Hilbert problems

The Hilbert problems are a list of 23 problems in mathematics . They were presented by the German mathematician David Hilbert on August 8, 1900 at the International Congress of Mathematicians in Paris and were unsolved at that time.

David Hilbert (1886)

history

Prehistory and background

Hilbert had been invited to give a lecture for the second international congress of mathematicians in Paris in August 1900. He decided not to give a “lecture” in which he would lecture and appreciate what had been achieved so far in mathematics, nor to reply to the lecture by Henri Poincaré at the first international mathematicians' congress in 1897, which was about the relationship between mathematics and physics had presented. Instead, his lecture was intended to offer a programmatic outlook on future mathematics in the coming century . This objective is expressed in his introductory words:

“Who of us would not like to lift the veil under which the future lies, to glimpse the impending advances in our science and the mysteries of its development over the centuries to come! What special goals will the leading mathematical minds of the coming generations pursue? What new methods and new facts will the new centuries discover - in the wide and rich field of mathematical thought? "

He therefore used the congress as an opportunity to compile a thematically broad list of unsolved mathematical problems. As early as 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 for suggestions as to which areas a corresponding lecture should cover; both read the manuscript and commented on it before the lecture. Hilbert did not finally write down his list until immediately before the congress - so it does not appear in the official congress program. Originally the lecture was supposed to be given at the opening, but Hilbert was still working on it at the time.

Fewer mathematicians came to the congress than expected (around 250 instead of the expected 1000). Hurwitz and Felix Klein were not present, but Minkowski. Hilbert was President of the Algebra and Number Theory Section, which met from August 7th (the second day of the conference) to August 10th. Hilbert's lecture took place in Sections 5 and 6 (Bibliography, History, Teaching and Methods, Moritz Cantor's Presidency ) on Wednesday, August 8th, in the morning in the Sorbonne . Due to lack of time, he initially only presented ten problems (No. 1, 2, 6, 7, 8, 13, 16, 19, 21, 22). Those present received a French summary of the list, which appeared shortly afterwards in the Swiss magazine L'Enseignement Mathématique . The complete German original article appeared a short time later in the news of the Königliche Gesellschaft der Wissenschaften zu Göttingen and in 1901 with some additions in the archive of mathematics and physics .

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

Problems of work

Mathematics at the turn of the century was not yet well established. The tendency to replace words with symbols and vague concepts with strict axiomatics was not yet very pronounced and was only to allow the next generation of mathematicians to formalize their subject more strongly. Hilbert could not yet fall back on the Zermelo-Fraenkel set theory , terms such as topological space and the Lebesgue integral or the Church-Turing thesis . The functional analysis , which itself among other things by Hilbert with the introduction of the eponymous Hilbert space was established, had not yet as a mathematical field of the calculus of variations separated.

Many of the problems in Hilbert's list are - partly for this reason - not formulated in such a precise and restricted manner that they could clearly be solved by the publication of a proof . Some problems are less concrete questions than requests to research in certain areas; for other problems, the questions are too vague to say exactly what Hilbert would have considered the solution.

A mistake by Hilbert, however, which does not affect the formulation of the problems, can be found in the introduction to the article. There he expresses his conviction that every problem must be fundamentally solvable:

“This conviction of the solubility of every mathematical problem is a powerful incentive for us during our work; we hear the constant call in us: there is the problem, look for the solution. You can find it through pure thinking; because there is no ignorance in mathematics ! "

Hilbert's fundamental epistemological optimism had to be somewhat relativized. At the latest in 1931 with the discovery of Gödel's incompleteness theorem and Turing's proof of 1936 that the decision problem cannot be solved, Hilbert's († 1943) approach can be regarded as too narrow in its original formulation. However, this does not devalue the list, because negative solutions, such as the tenth problem, sometimes lead to a great gain in knowledge.

The choice of problems is partly a very personal selection by Hilbert and grew out of his own work, although, as mentioned, he consulted with his close friend Minkowski and Hurwitz (who was known for the versatility of his mathematical work and his encyclopedic overview). Ivor Grattan-Guinness names a few noticeable gaps. On the one hand the great Fermat conjecture and the three-body problem (which Poincaré worked on a lot), which he mentions in the introduction as prime examples of mathematical problems, but does not include them in his list. Applied mathematics is rarely represented (at most problem 6 could be classified there), just as little numerical mathematics (mentioned only briefly in problem 13, the core of which lies elsewhere) and the sub-area of ​​analysis later called functional analysis, on which Hilbert himself intensively from 1903 to 1910 worked. The electrodynamics of moving bodies (prehistory of the theory of relativity) was also missing and was a very active field of research at that time, in which Poincaré also worked and on which Joseph Larmor , who also presided over a section at the congress, published an important book in the same year (Aether and Matter) . On the other hand, Grattan-Guinness finds the omission of mathematical logic, statistics and matrix theory (linear algebra) understandable, since they were not as prominent then as they are today. In contrast, in his lecture at the International Congress of Mathematicians in 1908 on the future of mathematics, in a sense an answer to Hilbert, Poincaré placed a lot of emphasis on applications, emphasizing the future development of topology ("geometry of the situation") as a central concern of mathematics (With Hilbert it appears in problems 5 and 16) and also emphasized the importance of set theory ("Cantorism"), with Hilbert represented in problem 1. Overall, however, his portrayal was much more vague and sketchy than Hilbert's.

Influence of the list

According to Charlotte Angas Scott , the immediate reaction at the congress was disappointing, possibly due to Hilbert's dry style of presentation or language problems (Hilbert lectured in German, but had previously had a summary distributed in French). Giuseppe Peano spoke up to remark that his school ( Cesare Burali-Forti , Mario Pieri , Alessandro Padoa ) had essentially solved the problem of the foundation of arithmetic and that his student Alessandro Padoa was giving a lecture on the same congress would. Rudolf Mehmke , who was also present at the lecture , made a comment about progress through numerical (nomographic) methods in problem 13, especially in the equation of the 7th degree. No reaction is known from Poincaré, and he was probably not present at Hilbert's lecture. After Ivor Grattan-Guinness, he was more interested in applied questions at the time and also less interested in the axiomatic approach. At the same congress he gave one of the two closing lectures on August 11th on the role of intuition and logic in mathematics and emphasized the role of intuition. Later, however, he took up the problem of uniformity (Hilbert's Problem 22) and in his lecture on the future of mathematics at the International Congress of Mathematicians in Rome in 1908 he also included the problem of boundary cycles (part of Problem 16, in which Hilbert explicitly referred to Poincaré took) into his own list of problems. There he also praised Hilbert for his work on the axiomatic method and the Dirichlet problem. When the conference proceedings were published in 1902, the importance of Hilbert's lecture was expressly recognized and it was therefore printed outside of its section at the beginning, immediately followed by Poincaré's lecture.

Hilbert's list was intended to influence the further development of mathematics. Favored by the fact that Hilbert was one of the most renowned mathematicians of his generation, this plan worked: It promised considerable fame to solve one of the problems in parts, so that more and more mathematicians were concerned with the topics from Hilbert's lecture and thus - themselves if they failed - further developed the relevant sub-areas. The presentation of this list thus had a significant influence on the development of mathematics in the 20th century.

Although there have been multiple attempts to replicate this success, no other collection of problems and conjectures has had a comparable impact on the development of mathematics. The Weil conjectures , named after the mathematician André Weil , were influential but limited to a sub-area of ​​number theory , and similar lists by John von Neumann at the 1954 International Congress of Mathematicians were explicitly based on the model of Hilbert's list (with little influence, the lecture was not even published) and from Stephen Smale on ( Smale problems ). In 2000, the Clay Mathematics Institute awarded prizes of $ 1 million each for solving seven important problems . However, the fame of Hilbert's article remains unique to date.

The problems

At the beginning of his list, Hilbert put questions about axiomatic set theory and other axiomatic considerations. In his opinion, it was particularly important that the mathematical community gain clarity about the fundamentals of mathematics in order to be able to better understand more in-depth statements. This concerned not only the axiomatic foundations of geometry, about which Hilbert himself had published a book shortly before (1899), but also physics. Some questions of number theory follow , which are supplemented by algebraic topics and finally by problems from function theory and the calculus of variations or analysis.

