved in principle:

"This conviction of the solubility of every mathematical problem is a powerful incentive to the worker; We hear within us the perpetual call: There is the problem, seeking the solution. You can find it by pure thought; because in mathematics there is no Ignorabimus! "

The basic epistemological optimism Hilbert had to be something in perspective. At the latest in 1931 with the discovery of Godel's incompleteness theorem and Turing's proof from 1936 that the decision problem is not solvable, this approach can be considered Hilbert († 1943) adopted in the original formulation as too narrow. However, this devalues the list not because negative solutions, such as the tenth problem, sometimes lead to large insights.

Influence of the list

Hilbert's list was intended to influence the further development of mathematics. Helped by the fact that Hilbert was one of the most renowned mathematicians of his generation, was this plan: It promised considerable fame, one of the problems to solve in parts, so that more and more mathematicians dealt with the issues of Hilbert's lecture and thus - even if they failed - the corresponding partial areas more developed. The idea of this list thus exerted a significant influence on the development of mathematics in the 20th century.

Although multiple attempts were to repeat this success, no other collection of problems and conjectures had a comparable influence on the development of mathematics. But the Weil conjectures, named after the mathematician André Weil, and a similar list by John von Neumann came to greater prominence. In 2000, praised the Clay Mathematics Institute prize money of one million U.S. dollars for the solution of seven important problems (see Millennium problems). The fame of Hilbert's article remains but so far unique.

The problems

Hilbert presented questions of axiomatic set theory and other axiomatic considerations at the beginning of his list. In his eyes, it was particularly important that the mathematical community clarify the foundations of mathematics gives to better understand deeper statements. Here are some questions of number theory, supplemented by algebraic topics and finally by problems from the theory of functions.

Legend:

Problems are largely solved, are highlighted in green.
Problems are partially solved, are highlighted in yellow.
Problems that remain unresolved are highlighted in red.

Hilbert's first problem

Question: Is there an uncountable subset of the real numbers which is really smaller in thickness than the real numbers?

Solution: Undecidable in the classical axiom system.

In set theory, mathematicians go today mostly of ZFC, the Zermelo - Fraenkel axioms with the axiom of choice from (the latter is sometimes omitted), the sound all the mathematical considerations formally. One can show that, based on this number of sets have the same thickness, such as the set of real numbers, the set of complex numbers, the interval [0, 1] or the power set of natural numbers. The continuum hypothesis states that, all amounts which are not countable, ie not in a 1:1 relationship with the natural numbers can be brought, have at least the cardinality of the real numbers.

Kurt Gödel showed that the continuum hypothesis to ZFC is relatively consistent in 1939: If ZFC does not lead to contradiction, so this property is preserved when one adds the axiom system to the continuum hypothesis. Paul Cohen was finally able to show in 1963 that the negation of the continuum hypothesis is relatively consistent to ZFC, so it can not be inferred from ZFC. It follows that the continuum hypothesis is independent of the classical axiom system and can be used when needed as a new axiom.

A related question that Hilbert has added in the formulation of his problem is whether a well-ordering of the real numbers exist. Ernst Zermelo was able to prove that this is on the basis of ZFC is indeed the case. Without the axiom of choice, so the system ZF, the statement can not be shown.

Hilbert's second problem

Question: Are the arithmetical axioms consistent?

Solution: After the incompleteness theorem of Kurt Gödel, this question can not be answered with the help of the axioms of arithmetic.

Giuseppe Peano in 1889 had an arithmetic axiom system described, which should establish the foundations of mathematics. Hilbert was convinced that it should thus be possible to show that, starting no contradiction can only be generated by this basis. However, these hopes destroyed Kurt Gödel when he in 1930 with his incompleteness theorem showed that this is not only using the Peano axioms is possible.

Hilbert's third problem

Question: Are any two tetrahedra with equal bases and equal heights always equidecomposable or can they complement each other with congruent polyhedra to dissection same bodies?

Solution: Neither the former nor the latter is the case.

Two hot body equidecomposable if one can thus decompose into finitely many parts, that the individual parts together again for the second body. In the two - dimensional plane is that polygons if and only have the same area, if they are equivalent by dissection. The question suggests itself whether this result is also valid in 3 -dimensional space.

Max Dehn, a student of Hilbert, in 1900 this question was, shortly after the release of the 23 problems, answer " No". He assigned to each polyhedron to a Dehn invariant called number. This has the property that it exactly then is the same for two polyhedra of equal volume if they are equivalent by dissection. With the observation that each cube is the Dehn invariant 0 and every regular tetrahedron has a different from 0 Dehn invariant, then the assertion follows. The problem is the first list of Hilbert, which was dissolved.

