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Hilbert's problems

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David Hilbert

Hilbert's problems are 23 problems in mathematics published by German mathematician David Hilbert in 1900. They were all unsolved at the time, and several proved to be very influential for 20th-century mathematics. Hilbert presented ten of the problems (1, 2, 6, 7, 8, 13, 16, 19, 21, and 22) at the Paris conference of the International Congress of Mathematicians, speaking on August 8 at the Sorbonne. The complete list of 23 problems was published later, in English translation in 1902 by Mary Frances Winston Newson in the Bulletin of the American Mathematical Society.[1] Earlier publications (in the original German) appeared in Archiv der Mathematik und Physik.[2]

List of Hilbert's Problems

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The following are the headers for Hilbert's 23 problems as they appeared in the 1902 translation in the Bulletin of the American Mathematical Society.[1]

1. Cantor's problem of the cardinal number of the continuum.
2. The compatibility of the arithmetical axioms.
3. The equality of the volumes of two tetrahedra of equal bases and equal altitudes.
4. Problem of the straight line as the shortest distance between two points.
5. Lie's concept of a continuous group of transformations without the assumption of the differentiability of the functions defining the group.
6. Mathematical treatment of the axioms of physics.
7. Irrationality and transcendence of certain numbers.
8. Problems of prime numbers (The "Riemann Hypothesis").
9. Proof of the most general law of reciprocity in any number field.
10. Determination of the solvability of a Diophantine equation.
11. Quadratic forms with any algebraic numerical coefficients
12. Extensions of Kronecker's theorem on Abelian fields to any algebraic realm of rationality
13. Impossibility of the solution of the general equation of 7th degree by means of functions of only two arguments.
14. Proof of the finiteness of certain complete systems of functions.
15. Rigorous foundation of Schubert's enumerative calculus.
16. Problem of the topology of algebraic curves and surfaces.
17. Expression of definite forms by squares.
18. Building up of space from congruent polyhedra.
19. Are the solutions of regular problems in the calculus of variations always necessarily analytic?
20. The general problem of boundary values (Boundary value problems in PD )
21. Proof of the existence of linear differential equations having a prescribed monodromy group.
22. Uniformization of analytic relations by means of automorphic functions.
23. Further development of the methods of the calculus of variations.

Nature and influence of the problems

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Hilbert's problems ranged greatly in topic and precision. Some of them, like the 3rd problem, which was the first to be solved, or the 8th problem (the Riemann hypothesis), which still remains unresolved, were presented precisely enough to enable a clear affirmative or negative answer. For other problems, such as the 5th, experts have traditionally agreed on a single interpretation, and a solution to the accepted interpretation has been given, but closely related unsolved problems exist. Some of Hilbert's statements were not precise enough to specify a particular problem, but were suggestive enough that certain problems of contemporary nature seem to apply; for example, most modern number theorists would probably see the 9th problem as referring to the conjectural Langlands correspondence on representations of the absolute Galois group of a number field.[3] Still other problems, such as the 11th and the 16th, concern what are now flourishing mathematical subdisciplines, like the theories of quadratic forms and real algebraic curves.

There are two problems that are not only unresolved but may in fact be unresolvable by modern standards. The 6th problem concerns the axiomatization of physics, a goal that 20th-century developments seem to render both more remote and less important than in Hilbert's time. Also, the 4th problem concerns the foundations of geometry, in a manner that is now generally judged to be too vague to enable a definitive answer.

The 23rd problem was purposefully set as a general indication by Hilbert to highlight the calculus of variations as an underappreciated and understudied field. In the lecture introducing these problems, Hilbert made the following introductory remark to the 23rd problem:

"So far, I have generally mentioned problems as definite and special as possible, in the opinion that it is just such definite and special problems that attract us the most and from which the most lasting influence is often exerted upon science. Nevertheless, I should like to close with a general problem, namely with the indication of a branch of mathematics repeatedly mentioned in this lecture—which, in spite of the considerable advancement lately given it by Weierstrass, does not receive the general appreciation which, in my opinion, is its due—I mean the calculus of variations."

The other 21 problems have all received significant attention, and late into the 20th century work on these problems was still considered to be of the greatest importance. Paul Cohen received the Fields Medal in 1966 for his work on the first problem, and the negative solution of the tenth problem in 1970 by Yuri Matiyasevich (completing work by Julia Robinson, Hilary Putnam, and Martin Davis) generated similar acclaim. Aspects of these problems are still of great interest today.