Legend:

• Problems that have largely been solved are highlighted in green.
• Problems that have been partially solved are highlighted in yellow.
• Problems that are still unsolved are highlighted in red.

Hilbert's first problem

Question: Is there an uncountable subset of real numbers that is really smaller than the real numbers in terms of its power ?

Solution: Undecidable in the classical system of axioms.

In set theory today, mathematicians mostly start from ZFC , the Zermelo-Fraenkel axiom system with axiom of choice (the latter is sometimes omitted), which formally provides a basis for all mathematical considerations. One can show that on this basis many sets have the same power, for example the set of real numbers, the set of complex numbers , the (real) interval or the power set of natural numbers . The continuum hypothesis now states that all sets that can no longer be counted , that is, cannot be brought into a 1: 1 relationship with the natural numbers , have at least the power of the real numbers. ${\displaystyle [0,1]}$

Kurt Gödel was able to show in 1939 that the continuum hypothesis for ZFC is relatively free of contradictions: If ZFC does not lead to a contradiction, this property is retained if the system of axioms is supplemented by the continuum hypothesis. Paul Cohen was finally able to show in 1963 that the negation of the continuum hypothesis is also relatively consistent with ZFC, so it cannot be deduced from ZFC. It follows that the continuum hypothesis is independent of the classical system of axioms and can be used as a new axiom if necessary. To prove this, Cohen developed one of the most important methods of axiomatic set theory, the forcing method, which was also used in the investigation of the independence of many other theorems in ZFC.

A related question that Hilbert added in the formulation of his problem is whether there is a well-ordering of the real numbers. Ernst Zermelo was able to prove that this is actually the case on the basis of ZFC. Without the axiom of choice, i.e. in the ZF system, the statement cannot be shown.

• Donald A. Martin : Hilbert's first problem: the continuum hypothesis . In: F. Browder: Mathematical Developments Arising from Hilbert's Problems . AMS, Part 1, 1976, pp. 81-92.

→ Main article : Continuum hypothesis

Hilbert's second problem

Question: Are the arithmetic axioms free of contradictions ?

Solution: According to Kurt Gödel's incompleteness theorem , this question cannot be answered with the help of arithmetic axioms.

In 1889 Giuseppe Peano had described an arithmetic system of axioms that was supposed to establish the foundation of mathematics. Hilbert was convinced that it should be possible to show that only starting from this basis in a finite number of steps (with finite methods) no contradiction can be generated. However, Kurt Gödel destroyed this hope when he showed with his incompleteness theorem in 1930 that this is not possible using only the Peano axioms . With transfinite methods, which were not permitted according to Hilbert's original program, Gerhard Gentzen succeeded in proving the consistency of arithmetic in 1936 .

• Georg Kreisel : What have we learned of Hilbert's second problem? In: F. Browder: Mathematical Developments Arising from Hilbert's Problems . AMS, Part 1, 1976, pp. 93-130.

→ Main article : Hilbert program

Hilbert's third problem

Question: Are any two tetrahedra with the same base area and the same heights always having the same decomposition or can they be supplemented with congruent polyhedra to form bodies with the same decomposition?

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

Two bodies are called equal if one can be broken down into a finite number of parts so that the individual parts can be put back together to form the second body. In the two-dimensional plane it is true that polygons have the same area if they have the same decomposition. There is an elementary theory based on the division into triangles of the area of ​​simple figures bounded by straight sides (polygons), and one does not have to rely on non-elementary methods such as the exhaustion method , which requires a boundary crossing and for surfaces with curved edges Application comes. So the question arises as to whether this result also applies in three-dimensional space.

Max Dehn , a student of Hilbert, was able to answer this question with "No" as early as 1900, shortly after the 23 problems were published. To do this, he assigned a number called a stretch invariant to each polyhedron . In addition to the volume, there was another number assigned to polyhedra, which remained the same (invariant) when the polyhedra were broken down. It depended on the angles of neighboring sides in the polyhedron and its edge lengths (and is defined as the sum of the tensor products of edge length and angle of the sides that meet an edge over all edges of the polyhedron). With the observation that every cube has the Dehn invariant and every regular tetrahedron has a different Dehn invariant, the statement follows. The problem is the first on Hilbert's list to be resolved. ${\displaystyle 0}$ ${\displaystyle 0}$

While Dehn showed that for equality of decomposition in three-dimensional Euclidean space the equality of the expansion numbers is necessary (this was already clear for the volume), JP Sydler showed in 1965 that this is also sufficient : two polyhedra are equal to decomposition if and only if volume and expansion number are the same. For more than four dimensions (for four dimensions a similar theorem can be proven with the help of Hadwiger invariants instead of Dehn invariants, generalizations of Dehn invariants on higher dimensions introduced by Hugo Hadwiger ) or, for example, non-Euclidean space, no comparable result is known . If one restricts the movements to translations, however, the equality of decomposition of polyhedra can be characterized with the help of the Hadwiger invariants in any dimensions.

• CH Saw: Hilbert's third problem: scissors congruence . Pitman, 1979.
• VG Boltianskii: Hilbert's third problem . Wiley, 1978.

Hilbert's fourth problem

Question: How can the metrics in which all straight lines are geodesic be characterized?

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

For over 2000 years, geometry was taught using Euclid's five axioms . Towards the end of the nineteenth century people began to investigate the consequences of adding and removing different axioms. Lobatschewski examined a geometry in which the axiom of parallels does not hold, and Hilbert examined a system in which the Archimedean axiom was absent. Hilbert finally examined the axiomatic foundations of geometry in detail in his book of the same name. In his 23 problems he finally called for a “list and systematic treatment of the [...] geometries” that satisfy a certain system of axioms, in which the shortest connection between two points is always the straight line between the points. The problem corresponds to the investigation of geometries that are as close as possible to the usual Euclidean geometry. In Hilbert's system of axioms of Euclidean geometry the axioms of incidence, arrangement and continuity are retained, but the axioms of congruence are weakened: the strong axiom of congruence III-6 (triangular congruence) is no longer assumed, but that the length of the sides in a triangle is less than or equal to Is the sum of the lengths of the other two (which is equivalent to the fact that the straight line is the shortest connection between two points). In Euclid's theorem that the straight line is the shortest connection between two points was derived with the triangular congruence theorem. Hilbert found an example of such a geometry close to Euclidean geometry with the new postulates in the geometry of numbers by Hermann Minkowski and Hilbert himself gave another example.

As early as 1901, Georg Hamel , a student of Hilbert, was able to make important statements about such systems in his dissertation, which he published in 1903. In the case of the plane, he was able to specify and classify a whole series of such geometries, of which the Hilbert and Minkowski geometries mentioned are typical examples. According to Isaak Moissejewitsch Jaglom , Hamel solved the fourth Hilbert problem in a certain way, with the restriction that he used analytical methods of the calculus of variations, which are less desirable in basic geometric research because they make additional assumptions (differentiability requirements). In the decades to come, papers were repeatedly published that contributed further results to Hilbert's fourth problem. Among other things, Herbert Busemann dealt extensively with the geometries in question and wrote a monograph on it. According to Busemann, Hilbert put the problem too far, probably because he did not understand how many such geometries there were, and additional restrictions (axioms) are to be assumed. Busemann's method was expanded by Alexei Wassiljewitsch Pogorelow , who published a monograph on the fourth problem in 1979.

• Herbert Busemann: The Geometry of Geodesics . Academic Press 1955, Dover 2005.
• Herbert Busemann: Problem IV: Desarguesian spaces . In: F. Browder: Mathematical Developments Arising from Hilbert's Problems . AMS, Part 1, 1976, pp. 131-141.
• AV Pogorelov: Hilbert's fourth problem . Winston & Wiley, 1979.

Hilbert's fifth problem

Question: Is a locally Euclidean, topological group a Lie group , in which the group operations are also differentiable?

Solution: yes.

At the end of the 19th century, Sophus Lie and Felix Klein endeavored to axiomatize geometry using group theoretic means, but based on assumptions about the differentiability of certain functions. Hilbert asked himself in what way the theory still holds without these assumptions. Since the field of algebraic topology did not develop until the 20th century, the formulation of the problem has changed over time. Hilbert's original version only referred to continuous transformation groups.