WF Kagan: On the transformation of the polyhedron. In: Mathematische Annalen. Springer, Berlin 57.1903, pp. 421-424. ISSN 0025-5831

Hilbert's fourth problem

Question: How can we characterize the metrics in which all lines are geodesics?

Solution: Today, there are numerous publications that deal with the characterization of such metrics. Hilbert's problem, however, is too vague, that one could find a clear solution.

Over 2000 years geometry has been taught according to the five axioms of Euclid. Towards the end of the 19th century, they began to investigate what has to add and remove various axioms for consequences. So Lobachevsky examined a geometry in which the parallel axiom does not apply and Hilbert considered a system in which lacked the Archimedean axiom. In his 23 problems he finally called a " preparation and systematic treatment of [ ... ] geometries " on that satisfy a given axiom system, in particular, the shortest distance between two points is always straight line between the points.

1901 could Georg Hamel, a student of Hilbert, in his thesis show important statements about such systems, which he published in 1903. In the coming decades, work has been published over and over again, which contributed more results to Hilbert's fourth problem.

Hilbert's fifth problem

Question: Can an arbitrary locally Euclidean topological group coordinates are chosen so that they have the structure of a Lie group?

Solution: Yes.

Sophus Lie and Felix Klein sought at the end of the 19th century, to axiomatize geometry with group theoretical means, but it went from assumptions about the differentiability of certain functions. Hilbert wondered now, the way in which the theory without these conditions has stock. As the field of algebraic topology has developed only in the 20th century, the formulation of the problem has changed over time. Hilbert's original version referred only to continuous transformation groups.

Another formulation of the problem is the following: Consider a group G with neutral element e, an open set U in Euclidean space that contains e and a continuous map F: V × V → U that of on the open subset V U satisfies the group axioms. The question then is whether F on a neighborhood of e, so is smooth infinitely differentiable. After John von Neumann could solve and Lev Pontryagin special cases, Andrew Gleason, Deane Montgomery and Leo Zippin succeeded in the 1950s, the final clarification of the problem.

A. Gleason: Groups without small subgroups. In: Annals of Mathematics. University Press, Princeton 56.1952, pp. 193-212. ISSN 0003- 486x
D. Montgomery, L. Zippin: Small groups of finite - dimensional groups. In: Annals of Mathematics. University Press, Princeton 56.1952, pp. 213-241. ISSN 0003- 486x

Hilbert's sixth problem

Question: How can the physics be axiomatized?

Solution: Unknown.

Originally, Hilbert at this point to an axiomatic treatment of probability theory and mechanics. Meanwhile, there is the theory of relativity and quantum physics much deeper insights into the structure of the universe, but a general axiomatic formulation of physics is not in sight.

Hilbert's seventh problem

Question: Is the power of αβ always transcendent, if α algebraically ( α ≠ 0, α ≠ 1) and β irrational and algebraically?

Solution: Yes.

A number is called algebraic if it is zero of a polynomial with integer coefficients, otherwise it is called transcendent. The square root of 2, for example, a number that is not rational, as the root of x2 - 2 but still algebraically. Real numbers that are not algebraic more (and transcendent ) are, for example, π the loop number or e Euler's number

To Hilbert's times there were already some results on the transcendence of different numbers. The above problem seemed especially difficult, and he hoped for from its solution a deeper understanding of the nature of numbers. After the problem was first solved for some special cases (Alexander Gelfond 1929 Rodion Kuzmin 1930), Alexander Gelfond 1934 could prove the statement. A short time later, Theodor Schneider continued to improve the sentence, so that the answer to Hilbert's seventh problem is known as Gelfond -Schneider theorem of today.

Hilbert's seventh problem is a special case of the broader geometric assertion: If an isosceles triangle the ratio of the angle at the base of the angle is irrational and algebraically at the top, then the ratio between base and leg is transcendent.

A generalization of Hilbert's question would be answered by a proof or a refutation of the conjecture of Schanuel, Stephen Schanuel which established in the 1960s.

Hilbert's eighth problem

Question: Do all non-trivial zeros of the Riemann zeta function the real part ½? Is every even number greater than 2 represented as the sum of two primes?

Solution: Unknown.

These two problems are known as Riemann Hypothesis and Goldbach's conjecture and two of the most popular unsolved problems of mathematics. For the first question have already been calculated over a trillion zeros with real part ½ and it found none that would falsify the conjecture and the second question was already tested up to an order of 1017. Formal proof to show but you could not see until today (2014).

Under the heading " prime problems " Hilbert has collected more questions related to prime numbers. So he calls, for example, ( also to still unresolved ) question whether there are infinitely many twin primes and whether the equation ax by c = 0 with arbitrary integer, with each coprime coefficients a, b and c always prime solutions x, y has.

Hilbert's ninth problem

Question: How can the law of reciprocity be generalized to arbitrary number fields?