Knowability

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Following Gottlob Frege and Bertrand Russell, Hilbert sought to define mathematics logically using the method of formal systems, i.e., finitistic proofs from an agreed-upon set of axioms.[4] One of the main goals of Hilbert's program was a finitistic proof of the consistency of the axioms of arithmetic: that is his second problem.[a]

However, Gödel's second incompleteness theorem gives a precise sense in which such a finitistic proof of the consistency of arithmetic is provably impossible. Hilbert lived for 12 years after Kurt Gödel published his theorem, but does not seem to have written any formal response to Gödel's work.[b][c]

Hilbert's tenth problem does not ask whether there exists an algorithm for deciding the solvability of Diophantine equations, but rather asks for the construction of such an algorithm: "to devise a process according to which it can be determined in a finite number of operations whether the equation is solvable in rational integers". That this problem was solved by showing that there cannot be any such algorithm contradicted Hilbert's philosophy of mathematics.

In discussing his opinion that every mathematical problem should have a solution, Hilbert allows for the possibility that the solution could be a proof that the original problem is impossible.[d] He stated that the point is to know one way or the other what the solution is, and he believed that we always can know this, that in mathematics there is not any "ignorabimus" (statement whose truth can never be known).[e] It seems unclear whether he would have regarded the solution of the tenth problem as an instance of ignorabimus: what is proved not to exist is n

On the other hand, the status of the first and second problems is even more complicated: there is no clear mathematical consensus as to whether the results of Gödel (in the case of the second problem), or Gödel and Cohen (in the case of the first problem) give definitive negative solutions or not, since these solutions apply to a certain formalization of the problems, which is not necessarily the only possible one.[f]

The 24th problem

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Main article: Hilbert's twenty-fourth problem

Hilbert originally included 24 problems on his list, but decided against including one of them in the published list. The "24th problem" (in proof theory, on a criterion for simplicity and general methods) was rediscovered in Hilbert's original manuscript notes by German historian Rüdiger Thiele in 2000.[7]

Follow-ups

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Since 1900, mathematicians and mathematical organizations have announced problem lists but, with few exceptions, these have not had nearly as much influence nor generated as much work as Hilbert's problems.

One exception consists of three conjectures made by André Weil in the late 1940s (the Weil conjectures). In the fields of algebraic geometry, number theory and the links between the two, the Weil conjectures were very important.[8][9] The first of these was proved by Bernard Dwork; a completely different proof of the first two, via ℓ-adic cohomology, was given by Alexander Grothendieck. The last and deepest of the Weil conjectures (an analogue of the Riemann hypothesis) was proved by Pierre Deligne. Both Grothendieck and Deligne were awarded the Fields medal. However, the Weil conjectures were, in their scope, more like a single Hilbert problem, and Weil never intended them as a programme for all mathematics. This is somewhat ironic, since arguably Weil was the mathematician of the 1940s and 1950s who best played the Hilbert role, being conversant with nearly all areas of (theoretical) mathematics and having figured importantly in the development of many of them.

Paul Erdős posed hundreds, if not thousands, of mathematical problems, many of them profound. Erdős often offered monetary rewards; the size of the reward depended on the perceived difficulty of the problem.[10]

The end of the millennium, which was also the centennial of Hilbert's announcement of his problems, provided a natural occasion to propose "a new set of Hilbert problems". Several mathematicians accepted the challenge, notably Fields Medalist Steve Smale, who responded to a request by Vladimir Arnold to propose a list of 18 problems.

At least in the mainstream media, the de facto 21st century analogue of Hilbert's problems is the list of seven Millennium Prize Problems chosen during 2000 by the Clay Mathematics Institute. Unlike the Hilbert problems, where the primary award was the admiration of Hilbert in particular and mathematicians in general, each prize problem includes a million-dollar bounty. As with the Hilbert problems, one of the prize problems (the Poincaré conjecture) was solved relatively soon after the problems were announced.

The Riemann hypothesis is noteworthy for its appearance on the list of Hilbert problems, Smale's list, the list of Millennium Prize Problems, and even the Weil conjectures, in its geometric guise. Although it has been attacked by major mathematicians of our day, many experts believe that it will still be part of unsolved problems lists for many centuries. Hilbert himself declared: "If I were to awaken after having slept for a thousand years, my first question would be: Has the Riemann hypothesis been proved?"[11]

In 2008, DARPA announced its own list of 23 problems that it hoped could lead to major mathematical breakthroughs, "thereby strengthening the scientific and technological capabilities of the DoD".[12][13] The DARPA list also includes a few problems from Hilbert's list, e.g. the Riemann hypothesis.