A more detailed formulation of the problem is as follows: Consider a group with a neutral element , an open set in Euclidean space that contains, and a continuous mapping that satisfies the group axioms on the open subset of . The question is then whether on a neighborhood of smooth , i.e. infinitely often , is differentiable . After John von Neumann (1933, solution for compact groups), Lew Pontryagin (1939, solution for Abelian groups) and Claude Chevalley (solvable topological groups, 1941) were able to solve special cases (and other mathematicians were able to solve the problem for dimensions up to four), succeeded Andrew Gleason , Deane Montgomery and Leo Zippin in the 1950s, the final clarification of the problem. They even proved that locally Euclidean topological groups are real-analytic. ${\displaystyle G}$ ${\displaystyle e}$ ${\displaystyle U}$ ${\displaystyle e}$ ${\displaystyle F\colon V\times V\to U}$ ${\displaystyle V}$ ${\displaystyle U}$ ${\displaystyle F}$ ${\displaystyle e}$

The proof was very technical and complicated. Joram Hirschfeld gave a simpler proof within the framework of the nonstandard analysis . The problem was very fashionable in the period after World War II and the solution found in 1952 practically ended the research area after Jean-Pierre Serre , who was then trying to solve it himself.

The question is open: Is a locally compact topological group, whose group operations work faithfully on a topological manifold, a Lie group? (Hilbert-Smith conjecture after Hilbert and Paul A. Smith ). An example would be the p-adic integers. It does not apply to these that they have no small subgroups - a condition which, according to Gleason, Montgomery and Zippin, characterizes the Lie groups among the locally compact topological groups. A topological group has no small subgroups if there is a neighborhood of the unit that does not contain subgroups greater than . Some mathematicians see the Hilbert-Smith conjecture as the actually correct formulation of the Hilbert problem. ${\displaystyle e}$ ${\displaystyle \{e\}}$

• A. Gleason: Groups without small subgroups . Annals of Mathematics , Volume 56, 1952, pp. 193-212.
• D. Montgomery, L. Zippin: Small groups of finite-dimensional groups. Annals of Mathematics, Volume 56, 1952, pp. 213-241.
• I. Kaplansky: Lie algebras and locally compact groups . University of Chicago Press, 1964.
• CT Yang: Hilbert's fifth problem and related problems on transformation groups . In: F. Browder: Mathematical Developments Arising from Hilbert's Problems . AMS, Part 1, 1976, pp. 142-146.

Hilbert's sixth problem

Question: How can physics be axiomatized?

Solution: Unknown.

According to Leo Corry , the sixth problem does not take on an outsider role in the list of problems for Hilbert, as is often assumed, but corresponded in a central way to his interests over a long period of time (at least from 1894 to 1932). This program includes, for example, his well-known derivation of the field equations of general relativity from a principle of variation (1916). According to Corry, there is also a misunderstanding of Hilbert's conception of his program of axiomatization, which was mainly based on Hilbert's later program for the foundation of mathematics, which in connection with physics primarily served to clarify the logical structure of established theories. At the time of his lecture, Hilbert was still in the 19th century tradition of wanting to reduce physics to mechanics, and his formulation at that time concentrated on mechanics, heavily influenced by Heinrich Hertz's research on the fundamentals of mechanics and by Ludwig Boltzmann (transition from statistical mechanics to continuum mechanics). Later on, Hilbert's interest went much further than that, at the latest by 1905 he also extended it to include electrodynamics, which he had not explicitly mentioned in his list of problems. In 1905 he gave a lecture on the axiomatization of physics, in which he included thermodynamics and electrodynamics, among other things. His endeavors to axiomatize geometry were also motivated to give a fundamentally empirical theory a strict foundation (and to simplify it). Since he also included probability theory, Andrei Kolmogorow's axiomatization of it can be seen as a contribution to Hilbert's program.

There have always been approaches to axiomatizations in sub-areas of physics, for example thermodynamics ( Constantin Caratheodory ), quantum field theory ( Arthur Wightman and Wightman axioms, Rudolf Haag , Daniel Kastler , Huzihiro Araki and Haag-Kastler axioms, Osterwalder- Schrader axioms) , Topological quantum field theory , conformal field theories and physicists who dealt with the basic structure of physical theories such as Günther Ludwig .

• Joseph Kouneiher (Ed.): Foundations of Mathematics and Physics One Century after Hilbert: New Perspectives, Springer 2018
• Leo Corry: Hilbert's sixth problem: between the foundations of geometry and the axiomatization of physics . Phil. Trans. R. Soc. A 376 (2118), 2018, 20170221; doi : 10.1098 / rsta.2017.0221 .
• Arthur Wightman: Hilbert's sixth problem: mathematical treatment of the axioms of physics . In: F. Browder: Mathematical Developments Arising from Hilbert's Problems . AMS, Part 1, 1976, pp. 147-240.

Hilbert's seventh problem

Question: Is the power always transcendent when it is algebraic and irrational and algebraic? ${\displaystyle \alpha ^{\beta }}$ ${\displaystyle \alpha }$ ${\displaystyle (\alpha \neq 0,\alpha \neq 1)}$ ${\displaystyle \beta }$

Solution: yes.

A complex number is called algebraic if it is the zero of a polynomial with integer coefficients, otherwise it is called transcendent . For example, the root of 2 is a number that is no longer rational , but is still algebraic as a zero of . Real numbers that are no longer algebraic (and thus transcendent ) are, for example, the circle number or Euler's number . ${\displaystyle x^{2}-2}$ ${\displaystyle \pi }$ ${\displaystyle e}$

In Hilbert's time there were already some results about the transcendence of different numbers. The above problem seemed particularly difficult to him, and he hoped that its solution would provide a deeper understanding of the nature of numbers. After the problem was first solved for a few special cases ( Alexander Gelfond 1929, Rodion Kusmin 1930), Alexander Gelfond was able to prove the statement in 1934. A short time later, Theodor Schneider improved the sentence further, so that the answer to Hilbert's seventh problem is now known as the Gelfond-Schneider Theorem .

Hilbert's seventh problem can also be understood as a statement about pairs of logarithms of algebraic numbers (namely that from their linear independence over the rational numbers the linear independence over the algebraic numbers follows). In this formulation, the sentence has been considerably expanded by Alan Baker .

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

• Robert Tijdeman : Hilbert's seventh problem: the Gelfond-Baker method and applications . In: F. Browder: Mathematical Developments Arising from Hilbert's Problems . AMS, Part 1, 1976, pp. 241-268.

→ Main article : Theorem of Gelfond-Schneider

Hilbert's eighth problem

Question: Do all nontrivial zeros of the Riemann zeta function have the real part ? Can every even number be greater than the sum of two prime numbers? ${\displaystyle {\tfrac {1}{2}}}$ ${\displaystyle 2}$

Solution: Unknown.

The two problems mentioned are known as the Riemann Hypothesis and Goldbach Hypothesis and are two of the most popular unsolved problems in mathematics. For the first question, over a trillion zeros have already been calculated and none found that would falsify the assumption. The second question has already been examined up to the order of magnitude . To this day, however, no evidence has been found. The proof of the analogue of the Riemann conjecture for curves over finite fields by Pierre Deligne , part of the Weil conjectures, was regarded as a significant advance . ${\displaystyle 10^{18}}$

Under the heading "Prime number problems" Hilbert has put together more questions that are connected with prime numbers . He (also still unresolved) calls, for example, the question of whether there are infinitely many twin primes are and whether the equation with arbitrary integer among themselves prime coefficient , and always prime solutions , has a slight variation on the Goldbach conjecture and well unsolved. ${\displaystyle ax+by+c=0}$ ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle c}$ ${\displaystyle x}$ ${\displaystyle y}$

• Enrico Bombieri : Hilbert's 8th problem: an analogue . In: F. Browder: Mathematical Developments Arising from Hilbert's Problems . AMS, Part 1, 1976, pp. 269-274.
• Hugh Montgomery : Problems concerning prime numbers (Hilbert's problem 8) . In: F. Browder: Mathematical Developments Arising from Hilbert's Problems . AMS, Part 1, 1976, p. 307.