Solution: Known only in the abelian case.

Since Fermat was known that the prime factorization of each number that is 1 greater than a square number, only the number 2 and numbers of the form 4k 1 exists. A more general form that uses the Legendre symbol, Gauss was able to prove:

This reciprocity should now be generalized to general number fields. With the development of class field theory to the necessary funds were available, so Emil Artin was able to solve the problem in the abelian case - even if it does not succeed to provide an explicit formula. This took Igor Safarevic 1948 after. A further generalization to non- abelian case has not yet been achieved.

Hilbert's tenth problem

Question: Give a procedure that decides for an arbitrary Diophantine equation whether it is solvable.

Solution: There is no such method.

Diophantine equations are equations of the form f ( x1, x2, ..., xn ) = 0, where f is a polynomial in multiple variables and with integer coefficients and integers only be considered solutions. A well-known example is the equation x2 y2 - z2 = 0, which is related to the Pythagorean theorem. Diophantine equations play in the history of mathematics an important role, and many great mathematicians have dealt extensively with such formulas.

Although special cases could always be solved, but a general solution seemed to mathematicians in the 19th century unreachable remote. Therefore, Hilbert asked only how to check whether a given Diophantine equation has integer solutions at all without being able to specify it exactly. However, this problem is so severe that in 1970 Yuri Matijassewitsch could prove that such a procedure for the general case does not exist.

Hilbert's eleventh problem

Question: How can the theory of quadratic forms can be generalized to arbitrary algebraic number field?

Solution: The theory was developed extensively in the 20th century.

A square shape is a function of the form p ( x) = x Tax, x being a vector and A is a symmetric matrix. In the decades after Hilbert's lecture numerous results have been published that deal in depth with the topic. As a central result of this include the local-global principle, the Helmut Hasse in 1923 formulated.

Hilbert's twelfth problem

Question: How can the set of Kronecker -Weber generalize to arbitrary number fields?

Solution: Unknown.

The set of Kronecker -Weber states that the maximum abelian extension of the field of rational numbers is formed by adjoining all roots of unity. The generalization of this theorem Hilbert measure of great importance. Although there were in the field in the 20th century, many advances, to a solution of Hilbert's twelfth problem, however, was not yet there.

Hilbert's thirteenth problem

Question: Can the solution of the equation x7 ax3 be constructed bx2 x 1 = 0 with a finite number of continuous functions that depend on two variables?

Solution: Yes.

A more general version of this question is: Is there continuous functions in three variables that can not be represented as a concatenation of a finite number of continuous functions in two variables? Andrei Kolmogorov, Vladimir Arnold and 1957 could show that this is the case, in fact, and thus contradicted Hilbert, who had predicted a different result in his presentation.

Hilbert's fourteenth problem

Question: Are certain rings ( see below) finitely generated?

Solution: No.

In the fourteenth problem describes Hilbert special rings: where K [x1, ..., xn ] is a polynomial ring over a field K, L be a subfield of the field of rational functions in n variables, and R be the intersection

The question then is whether the constructed rings are always finitely generated, ie whether there is a finite subset of the ring, which generate R.

Until the 1950s, you could by some special cases, particularly the cases n = 1 and n = 2 prove that the constructed rings are actually finitely generated. Thus, the results suggested that this statement could also apply to all the rings of the type described. Surprisingly, therefore, was the result of Masayoshi Nagata, 1957 a counterexample stated, in which this is not the case, and thus negatively solved the problem.

Masayoshi Nagata: On the 14 - th Problem of Hilbert. In: American Journal of Mathematics. The Johns Hopkins University Press, Baltimore 81.1959, pp. 766-772. ISSN 0002-9327

Hilbert's fifteenth problem

Question: How can be concretized Schubert 's enumerative and formally justified?

Solution: The results of Schubert have been well understood in the mid-20th century.

With the advancement of algebraic geometry were gradually mathematical tools available with which Hermann Schubert's work could be formalized.

Hilbert's sixteenth problem

Question: What can be said about the mutual position of algebraic curves?

Solution: There were different results are obtained, but many questions remain unanswered.

Algebraic curves are subsets of the plane are determined by polynomial equations. These include, for example, the unit circle (x2 y2 = 1) or simple straight line ( ax b = y). Carl Harnack was able to show in 1876 that such quantities of polynomials of degree n (also curves of order n ) shall consist of at most ½ ( n - 1 ) (n - 2 ) 1 parts survive. He could also construct examples which also achieve ("M -curve "), this maximum number.

Hilbert noted that these parts can not be arranged arbitrarily in the plane. So he guessed, for example, that the eleven components of M- curves sixth order always lie so that nine components are located inside a loop and the last component outside this loop runs (or vice versa) and challenged in the first part of the sixteenth problem to on to investigate correlations of this type closer.