Summary

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Of the cleanly formulated Hilbert problems, numbers 3, 7, 10, 14, 17, 18, 19, and 20 have resolutions that are accepted by consensus of the mathematical community. Problems 1, 2, 5, 6,[g] 9, 11, 12, 15, 21, and 22 have solutions that have partial acceptance, but there exists some controversy as to whether they resolve the problems.

That leaves 8 (the Riemann hypothesis), 13 and 16[h] unresolved, and 4 and 23 as too vague to ever be described as solved. The withdrawn 24 would also be in this class.

Table of problems

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Hilbert's 23 problems are (for details on the solutions and references, see the articles that are linked to in the first column):

Problem Brief explanation Status Year solved
1st The continuum hypothesis (that is, there is no set whose cardinality is strictly between that of the integers and that of the real numbers) Proven to be impossible to prove or disprove within Zermelo–Fraenkel set theory with or without the axiom of choice (provided Zermelo–Fraenkel set theory is consistent, i.e., it does not contain a contradiction). There is no consensus on whether this is a solution to the problem. 1940, 1963
2nd Prove that the axioms of arithmetic are consistent. There is no consensus on whether results of Gödel and Gentzen give a solution to the problem as stated by Hilbert. Gödel's second incompleteness theorem, proved in 1931, shows that no proof of its consistency can be carried out within arithmetic itself. Gentzen proved in 1936 that the consistency of arithmetic follows from the well-foundedness of the ordinal ε0. 1931, 1936
3rd Given any two polyhedra of equal volume, is it always possible to cut the first into finitely many polyhedral pieces that can be reassembled to yield the second? Resolved. Result: No, proved using Dehn invariants. 1900
4th Construct all metrics where lines are geodesics. Too vague to be stated resolved or not.[i] —
5th Are continuous groups automatically differential groups? Resolved by Andrew Gleason, assuming one interpretation of the original statement. If, however, it is understood as an equivalent of the Hilbert–Smith conjecture, it is still unsolved. 1953?
6th Mathematical treatment of the axioms of physics:

(a) axiomatic treatment of probability with limit theorems for foundation of statistical physics

(b) the rigorous theory of limiting processes "which lead from the atomistic view to the laws of motion of continua"

Unresolved, or partially resolved, depending on how the original statement is interpreted.[14] Items (a) and (b) were two specific problems given by Hilbert in a later explanation.[1] Kolmogorov's axiomatics (1933) is now accepted as standard for the foundations of probability theory. There is some success on the way from the "atomistic view to the laws of motion of continua",[15], but the transition from classical to quantum physics means that there would have to be two axiomatic formulations, with a clear link between them. John von Neumann made an early attempt to place Quantum Mechanics on a rigorous mathematical basis in his book Mathematical Foundations of Quantum Mechanics,[16] but subsequent developments have occurred, further challenging the axiomatic foundations of quantum physics. 1933–2002?
7th Is ab transcendental, for algebraic a ≠ 0,1 and irrational algebraic b ? Resolved. Result: Yes, illustrated by the Gelfond–Schneider theorem. 1934
8th The Riemann hypothesis ("the real part of any non-trivial zero of the Riemann zeta function is 1/2") and other prime-number problems, among them Goldbach's conjecture and the twin prime conjecture Unresolved. —
9th Find the most general law of the reciprocity theorem in any algebraic number field. Partially resolved. Solved by Emil Artin in 1927 for Abelian extensions of the rational numbers during the development of class field theory. The non-abelian case remains unsolved, if one interprets that as meaning non-abelian class field theory. —
10th Find an algorithm to determine whether a given polynomial Diophantine equation with integer coefficients has an integer solution. Resolved. Result: Impossible; Matiyasevich's theorem implies that there is no such algorithm. 1970
11th Solving quadratic forms with algebraic numerical coefficients. Partially resolved.[17] —
12th Extend the Kronecker–Weber theorem on Abelian extensions of the rational numbers to any base number field. Partially resolved.[18] —
13th Solve 7th-degree equation using algebraic (variant: continuous) functions of two parameters. Unresolved. The continuous variant of this problem was solved by Vladimir Arnold in 1957 based on work by Andrei Kolmogorov (see Kolmogorov–Arnold representation theorem), but the algebraic variant is unresolved.[j] —
14th Is the ring of invariants of an algebraic group acting on a polynomial ring always finitely generated? Resolved. Result: No, a counterexample was constructed by Masayoshi Nagata. 1959
15th Rigorous foundation of Schubert's enumerative calculus. Partially resolved.[23] Haibao Duan and Xuezhi Zhao claimed that this problem is actually resolved. —
16th Describe relative positions of ovals originating from a real algebraic curve and as limit cycles of a polynomial vector field on the plane. Unresolved, even for algebraic curves of degree 8. —
17th Express a nonnegative rational function as quotient of sums of squares. Resolved. Result: Yes, due to Emil Artin. Moreover, an upper limit was established for the number of square terms necessary. 1927
18th (a) Are there only finitely many essentially different space groups in n-dimensional Euclidean space? Resolved. Result: Yes (by Ludwig Bieberbach) 1910
(b) Is there a polyhedron that admits only an anisohedral tiling in three dimensions? Resolved. Result: Yes (by Karl Reinhardt). 1928
(c) What is the densest sphere packing? Widely believed to be resolved, by computer-assisted proof (by Thomas Callister Hales). Result: Highest density achieved by close packings, each with density approximately 74%, such as face-centered cubic close packing and hexagonal close packing.[k] 1998
19th Are the solutions of regular problems in the calculus of variations always necessarily analytic? Resolved. Result: Yes, proven by Ennio De Giorgi and, independently and using different methods, by John Forbes Nash. 1957
20th Do all variational problems with certain boundary conditions have solutions? Partially resolved. A significant topic of research throughout the 20th century, resulting in solutions for some cases.[24][25][26] ?
21st Proof of the existence of Fuchsian linear differential equations having a prescribed monodromy group Partially resolved. Result: Yes/no/open, depending on more exact formulations of the problem.[27][28][29][30][31] ?
22nd Uniformization of analytic relations by means of automorphic functions Partially resolved. Uniformization theorem ?
23rd Further development of the calculus of variations Too vague to be stated resolved or not. —