→ Main articles : Riemann Hypothesis  and Goldbach Hypothesis

Hilbert's ninth problem

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

Solution: Known only in the Abelian case .

The quadratic reciprocity law proved by Gauss (formulated with the Legendre symbol ):

${\displaystyle \left({\frac {p}{q}}\right)\left({\frac {q}{p}}\right)=(-1)^{{\frac {p-1}{2}}{\frac {q-1}{2}}}}$

gives criteria for the solution of quadratic equations in modular arithmetic and with his generalizations played a central role in algebraic number theory. In the 19th century, various higher reciprocity laws were already known, also from Hilbert in his number report , where he introduced Hilbert symbols in the formulation . Hilbert asked for a formulation and a proof for general algebraic number fields . With the development of class field theory beginning with Teiji Takagi , the necessary means were available so that Emil Artin could solve the problem in the Abelian expansion of algebraic number fields ( Artin's law of reciprocity , 1924), and Helmut Hasse also proved reciprocity theorems in class field theory. In 1948 , Igor Schafarewitsch made significant progress on the question of explicit formulas for this reciprocity law , with Helmut Brückner , Sergei Wladimirowitsch Vostokow and Guy Henniart simplifying and expanding its results. A further generalization to the non-Abelian case could not yet be achieved and is one of the main problems of algebraic number theory, also connected with Hilbert's 12th problem.

• John T. Tate : The general reciprocity law . In: F. Browder: Mathematical Developments Arising from Hilbert's Problems . AMS, Part 2, 1976, pp. 311-323.

Hilbert's tenth problem

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

Solution: There is no such procedure.

Diophantine equations are equations of the form , where a polynomial is in multiple variables and with integer coefficients and only whole numbers are considered as solutions. A well-known example is the equation related to the Pythagorean theorem . Diophantine equations play an important role in the history of mathematics, and many great mathematicians have studied such formulas extensively. ${\displaystyle f(x_{1},x_{2},\ldots ,x_{n})=0}$ ${\displaystyle f}$ ${\displaystyle x^{2}+y^{2}-z^{2}=0}$

Although special cases could always be solved, a general solution seemed inaccessible to mathematicians in the 19th century. Therefore Hilbert only asked how one can check whether a given Diophantine equation has integer solutions at all, without being able to state them precisely. However, this problem is still so difficult that it was not until 1970 that Yuri Matiyasevich was able to prove that such a procedure does not exist for the general case. Julia Robinson , Martin Davis and Hilary Putnam did the preparatory work .

When considering the algorithmic solvability, it is sufficient to consider Diophantine equations of the fourth or lower degree, to which the problem can be reduced ( Thoralf Skolem 1934). According to Matyasevich, there is no algorithm for the general Diophantine equation of the fourth degree. The question that remains unsolved is whether there is one for the general cubic equation. For quadratic and linear equations, however, Carl Ludwig Siegel showed in 1972 that such an algorithm exists.

If one looks at the ring of algebraic whole numbers instead of solutions in the whole numbers, there is such an algorithm according to Robert Rumely (1986).

• Martin Davis, Reuben Hersh: Hilbert's tenth problem . Scientific American, Volume 229, Nov. 1973.
• Martin Davis: Hilbert's tenth problem is unsolvable . American Mathematical Monthly, Volume 80, 1973, pp. 233-269.
• Martin Davis, Yuri Matiyasevich, Julia Robinson: Hilbert's tenth problem, Diophantine equations, positive aspects of a negative solution . In: F. Browder: Mathematical Developments Arising from Hilbert's Problems . AMS, Part 2, 1976, pp. 323-378.
• Yuri Matiyasevich: Hilbert's tenth problem . MIT Press, 1996.
• Alexandra Shlapentokh: Hilbert's tenth problem: Diophantine classes and extensions to global fields . Cambridge UP, 2006.

→ Main article : Diophantine equation

Hilbert's eleventh problem

Question: How can the theory of quadratic forms be generalized to any algebraic number fields?

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

A square shape is a function of the shape , where is a vector and a symmetric matrix . By the 19th century, extensive knowledge of quadratic forms had been gained over the rational numbers. Hilbert asked about extensions to any algebraic number fields and any number of variables. In the decades after Hilbert's lecture, numerous results have been published that deal in detail with the topic. The central result is the local-global principle that Helmut Hasse formulated in 1923 (theorem of Hasse-Minkowski). Then follows global solvability (over the field of rational numbers, a global field ) from local (over local fields , the field of p-adic and real numbers) for quadratic forms . Other contributions were made by Ernst Witt (geometric theory of square shapes) and Carl Ludwig Siegel (analytical theory). ${\displaystyle q(x)=x^{T}Ax}$ ${\displaystyle x}$ ${\displaystyle A}$

• Timothy O'Meara : Hilbert's eleventh problem: the arithmetic theory of quadratic forms . In: F. Browder: Mathematical Developments Arising from Hilbert's Problems . AMS, Part 2, 1976, pp. 379-400.

→ Main article : Square shape

Hilbert's twelfth problem

Question: How can Kronecker-Weber's theorem be generalized to any number fields?

Solution: Unknown.

Kronecker-Weber's theorem says that the maximum Abelian expansion of the field of rational numbers is created by the adjunction of all roots of unity (field of circular division). In this case, special values ​​of the exponential function are adjoint to the rational numbers, in general these can also be values ​​of other special functions such as elliptic functions (the connection between extensions of imaginary quadratic number fields and elliptic curves with complex multiplication was the subject of Kronecker's "Jugendtraum"), and one would like an explicit description of these extensions. Hilbert attached great importance to the generalization of this theorem. Although there were many advances in the field in the 20th century (for example, the so-called CM bodies according to Gorō Shimura and Yutaka Taniyama (their monograph appeared in 1961), which are associated with Abelian varieties with complex multiplication), to a solution by Hilberts However, the twelfth problem did not yet arise.

• Robert Langlands : Some contemporary problems with origins in the Jugendtraum (Hilbert's problem 12) . In: F. Browder: Mathematical Developments Arising from Hilbert's Problems . AMS, Part 2, 1976, pp. 401-418 ( online ).
• Norbert Schappacher : On the history of Hilbert's twelfth problem, in: Michele Audin (Hrsg.), Matériaux pour l'histoire des mathématiques au XXe siècle Actes du colloque à la mémoire de Jean Dieudonné (Nice 1996), SMF 1998

Hilbert's thirteenth problem

Question: Can the solution of the equation be constructed with the help of a finite number of continuous functions that depend on two variables? ${\displaystyle x^{7}+ax^{3}+bx^{2}+cx+1=0}$

Solution: yes.

The problem has its roots in the theory of algebraic equations, of which it has been known since Galois and Abel that the solutions of the fifth and higher degree equations cannot be given as a function of the coefficients with the elementary arithmetic operations and radical expressions. Even the reduction to standard forms, for example with Tschirnhaus transformations and the adjunction of further equations in one variable, did not generally bring the desired success. The fifth degree equation could be reduced to a standard form with one parameter, the sixth degree equation to one with two parameters, but the seventh degree equation only succeeded in reducing it to a normal form with three parameters a, b and c:

${\displaystyle f^{7}+a\,f^{3}+b\,f^{2}+c\,f+1=0}$

Hilbert assumed that this could not be reduced to two parameters, not even in the broad class of continuous functions. In this general form, whether there are continuous functions in three variables that cannot be represented as a concatenation of finitely many continuous functions in two variables, Hilbert's conjecture was refuted by Andrei Kolmogorow and Wladimir Arnold in 1957. Kolmogorov first showed that every continuous function of variables can be expressed by those of three variables by superposition, and his student Arnold improved this to two variables. The functions used do not even need to be differentiable and therefore not algebraic either. ${\displaystyle n}$

The conjecture remained open when one considers other classes that include the algebraic functions. In the case of analytical functions, Hilbert had already found in the case of three variables that there are those in three variables that cannot be represented by those in two variables, and Alexander Markowitsch Ostrowski proved in 1920 that those in two variables cannot generally be represented by those in one variable are representable. The question of whether p-times continuously differentiable functions of n variables can be represented by q-times differentiable of m variables was also examined. Vituschkin showed in 1955 that this is generally not possible for. can be understood as a measure of the complexity of p-times differentiable functions in n variables. ${\displaystyle {\frac {m}{q}}<{\frac {n}{p}}}$ ${\displaystyle {\frac {n}{p}}}$