This happened with the development of the topology of real algebraic manifolds. The pronounced by Hilbert conjecture was proved for the first time in the 1930s by Ivan Petrovsky Georgijewitsch. Other results relate to the three-dimensional equivalent of the question: Karl Rohn has already demonstrated in the 19th century that algebraic surfaces of the fourth order of a maximum of twelve areas may exist. If this limit can also be achieved, is so far not known.

Hilbert's seventeenth problem

Question: Can any rational function that everywhere it is defined, assumes non-negative values are represented as the sum of squares of rational functions?

Solution: Yes.

With Emil Artin algebraic means could solve the problem in 1927. To this end, he looked a certain class of bodies and showed properties that eventually led him to his result.

Hilbert's eighteenth problem

Question: Is there a finite number of essentially different space groups in n-dimensional Euclidean space?

Solution: Yes.

The first part of Hilbert's eighteenth problem is the mathematical formulation of a question from the crystallography. Many solids have a crystalline structure at the atomic level, which can be described mathematically with movement groups. You could already show early on that there are essentially different space groups in the plane and in the space 17 230. Ludwig Bieberbach was finally able to show that this number is always even in higher dimensions finite.

In the second part of the problem Hilbert asked whether there are polyhedra in 3-dimensional space, which do not occur as a fundamental domain of a movement group, which nevertheless, the entire space can be tiled seamlessly. That this is the case, it was the first time Karl Reinhardt 1928 show by giving an example.

Last queries Hilbert after the most space-saving way to arrange balls in the room. Already Johannes Kepler presented in 1611 on the assumption that the face-centered cubic packing and the hexagonal packing is optimal. However, this known as the Kepler conjecture statement turned out to be - though not surprisingly - very difficult to prove out. First published in 1998, Thomas Hales a computer-assisted proof, which is now (2010) tested and approved.

Hilbert's tenth problem nine

Question: Are all the solutions of Lagrangian functions analytically?

Solution: Under certain circumstances, yes.

Lagrangians are special partial differential equations, which are particularly in physics application. Hilbert had observed with some surprise that there are differential equations that only allow analytical solutions, ie those that can be represented locally by power series. Since different results also pointed at the important Lagrangians on this property, Hilbert took the question as to in his essay.

Already in 1903, Sergei Bernstein could solve the problem by proving the analyticity of a certain class of differential equations, which include the equations in question, provided that the third derivatives of the solutions exist and are bounded. Later Ivan Petrovsky could worsen the outcome and thus provide a more comprehensive solution to the nineteenth issue.

Hilbert's twentieth problem

Question: Under what conditions boundary value problems have solutions?

Solution: The existence of a solution can not be secured in each case by a restriction of the boundary values .

The twentieth issue is closely related to the nineteenth century and also has direct relevance to physics. There are now extensive results on the subject, so that the problem can be considered as solved.

Hilbert's twenty-first problem

Question: Is there always a system of differential equations with given fox between singularities and given monodromy group?

Solution: No.

After the question could be answered positively for some special cases first, finally succeeded Andrei Bolibruch 1994, in the general case to prove the contrary.

DV Anosov, AA Bolibruch: Aspects of Mathematics - The Riemann - Hilbert Problem. Vieweg, Braunschweig 1994. ISBN 3-528-06496- X

Hilbert's twenty-second problem

Question: How can analytic relations by means of automorphic functions are uniformisiert?

Solution: For equations with two variables solved with more variables there are still open questions.

In the uniformization you set itself the goal to parameterize algebraic curves in two variables, ie to replace the variables with functions that depend only on one variable. Thus, for example, the unit circle, which is given by x2 y2 = 1, parametrized by substituting for x and y cos α and sin α.

As Hilbert formulated his problems, there was already a partial solution by Henri Poincaré, Hilbert, however, not satisfied with that. 1907 could Poincaré and Koebe Paul independently solve the problem eventually - but only for the case with two variables. Generalizing the problem to more than two variables, so there are still unresolved issues in this area ( part of a program of William Thurston ).

Hilbert's twenty-third problem

Question: How can the methods of the calculus of variations to be developed further?

Solution: The problem is too vague, that one could specify a concrete solution.

The calculus of variations is in Hilbert's words, " the doctrine of varying the functions" and had in his view, a special importance. That's why he formulated in the last part of his lecture, no particular problem more, but generally called on the further development of this area. With the development and the extensive expansion of functional analysis this concern Hilbert was taken into account in the 20th century.

" Hilbert's twenty-fourth problem "

Hilbert's 24th problem is a mathematical problem, whose formulation was found in Hilbert's estate and which is sometimes named as a supplement to its list of 23 mathematical problems. Hilbert represents the question of criteria or evidence of whether a proof of the easiest for a mathematical problem.

392489