See also

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Notes

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  1. ^ See Nagel and Newman revised by Hofstadter (2001, p. 107),[5] footnote 37: "Moreover, although most specialists in mathematical logic do not question the cogency of [Gentzen's] proof, it is not finitistic in the sense of Hilbert's original stipulations for an absolute proof of consistency." Also see next page: "But these proofs [Gentzen's et al.] cannot be mirrored inside the systems that they concern, and, since they are not finitistic, they do not achieve the proclaimed objectives of Hilbert's original program." Hofstadter rewrote the original (1958) footnote slightly, changing the word "students" to "specialists in mathematical logic". And this point is discussed again on page 109[5] and was not modified there by Hofstadter (p. 108).[5]
  2. ^ Reid reports that upon hearing about "Gödel's work from Bernays, he was 'somewhat angry'. ... At first he was only angry and frustrated, but then he began to try to deal constructively with the problem. ... It was not yet clear just what influence Gödel's work would ultimately have" (p. 198–199).[6] Reid notes that in two papers in 1931 Hilbert proposed a different form of induction called "unendliche Induktion" (p. 199).[6]
  3. ^ Reid's biography of Hilbert, written during the 1960s from interviews and letters, reports that "Godel (who never had any correspondence with Hilbert) feels that Hilbert's scheme for the foundations of mathematics 'remains highly interesting and important in spite of my negative results' (p. 217). Observe the use of present tense – she reports that Gödel and Bernays among others "answered my questions about Hilbert's work in logic and foundations" (p. vii).[6]
  4. ^ This issue that finds its beginnings in the "foundational crisis" of the early 20th century, in particular the controversy about under what circumstances could the Law of Excluded Middle be employed in proofs. See much more at Brouwer–Hilbert controversy.
  5. ^ "This conviction of the solvability of every mathematical problem is a powerful incentive to the worker. We hear within us the perpetual call: There is the problem. Seek its solution. You can find it by pure reason, for in mathematics there is no ignorabimus." (Hilbert, 1902, p. 445)
  6. ^ Nagel, Newman and Hofstadter discuss this issue: "The possibility of constructing a finitistic absolute proof of consistency for a formal system such as Principia Mathematica is not excluded by Gödel's results. ... His argument does not eliminate the possibility ... But no one today appears to have a clear idea of what a finitistic proof would be like that is not capable of being mirrored inside Principia Mathematica (footnote 39, page 109). The authors conclude that the prospect "is most unlikely".[5]
  7. ^ Number 6 is now considered a problem in physics rather than in mathematics.
  8. ^ Some authors consider this problem as too vague to ever be described as solved, although there is still active research on it.
  9. ^ According to Gray, most of the problems have been solved. Some were not defined completely, but enough progress has been made to consider them "solved"; Gray lists the fourth problem as too vague to say whether it has been solved.
  10. ^ It is not difficult to show that the problem has a partial solution within the space of single-valued analytic functions (Raudenbush). Some authors argue that Hilbert intended for a solution within the space of (multi-valued) algebraic functions, thus continuing his own work on algebraic functions and being a question about a possible extension of the Galois theory (see, for example, Abhyankar[19] Vitushkin,[20] Chebotarev,[21] and others). It appears from one of Hilbert's papers[22] that this was his original intention for the problem. The language of Hilbert there is "Existenz von algebraischen Funktionen" ("existence of algebraic functions"). As such, the problem is still unresolved.
  11. ^ Gray also lists the 18th problem as "open" in his 2000 book, because the sphere-packing problem (also known as the Kepler conjecture) was unsolved, but a solution to it has now been claimed.