The resolvent problem asks for the minimum k so that the solutions of an algebraic equation of the nth degree can be expressed by superposing algebraic functions of k variables. For is . In a work from 1926, Hilbert assumed that each for and found that . Anders Wiman showed that for true More results achieved Nikolai Chebotaryov , for example for . ${\displaystyle n=5}$ ${\displaystyle k=1}$ ${\displaystyle k=2,3,4}$ ${\displaystyle n=6,7,8}$ ${\displaystyle n=9}$ ${\displaystyle k=4}$ ${\displaystyle n\geq 9}$ ${\displaystyle k\leq n-5}$ ${\displaystyle k\leq n-6}$ ${\displaystyle n\geq 21}$

• George G. Lorentz : The 13th problem of Hilbert . In: F. Browder: Mathematical Developments Arising from Hilbert's Problems . AMS, Part 2, 1976, pp. 419-430.
• Jean-Pierre Kahane : Le 13èmeproblemème de Hilbert: un carrefour de l'algèbre, de l'analyse et de la géométrie . In: Cahiers du seminaire d'histoire des mathématiques . Volume 3, 1982, pp. 1-25 ( online ).
• Anatoly Georgijewitsch Wituschkin : On the thirteenth Hilbert problem . In: P. Alexandrov (ed.): The Hilbert problems . Harri Deutsch, 1998.

Hilbert's fourteenth problem

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

Solution: no.

In the fourteenth problem, Hilbert describes special rings: Let a polynomial ring over a body , a sub-body of the body of the rational functions in variables and be the intersection ${\displaystyle K[x_{1},\ldots ,x_{n}]}$ ${\displaystyle K}$ ${\displaystyle L\subseteq K(x_{1},\ldots ,x_{n})}$ ${\displaystyle n}$ ${\displaystyle R}$

${\displaystyle R:=L\cap K[x_{1},\ldots ,x_{n}].}$

The question then is whether the rings constructed in this way are always finitely generated , i.e. whether there is a finite subset of the ring that generates. ${\displaystyle R}$

The problem originated in the circle of the invariant theory (rings of under the action of certain groups of invariant polynomials) flourishing at the end of the 19th century, in which Hilbert himself had caused a stir in 1890, by proving the finite producibility of the polynomial invariant rings in the case of some classic semi-simple ones Lie groups (like the general and special linear group) and consideration over the complex numbers. In doing so, he used the basic theorem he had proven . This was by Hermann Weyl was later extended to all semi-simple Lie groups. Oscar Zariski formulated the problem in the context of algebraic geometry.

Up until the 1950s, it was possible to prove for some special cases, in particular cases and (Oscar Zariski), that the rings constructed in this way are actually finite. The results therefore suggested that this statement could also apply to all rings of the type described. Therefore, the result of Masayoshi Nagata came as a surprise. In 1957 he gave a counterexample in which this was not the case, and thus solved the problem negatively. ${\displaystyle n=1}$ ${\displaystyle n=2}$

• Masayoshi Nagata: On the 14th problem of Hilbert . American Journal of Mathematics, Volume 81, 1959, pp. 766-772, ISSN  0002-9327 .
• David Mumford : Hilbert's fourteenth problem - the finite generation of subrings such as rings of invariants . In: F. Browder: Mathematical Developments Arising from Hilbert's Problems . AMS, Part 2, 1976, pp. 431-444.

Hilbert's fifteenth problem

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

Solution: Despite advances in the 20th century, the problem cannot be regarded as resolved.

Schubert's counting calculus dates back to the 19th century and concerns intersections of algebraic varieties. It was taken up by the Italian school of algebraic geometry ( Francesco Severi and others), but they used non-strict methods (heuristic arguments for continuity for the invariance of the intersection numbers). With the further development of algebraic geometry in the 20th century, mathematical aids gradually became available with which Hermann Schubert's work could be formalized (including the theory of multiplicities by Alexander Grothendieck , Pierre Samuel , topological work by René Thom , contributions by, among others Steven Kleiman , William Fulton , Robert MacPherson , Michel Demazure ). However, the problem cannot be regarded as solved.

• Steven Kleiman: Rigorous foundation of Schubert's enumerative calculus . In: F. Browder: Mathematical Developments Arising from Hilbert's Problems . AMS, Part 2, 1976, pp. 445-482.

Hilbert's sixteenth problem

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

Solution: Various results could be achieved, but many questions remain open.

Algebraic curves are subsets of the plane that are determined by polynomial equations. These include, for example, the unit circle ( ) or simple straight lines ( ). In 1876 Axel Harnack was able to show that such sets of polynomials of degree (also called curves -th order) can consist of at most parts (connected components) that have the shape of closed curves (ovals) (since the projective plane is possibly considered under Inclusion of the point in infinity). He was also able to construct examples that also achieve this maximum number (“M curves”). ${\displaystyle x^{2}+y^{2}=1}$ ${\displaystyle ax+b=y}$ ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle {\tfrac {1}{2}}(n-1)(n-2)+1}$

Hilbert treated the case using different methods than Harnack in 1891 and found additional configurations that could not be found by Harnack's construction methods. He found that the parts cannot be arranged anywhere in the plane. For example, he assumed that the eleven components of sixth-order M-curves always lie in such a way that nine components are inside a loop and the last component runs outside this loop (or vice versa in the Harnack configuration nine components are outside and one Component in another) and asked in the first part of the sixteenth problem to investigate relationships of this kind. ${\displaystyle n=6}$

This happened with the development of the topology of real algebraic manifolds . Ivan Georgijewitsch Petrowski recognized the role of topological invariants in the problem in the 1930s (and, independently, also Hilbert's student Virginia Ragsdale ) and in 1949, together with Olga Oleinik, he proved inequalities for the problem, in which the Euler characteristic was incorporated. The assumption made by Hilbert for curves of the sixth degree was refuted in 1969 by DA Gudkov in his habilitation thesis after he had thought in his dissertation in 1954 that he had found proof. In his habilitation, his supervisor disliked the fact that the resulting figure of all configurations was not symmetrical and he finally found in the maximum case an additional configuration that Hilbert had missed: five ovals in another and five outside. He completed the classification (except for isotopy) of the non-singular planar algebraic projective curves of degree 6.

Since Hilbert, the procedure for M-curves has consisted of the deformation of non-singular output curves (Hilbert-Rohn-Gudkov method), but required an advanced singularity theory that did not yet exist at Hilbert's time. Gudkov hypothesized that in the case of plane curves of even degrees, the maximum number of ovals applies ( is the number of even ovals, that is, contained in an even number of ovals, and the number of odd ovals). Wladimir Arnold proved a partial result in 1971 ( ) and at the same time formulated the problem in such a way (by complexification and consideration on the Riemann sphere) that the actual topological reason for the restriction of the configurations became clear. Soon Wladimir Abramowitsch Rochlin published a proof of the rest of Gudkov's conjecture, but soon found that it was wrong and so was the conjecture. But he found a generalized version (with a congruence modulo 16 instead of 8) and proved it. Arnold himself and others also proved inequalities (for numeric invariants related to the position of the ovals). The case of the classification of curves of the seventh degree was solved in 1979 by Oleg Viro , so that the case of the classification of planar projective non-singular algebraic curves down to isotopy is solved (with significant advances in the case of M-curves in ), with the simple Cases were resolved as early as the 19th century. ${\displaystyle d=2k}$ ${\displaystyle p-o=k^{2}\,mod\,8}$ ${\displaystyle p}$ ${\displaystyle o}$ ${\displaystyle p-o=k^{2}\,mod\,4}$ ${\displaystyle n\leq 7}$ ${\displaystyle n=8}$ ${\displaystyle n\leq 5}$

Other results mentioned by Hilbert relate to the three-dimensional equivalent of the question: Karl Rohn showed as early as the 19th century that fourth-order algebraic surfaces can consist of a maximum of twelve surfaces. The exact upper limit was not known at the time. VM Kharlamov proved in 1972 that it is 10 and he completed these studies of non-singular quartic surfaces in three dimensions by 1976. The problems explicitly posed by Hilbert were thus solved by the Leningrad School (DA Gudkov, VM Kharlamov, Wladimir Arnold, Wladimir Abramowitsch Rochlin ) finally resolved between 1969 and 1972.