References

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  1. ^ a b c Hilbert, David (1902). "Mathematical Problems". Bulletin of the American Mathematical Society. 8 (10): 437–479. doi:10.1090/S0002-9904-1902-00923-3.
  2. ^ Hilbert, David (1900). "Mathematische Probleme". Göttinger Nachrichten: 253–297. and Hilbert, David (1901). "[no title cited]". Archiv der Mathematik und Physik. 3. 1: 44–63, 213–237.
  3. ^ Weinstein, Jared (2015-08-25). "Reciprocity laws and Galois representations: recent breakthroughs". Bulletin of the American Mathematical Society. 53 (1). American Mathematical Society (AMS): 1–39. doi:10.1090/bull/1515. ISSN 0273-0979.
  4. ^ van Heijenoort, Jean, ed. (1976) [1966]. From Frege to Gödel: A source book in mathematical logic, 1879–1931 ((pbk.) ed.). Cambridge MA: Harvard University Press. pp. 464ff. ISBN 978-0-674-32449-7. A reliable source of Hilbert's axiomatic system, his comments on them and on the foundational 'crisis' that was on-going at the time (translated into English), appears as Hilbert's 'The Foundations of Mathematics' (1927).
  5. ^ a b c d Nagel, Ernest; Newman, James R.; Hofstadter, Douglas R. (2001). Hofstadter, Douglas R. (ed.). Gödel's proof (Rev. ed.). New York: New York University Press. ISBN 978-0-8147-5816-8.
  6. ^ a b c Reid, Constance (1996). Hilbert. New York, NY: Springer-Verlag. ISBN 978-0387946740.
  7. ^ Thiele, Rüdiger (January 2003). "Hilbert's twenty-fourth problem" (PDF). American Mathematical Monthly. 110: 1–24. doi:10.1080/00029890.2003.11919933. S2CID 123061382.
  8. ^ Weil, André (1949). "Numbers of solutions of equations in finite fields". Bulletin of the American Mathematical Society. 55 (5): 497–508. doi:10.1090/S0002-9904-1949-09219-4. ISSN 0002-9904. MR 0029393.
  9. ^ Browder, Felix E. (1976). Mathematical developments arising from Hilbert problems. Providence: American Mathematical Society. ISBN 0-8218-1428-1. OCLC 2331329.
  10. ^ Chung, Fan R. K.; Graham, Ronald L. (1999-06-01). Erdös on Graphs: his legacy of unsolved problems. Natick, Mass: A K Peters/CRC Press. ISBN 978-1-56881-111-6. OCLC 42809520.
  11. ^ Clawson, Calvin C. (8 December 1999). Mathematical Mysteries: The beauty and magic of numbers. Basic Books. p. 258. ISBN 9780738202594. LCCN 99-066854.
  12. ^ Cooney, Michael (30 September 2008). "The world's 23 toughest math questions". Network World. Retrieved 7 April 2024.
  13. ^ "DARPA Mathematical Challenges". 2008-09-26. Archived from the original on 2019-01-12. Retrieved 2021-03-31.
  14. ^ Corry, L. (1997). "David Hilbert and the axiomatization of physics (1894–1905)". Arch. Hist. Exact Sci. 51 (2): 83–198. doi:10.1007/BF00375141. S2CID 122709777.
  15. ^ Gorban, A. N.; Karlin, I. (2014). "Hilbert's 6th Problem: Exact and approximate hydrodynamic manifolds for kinetic equations". Bulletin of the American Mathematical Society. 51 (2): 186–246. arXiv:1310.0406. doi:10.1090/S0273-0979-2013-01439-3.
  16. ^ Von Neumann, John (2018). Wheeler, Nicholas A. (ed.). Mathematical foundations of quantum mechanics. Translated by Beyer, Robert T. Princeton Oxford: Princeton University Press. ISBN 978-0-691-17856-1.
  17. ^ Hazewinkel, Michiel (2009). Handbook of Algebra. Vol. 6. Elsevier. p. 69. ISBN 978-0080932811.
  18. ^ Houston-Edwards, Kelsey (25 May 2021). "Mathematicians Find Long-Sought Building Blocks for Special Polynomials".
  19. ^ Abhyankar, Shreeram S. (1997). Hilbert's Thirteenth Problem (PDF). Séminaires et Congrès. Vol. 2. Société Mathématique de France.
  20. ^ Vitushkin, Anatoliy G. (2004). "On Hilbert's thirteenth problem and related questions". Russian Mathematical Surveys. 59 (1). Russian Academy of Sciences: 11–25. Bibcode:2004RuMaS..59...11V. doi:10.1070/RM2004v059n01ABEH000698. S2CID 250837749.
  21. ^ Morozov, Vladimir V. (1954). "О некоторых вопросах проблемы резольвент" [On certain questions of the problem of resolvents]. Proceedings of Kazan University (in Russian). 114 (2). Kazan University: 173–187.
  22. ^ Hilbert, David (1927). "Über die Gleichung neunten Grades". Math. Ann. 97: 243–250. doi:10.1007/BF01447867. S2CID 179178089.
  23. ^ Kleiman, S.L.; Laksov, Dan (1972). "Schubert Calculus". American Mathematical Monthly. 79 (10). American Mathematical Society: 1061–1082. doi:10.1080/00029890.1972.11993188. ISSN 0377-9017.
  24. ^ Gilbarg, David; Trudinger, Neil S. (2001-01-12). Elliptic Partial Differential Equations of Second Order. Berlin New York: Springer Science & Business Media. ISBN 978-3-540-41160-4.
  25. ^ Serrin, James (1969-05-08). "The problem of Dirichlet for quasilinear elliptic differential equations with many independent variables". Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences. 264 (1153): 413–496. Bibcode:1969RSPTA.264..413S. doi:10.1098/rsta.1969.0033. ISSN 0080-4614.
  26. ^ Mawhin, Jean (1 January 1999). "Leray-Schauder degree: a half century of extensions and applications". Topological Methods in Nonlinear Analysis. 14 (2). Nicolaus Copernicus University in Toruń, Juliusz Schauder Center for Nonlinear Studies: 195–228. doi:10.12775/TMNA.1999.029. ISSN 1230-3429. Retrieved 8 April 2024.
  27. ^ Plemelj, Josip (1964). Radok., J. R. M. (ed.). Problems in the sense of Riemann and Klein. Interscience Tracts in Pure and Applied Mathematics. Vol. 16. New York-London-Sydney: Interscience Publishers John Wiley & Sons Inc. ISBN 9780470691250. MR 0174815.
  28. ^ Anosov, D. V.; Bolibruch, A. A. (1994). The Riemann-Hilbert problem. Aspects of Mathematics, E22. Braunschweig: Friedr. Vieweg & Sohn. doi:10.1007/978-3-322-92909-9. ISBN 978-3-528-06496-9. MR 1276272.
  29. ^ Bolibrukh, A. A. (1990). "The Riemann-Hilbert problem". Akademiya Nauk SSSR I Moskovskoe Matematicheskoe Obshchestvo. Uspekhi Matematicheskikh Nauk (in Russian). 45 (2): 3–47. Bibcode:1990RuMaS..45Q...1B. doi:10.1070/RM1990v045n02ABEH002350. ISSN 0042-1316. MR 1069347. S2CID 250853546.
  30. ^ Bolibrukh, A.A. (1992). "Sufficient conditions for the positive solvability of the Riemann-Hilbert problem". Matematicheskie Zametki (in Russian). 51 (2): 110–117. doi:10.1007/BF02102113. MR 1165460. S2CID 121743184.
  31. ^ Katz, N.M. (1976). "An Overview of Deligne's work on Hilbert's Twenty-First Problem". Proceedings of Symposia in Pure Mathematics. 28: 537–557. doi:10.1090/pspum/028.2/9904. ISBN 9780821814284.

Further reading

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