While the first part of Hilbert's 16th problem concerns the plane real algebraic geometry, the second part asks about the existence of an upper bound for the number of limit cycles of plane polynomial dynamic systems and statements about their relative position. The problem remains unsolved and has been added to Stephen Smale's list of math problems. Smale considers the problem to be the most difficult of the Hilbert problems alongside the Riemann conjecture. There has not even been significant progress in solving the problem, and not even for polynomials of degree the upper bound is known. It is only known that the number of limit cycles is finite ( July Sergeyevich Ilyashenko , Jean Écalle after a proof by Henri Dulac from 1923 turned out to be flawed). ${\displaystyle n=2}$

• Oleg Viro: The 16th Hilbert problem, a story of mystery, mistakes and solution . Presentation slides, Uppsala 2007 ( PDF; 2.9 MB ).

Hilbert's seventeenth problem

Question: Can every rational function , which assumes nonnegative values ​​wherever it is defined, be represented as the sum of squares of rational functions?

Solution: yes.

A function with the property that for all (at the points where it is defined, i.e. not diverging) is also referred to as definitive. ${\displaystyle f}$ ${\displaystyle f(x_{1},\cdot \cdot \cdot ,x_{n})\geq 0}$ ${\displaystyle x_{i}=a_{i}\in k}$

For variables, Hilbert himself solved the problem in 1893. ${\displaystyle n=2}$

The general problem was solved in a positive way by Emil Artin in 1927 . The work was the starting point of the theory of formally real bodies and ordered bodies in algebra (see also real closed bodies ), developed by Artin and Otto Schreier . He was also of importance for the development of real algebraic geometry.

Artin proved: If f is a definite rational function over the real, rational or real algebraic numbers (generally a subfield of the real numbers that only allows a single arrangement), then it is a sum of squares of rational functions: ${\displaystyle f=\sum _{i}{g_{i}}^{2}}$

Albrecht Pfister later proved that squares are sufficient for variables . ${\displaystyle n}$ ${\displaystyle 2^{n}}$

• Albrecht Pfister: Hilbert's seventeenth problem and related problems on definite forms . In: F. Browder: Mathematical Developments Arising from Hilbert's Problems. AMS, Part 2, 1976, pp. 507-524.
• N. Jacobson: Lectures on abstract algebra . Volume 3, Van Nostrand 1964, new edition Graduate Texts in Mathematics, Springer (textbook illustration of Artin's results).
• H. Benis-Sinaceur: De D. Hilbert a E. Artin: Les différents aspects du dix-septième Genealogie de Hilbert et les filiations conceptuelles de la théorie des corps réels clos . Arch. Hist. Exact Sci., Vol. 29, 1984, pp. 267-286.

Hilbert's eighteenth problem

Question: Are there only a finite number of essentially different space groups in -dimensional Euclidean space ? ${\displaystyle n}$

Solution: yes.

The first part of Hilbert's eighteenth problem is the mathematical formulation of a question from crystallography . Many solid substances have a crystalline structure at the atomic level, which can be described mathematically with groups of movements. It was shown early on that there are significantly different room groups on level 17 and room 230. Ludwig Bieberbach was finally able to show in 1910 that this number is always finite even in higher dimensions.

In the second part of the problem, Hilbert asks whether there are polyhedra in three-dimensional space that do not appear as the fundamental area of a movement group, but with which the entire space can still be tiled without gaps. Karl Reinhardt was able to show that this is the case for the first time in 1928 by giving an example. The area is an active research area (for example quasicrystals according to Roger Penrose , self-similar fractal tiling by William Thurston).

Finally, Hilbert asks about the most space-saving way of arranging balls in the room. Already in 1611 Johannes Kepler put forward the assumption that the cubic face-centered packing and the hexagonal packing are optimal. This statement, also known as Kepler's conjecture , turned out to be extremely difficult to prove, albeit unsurprisingly. It was not until 1998 that Thomas Hales published a computer-aided proof that has now (2010) been checked and approved. Closest packing of spheres in higher dimensions is still an active research area.

• John Milnor : Hilbert's problem 18: On crystallographic groups, fundamental domains and sphere packing . In: F. Browder: Mathematical Developments Arising from Hilbert's Problems . AMS, Part 2, 1976, pp. 491-506.

Hilbert's nineteenth problem

Question: Are all solutions to regular variation problems analytical ?

Solution: Yes, under certain conditions.

Hilbert found it remarkable that there are partial differential equations (such as the Laplace equation or the minimal area equation) that only allow analytical solutions, i.e. those that can be represented locally by power series . According to Hilbert, they are all related to variation problems (as solutions of the associated Euler-Lagrange equations ) that satisfy certain regularity conditions. Hilbert then formulated the problem as a regularity problem for elliptic partial differential equations with analytic coefficients.

As early as 1903, Sergei Bernstein was able to solve the problem by proving the analyticity of the solutions of a certain class of differential equations, which also include the equations in question, provided that the third derivatives of the solutions exist and are limited. Bernstein treated elliptic partial differential equations of the second order in two variables. Later, among others, Leon Lichtenstein , Eberhard Hopf , Ivan Petrovsky and Charles Morrey were able to generalize the result. A complete solution was then provided by Ennio de Giorgi and John Forbes Nash in the 1950s.

There are several generalizations of the problem by relaxing the constraints on the functionals of the variational problem. From the end of the 1960s, however, Vladimir Gilelewitsch Masja , Ennio de Giorgi and other counterexamples found here.

• Olga Oleinik : On the nineteenth Hilbert problem . In: Pavel S. Alexandrov (ed.): The Hilbert problems . Harri Deutsch, 1998, pp. 275-278.

Hilbert's twentieth problem

Question: Under which conditions do boundary value problems have solutions?

Solution: The existence of a solution cannot be guaranteed in every case by restricting the boundary values.

The twentieth problem is closely related to the nineteenth and is also directly related to physics. One of Hilbert's motivation was his preoccupation with and his rescue of Dirichlet's Principle (1904), the proof of the existence of the solution to a special problem of variation, which Bernhard Riemann used in his work on function theory, which, however, was discredited by Karl Weierstrass's criticism . The variation problem led to the Laplace equation, a special case of elliptic partial differential equations, which he treated as a solution to variation problems in the 19th problem. Here he asks for boundary conditions for the solutions of the partial differential equation that ensure the existence of a solution. The problem turned out to be extremely fruitful and there are extensive results on the subject so that the problem can be considered solved. The first important steps towards the solution came again from Sergei Bernstein around 1910, further advances, among others, from Jean Leray (1939).

• David Gilbarg , Neil Trudinger : Elliptic partial differential equations of the second order . Springer, 3rd edition 1998.
• James Serrin : The problem of Dirichlet for quasilinear elliptic differential equations with many independent variables . Philosophical Transactions of the Royal Society A, Volume 264, 1969, pp. 413-496.
• James Serrin : The Solvability of Boundary Value Problems . In: F. Browder: Mathematical Developments Arising from Hilbert's Problems . AMS, Part 2, 1976, pp. 507-524.
• Enrico Bombieri : Variational problems and elliptic equations (Hilbert's problem 20) . In: F. Browder: Mathematical Developments Arising from Hilbert's Problems . AMS, Part 2, 1976, pp. 525-536.

Hilbert's twenty-first problem

Question: Is there always a system of Fuchsian differential equations for given singularities and a given monodromic group ?

Solution: no.

Fuchs' differential equations are homogeneous linear differential equations of the nth order in the complex (viewed on the Riemann sphere , i.e. with the point at infinity ), in which the singular behavior of the coefficient functions is restricted in a certain way. It can be represented as an equivalent system of linear differential equations of the first order with a matrix of coefficient functions only with poles of the first order. If one continues a locally given solution around the k singular places , one obtains a transformation of the fundamental system of the solutions into itself by an n × n matrix , the monodromy matrix , when returning to the starting point . A homomorphism of the fundamental group of in the general linear group is obtained . The problem is: is there such a system of differential equations for k given singular places and an arbitrary subgroup of as a monodrome matrix? ${\displaystyle {\hat {\mathbb {C} }}}$ ${\displaystyle x_{0}}$ ${\displaystyle n}$ ${\displaystyle {\frac {df_{i}(z)}{dz}}=\sum _{j}A_{ij}(z)f_{j}(z)}$ ${\displaystyle A(z)}$ ${\displaystyle x_{i}}$ ${\displaystyle D}$ ${\displaystyle \pi _{1}}$ ${\displaystyle {\hat {\mathbb {C} }}-\{x_{0},\cdots \ x_{k}\}}$ ${\displaystyle GL(n,\mathbb {C} )}$ ${\displaystyle GL(n,\mathbb {C} )}$

After the question could initially be answered positively for some special cases (including Hilbert himself dealing with the problem and before that Poincaré and Ludwig Schlesinger ) and until the 1980s it was thought that Josip Plemelj had already found the solution in 1908 (in an affirmative sense ), using the theory of Fredholm's integral equations, a loophole was found in his proof in the early 1980s. Plemelj's proof does not apply to all Fuchs systems, but only with so-called regular singular places (polynomial growth of the function around the singular places), because Andrei Bolibruch found a counterexample in 1989. But Bolibruch found that there are such differential equations if one considers irreducible representations of the monodromic group, and classified all Fuchsian systems for which there is a monodromic representation for n = 3.

Various generalizations beyond Fuchs' differential equations were also considered (for example by Helmut Röhrl ). For regular singular points and generalizations of the concept of ordinary linear differential equations, Pierre Deligne succeeded in finding a general positive solution to the problem.

• DV Anosov , AA Bolibruch: Aspects of Mathematics - The Riemann-Hilbert problem. Vieweg, Braunschweig 1994, ISBN 3-528-06496-X .
• Helmut Röhrl : On the twenty-first Hilbert problem . In: Pavel S. Alexandrov (ed.): The Hilbert problems . Harri Deutsch, 1998 (deals with developments up to the 1960s).

Hilbert's twenty-second problem

Question: How can analytical relationships be made uniform by means of automorphic functions ?

Solution: Solved for equations with two variables, with more variables there are still open questions.

It is one of the most famous mathematical problems of the time, and much research was carried out on it in the second half of the 19th century and the beginning of the 20th century. With uniformization , the goal is to parameterize algebraic curves in two variables, i.e. to replace the variables with functions that only depend on one variable. Thus, for example, the unit circle through is given, parameterized by for and each and uses. The uniformization set we were looking for was a generalization of Riemann's mapping theorem on compact Riemann surfaces, and Felix Klein and Poincaré fought for a solution at the end of the 19th century, from which Poincaré initially emerged as the winner. Hilbert's proof was not satisfied. ${\displaystyle x^{2}+y^{2}=1}$ ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle \cos \alpha }$ ${\displaystyle \sin \alpha }$

In 1907 Poincaré and independently Paul Koebe were finally able to solve the problem - but only for the case with two variables. If one generalizes the problem to more than two variables, there are still unanswered questions in the area (part of a program by William Thurston ).

• Lipman Bers : On Hilbert's twenty-second problem . In: F. Browder: Mathematical Developments Arising from Hilbert's Problems . AMS, Part 2, 1976, pp. 559-609.

Hilbert's twenty-third problem

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

Solution: The problem is too vague to give a concrete solution.

The calculus of variations is, in Hilbert's words, “the doctrine of the variation of functions” and was of particular importance in his view. That is why he no longer formulates a specific problem in the last part of his lecture, but called for the further development of this area in general. With the development and extensive expansion of functional analysis , Hilbert's concern was taken into account in the 20th century, also in the area of ​​applications (for example the theory of optimal controls ). Hilbert's own later work on Dirichlet's principle stood at the beginning of the introduction of “direct methods” into the calculus of variations. Overviews of the development in the 20th century come from, among others, Stefan Hildebrandt and Guido Stampacchia .

→ Main article : calculus of variations

"Hilbert's twenty-fourth problem"

Hilbert's 24th problem is a mathematical problem, the formulation of which was found in Hilbert's papers and which is sometimes mentioned as an addition to his list of 23 mathematical problems. Hilbert asks the question of criteria or proofs of whether a proof is the simplest for a mathematical problem.

→ Main article : Hilbert's 24th problem

literature

• David Hilbert: Mathematical Problems . In: News of the Royal Society of Sciences in Göttingen, mathematical-physical class. Issue 3, 1900, pp. 253-297, ISSN  0369-6650 .
• David Hilbert: Sur lesproblemèmes futurs des mathématiques . Compte Rendu du deuxième congrès international des mathématiciens, Paris, Gauthier-Villars, 1902, pp. 58–114 (French translation by Léonce Laugel).
• David Hilbert: Mathematical problems . Bulletin of the American Mathematical Society, Volume 8, 1901, pp. 437-479 (English translation by Mary Newson).
• David Hilbert: Mathematical Problems . Archive of Mathematics and Physics, 3rd series, Volume 1, 1901, pp. 44–63, pp. 213–237.
• David Hilbert: Lecture “Mathematical Problems”. Held at the 2nd International Mathematicians Congress Paris 1900. In: Author collective under the editorship of Pavel S. Aleksandrov : The Hilbert problems (= Ostwald's classic of exact sciences. Vol. 252). 4th edition, reprint of the 3rd, unchanged edition. German, Thun u. a. 1998, ISBN 3-8171-3401-0 .
• Collective of authors under the editorship of Pavel S. Alexandrov : The Hilbert problems (= Ostwald's classic of exact sciences. Vol. 252). 4th edition, reprint of the 3rd, unchanged edition. German, Thun u. a. 1998, ISBN 3-8171-3401-0 .
  • Contributions by AG Vitushkin (13th problem), Evgeni Grigoryevich Skljarenko (EG Skljarenko, 5th problem), Boris Wladimirowitsch Gnedenko (6th problem), Alexander Ossipowitsch Gelfond (7th problem), Alexander Jessenin-Wolpin (1st and 2nd problem) Problem), Wladimir Grigoryevich Boltjanski (3rd problem), Isaak Moissejewitsch Jaglom (4th problem), Juri Linnik (8th problem), DK Fadeev (9th problem), Juri I. Chmelevskij (10th problem), Yuri Manin ( Problem 11, 12, 14, 15, 17), Olga Oleinik (Problem 16, 19), Boris Nikolajewitsch Delone (Problem 18), Alexander Grigorjewitsch Sigalow (AG Sigalov (1913–1969), Problem 19, 20), Helmut Röhrl ( Problem 21), Lev Ernestovich Elsgolc (Problem 23), Boris Wladimirowitsch Schabat (1917–1987, Problem 22), the state of the discussion is in many cases still at the end of the 1960s (the first edition appeared in 1971) with some additions by the German editors for example in solving the tenth problem.
• Felix E. Browder (Ed.): Mathematical Developments Arising from Hilbert's Problems (= Proceedings of Symposia in Pure Mathematics. Vol. 28). 2 volumes. American Mathematical Society, Providence RI 1976, ISBN 0-8218-9315-7 .
• Ivor Grattan-Guinness : A Sideways Look at Hilbert's Twenty-three Problems of 1900 . Notices AMS, August 2000 ( online ).
• Jeremy J. Gray : The Hilbert Challenge. Oxford University Press, Oxford u. a. 2000, ISBN 0-19-850651-1 .
• Jean-Michel Kantor : Hilberts's Problems and their Sequels . Mathematical Intelligencer, Volume 18, 1996, Issue 1, pp. 21-30.
• Rüdiger Thiele : Hilbert and his Twenty-Four Problems. In: Glen van Brummelen, Michael Kinyon (Eds.): Mathematics and the Historians Craft. The Kenneth O. May Lectures (= CMS Books in Mathematics. Vol. 21). Springer, New York NY 2005, ISBN 0-387-25284-3 , pp. 243-296.
• Benjamin H. Yandell: The honors class. Hilbert's problems and their solvers. AK Peters, Natick MA 2001, ISBN 1-56881-141-1 .

See also

• Unsolved problems in mathematics

Web links

Wikisource: Mathematical Problems  - Sources and Full Texts

• Math problems - sources, texts, works, translations, media on Wikilivres (also known as Bibliowiki )
• David Joyce's website on Hilbert's problems with links
• Hilbert's problems in the lexicon of mathematics on Spektrum.de

Individual evidence

1. ^ Ina Kersten : Hilbert's mathematical problems. ( Memento of the original from July 17, 2009 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. Bielefeld University, 2000. 
2. ^ David Hilbert: Mathematical Problems. Lecture given at the international mathematicians' congress in Paris in 1900. ( Memento of the original from April 8, 2012 on WebCite ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. 
3. ^ ^{A b} D. Hilbert: Mathematical Problems. Lecture given at the international mathematicians' congress in Paris 1900 . In: News from the Royal. Society of Sciences at Göttingen. Mathematical-physical class. Issue 3, 1900, pp. 253-297.
4. ↑ Constance Reid, Hilbert-Courant, Springer, 1986, p. 73.
5. ^ Hilbert: Problèmes mathématiques . In: L'enseignement mathématique . Volume 2, 1900, pp. 349-354 ( online ).
6. ↑ In the archive and also in the French version in the congress report published in 1902, he mentions, for example, problem 14, the progress that Adolf Hurwitz achieved in the invariant theory in 1897 (general proof of the finiteness of the invariants in the orthogonal group).
7. ^ Rüdiger Thiele: Hilbert's Twenty-Fourth Problem. (PDF; 197 kB) In: American Mathematical Monthly. Vol. 110, No. 1, January 2003, ISSN  0002-9890 , pp. 1-24.
8. ↑ Grattan Guinness, Notices AMS, August 2000, loc. cit.
9. ^ Charlotte Angas Scott: The International Congress of Mathematicians in Paris . Bulletin AMS, Volume 7, 1900, pp. 57-79. She called (p. 68) the subsequent discussion desultory ("half-hearted").
10. ↑ He referred to Volume 7, No. 1 of the Rivista di Matematica edited by him.
11. ↑ Padoa: Un nouveau système irreductible de postulats pour l'algèbre . ICM 1900. Padoa also went to Le Problem No. 2 de M. David Hilbert . In: L'enseignement mathématique . Volume 5, 1903, pp. 85-91 directly to Hilbert's lecture and his second problem.
12. ↑ Hilbert then added a citation from Maurice d'Ocagne to the printed version in the Archives for Mathematics and Physics . According to Grattan-Guinness: A sideways look at Hilbert's twenty-three problems of 1900 . Notices AMS, August 2000.
13. ↑ Hilbert noted in a letter to Adolf Hurwitz on August 25th that the conference was not very strong, neither in terms of quantity nor in terms of quality, and that Poincaré only attended the congress in obedience to his duties and was absent from the final banquet at which he was to preside. Quoted in Grattan-Guinness, Notices AMS, August 2000, p. 757.
14. ^ Compte Rendu du deuxième congrès international des mathématiciens . Paris, Gauthier-Villars, 1902, p. 24.
15. ^ Paul Cohen: Set theory and the continuum hypothesis . Benjamin 1963.
16. ↑ Dehn: About the volume . Mathematische Annalen, Volume 55, 1901, pp. 465-478. Simplified by WF Kagan : About the transformation of the polyhedron . Mathematische Annalen, Volume 57, 1903, pp. 421-424 and later by Hugo Hadwiger , who extended the Dehn invariant to higher dimensions, and Wladimir Grigorjewitsch Boltjanski .
17. ^ Sydler, Comm. Math. Helv., Volume 40, 1965, pp. 43-80. Simplified by Borge Jessen in Jessen: The algebra of polyhedra and Sydler's theorem . Math. Scand., Vol. 22, 1968, pp. 241-256.
18. ↑ Hilbert: About the straight line as the shortest connection between two points . Mathematische Annalen, Volume 46, 1896, P. 91-96 ( digitized version , SUB Göttingen ), reprinted in Hilbert: Fundamentals of Geometry . Teubner, 2nd edition 1903, p. 83.
19. ↑ Hamel: About the geometries in which the straight lines are the shortest . Mathematische Annalen, Volume 57, 1903, pp. 231-264.
20. ^ IM Jaglom: On the fourth Hilbert problem . In: Pavel S. Alexandrov (ed.): The Hilbert problems . Harri Deutsch, 1998.
21. ↑ Busemann, quoted in Yandell: The Honors Class . P. 138.
22. ↑ Béla Kerékjártó solved the two-dimensional case in 1931, Montgomery in 1948 in three and Montgomery and Zippin in 1952 in four dimensions.
23. ↑ J. Hirschfeld: The nonstandard treatment of Hilbert's fifth problem- . Trans. Amer. Math. Soc., Vol. 321, 1990, pp. 379-400.
24. ^ Serre, quoted from Jeremy Gray: The Hilbert problems 1900–2000 . ( Memento of the original from June 12, 2007 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. 
25. ^ Leo Corry: On the origins of Hilbert's sixth problem: physics and the empiricist approach to axiomatization . International Congress of Mathematicians, 2006.
26. ↑ Matyasevich: Hilbert's tenth problem . MIT Press 1993, p. 16.
27. ^ Siegel: On the theory of square shapes . Messages Ges. Wiss. Göttingen, Math.-Naturwiss. Class, 1972, No. 3, pp. 21-46.
28. ^ Encyclopedia of Mathematics: Local-global principles for the ring of algebraic integers .
29. ^ Encyclopedia of Mathematics: Quadratic forms .
30. ↑ Vitushkin: About higher dimensional variations . Moscow 1955 (Russian). Andrei Kolmogorov gave a simpler proof in the same year.
31. ^ Vitushkin: On the thirteenth Hilbert problem . In: P. Alexandrov (ed.): The Hilbert problems . Harri Deutsch, 1998, p. 211.
32. ↑ Hilbert: About the theory of algebraic forms . Mathematische Annalen, Volume 36, 1890, pp. 473-534.
33. ^ O. Zariski: Interpretations algebrico-geometriques du quatorzieme probleme de Hilbert . Bulletin des Sciences Mathematiques, Volume 78, 1954, pp. 155-168.
34. ^ Nagata: On the fourteenth problem of Hilbert . Proc. ICM 1958. Nagata: Lectures on the fourteenth problem of Hilbert . Tata Institute of Fundamental Research, Bombay 1965.
35. ^ Michael Kantor: Hilbert's problems and their sequels . Mathematical Intelligencer, 1996, No. 1, p. 25.
36. ^ Rokhlin: Congruences modulo sixteen in the sixteenth Hilbert problem . Functional Analysis and Applications, Vol. 6, 1972, pp. 301-306, Part 2, Vol. 7, 1973, pp. 91-92.
37. ↑ Artin: On the decomposition of definite functions into squares . Abh. Math. Seminar Hamburg, Volume 5, 1927, pp. 100-115.
38. ^ Artin, Schreier: Algebraic construction of real bodies . Abh. Math. Seminar Hamburg, Volume 5, 1927, pp. 85-99.
39. ↑ Pfister: To represent definite functions as the sum of squares . Inventiones Mathematicae, Volume 4, 1967, pp. 229-237.
40. ↑ Nicholas Katz : An overview of Deligne's work on Hilbert's twenty-first problem . In: F. Browder: Mathematical Developments Arising from Hilbert's Problems . AMS, Part 2, 1976, pp. 537-585.
41. ^ Deligne: Equations différentiels à points singuliers regulières . Lecture Notes in Mathematics, Springer 1970.
42. ^ Josef Bemelmans, Stefan Hildebrandt, Wolfgang Wahl: Partial differential equations and calculus of variations . In: Gerd Fischer et al: A century of mathematics 1890–1990. Festschrift for the anniversary of the DMV, Vieweg 1990, pp. 149–230.
43. ^ Stampacchia: Hilbert's twenty-third problem: extensions of the calculus of variations . In: F. Browder: Mathematical Developments Arising from Hilbert's Problems . AMS, Part 2, 1976, pp. 611-628.