Hilbert's problems
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Hilbert's problems are a list of 23 problems in mathematics put forth by German mathematician David Hilbert in the Paris conference of the International Congress of Mathematicians in 1900. The problems were all unsolved at the time, and several of them turned out to be very influential for twentieth-century mathematics. At this conference he presented 10 of the problems (1, 2, 6, 7, 8, 13, 16, 19, 21, and 22) and the list was published later.
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Status of the problems
Hilbert's 23 problems are:
| Problem 1 | solved1 | The continuum hypothesis |
| Problem 2 | solved | Are the axioms of arithmetic consistent? |
| Problem 3 | solved | Can two tetrahedra be proved to have equal volume (under certain assumptions)? |
| Problem 4 | too vague2 | Construct all metrics where lines are geodesics |
| Problem 5 | solved | Are continuous groups automatically differential groups? |
| Problem 6 | non-mathematical | Axiomatize all of physics |
| Problem 7 | solved | Is a b transcendental, for algebraic a ≠ 0,1 and irrational algebraic b ? |
| Problem 8 | open4 | The Riemann hypothesis and Goldbach's conjecture |
| Problem 9 | partially solved5 | Find most general law of reciprocity in any algebraic number field |
| Problem 10 | solved | Determination of the solvability of a diophantine equation |
| Problem 11 | solved | Quadratic forms with algebraic numerical coefficients |
| Problem 12 | solved | Algebraic number field extensions |
| Problem 13 | solved | Solve all 7-th degree equations using functions of two arguments |
| Problem 14 | solved | Proof of the finiteness of certain complete systems of functions |
| Problem 15 | solved | Rigorous foundation of Schubert's enumerative calculus |
| Problem 16 | open | Topology of algebraic curves and surfaces |
| Problem 17 | solved | Expression of definite rational function as quotient of sums of squares |
| Problem 18 | solved3 | Is there a non-regular, space-filling polyhedron? What's the densest sphere packing? |
| Problem 19 | solved | Are the solutions of Lagrangians always analytic? |
| Problem 20 | solved | Do all variational problems with certain boundary conditions have solutions? |
| Problem 21 | solved | Proof of the existence of linear differential equations having a prescribed monodromic group |
| Problem 22 | solved | Uniformization of analytic relations by means of automorphic functions |
| Problem 23 | solved | Further development of the calculus of variations |
Footnotes
- Cohen's independence result, showing the CH to be independent of ZFC (Zermelo-Frankel set theory, extended to include the axiom of choice) is often cited to justify the assertion that the first problem has been solved, although it may be the case that set theory should have additional axioms that are capable of settling the problem.
- According to Rowe & Gray (see reference below), most of the problems have been solved. Some were not completely defined, but enough progress has been made to consider them "solved"; Rowe & Gray lists the fourth problem as too vague to say whether it has been solved.
- Rowe & Gray also list the 18th problem as "open" in their 2000 book, because the sphere-packing problem (also known as the Kepler conjecture) was unsolved, but a solution to it has now been claimed (see reference below). Advances were made on problem 16 as recently as the 1990s.
- Problem 8 contains two famous problems, both of which remain unsolved. The first of them, the Riemann hypothesis, is one of the seven Millennium Prize Problems, which were intended to be the "Hilbert Problems" of the 21st century.
- Problem 9 has been solved in the Abelian case; the non-abelian case remains unsolved
The 24th problem
In preparing the problems Hilbert had 24 problems listed, but decided against one of the problems. The 24th problem was in proof theory on a criterion for simplicity and general methods. Discovery of this problem is due to Rüdiger Thiele.
External links
- Listing of the 23 problems, with descriptions of which have been solved (http://www.mathacademy.com/pr/prime/articles/hilbert_prob/index.asp?PRE=hilber&TAL=Y&TAN=Y&TBI=Y&TCA=Y&TCS=Y&TEC=Y&TFO=Y&TGE=Y&TNT=Y&TPH=Y&TST=Y&TTO=Y&TTR=Y&TAD=)
- Original text of Hilbert's talk, in German (http://www.mathematik.uni-bielefeld.de/~kersten/hilbert/rede.html)
- English translation of Hilbert's 1900 address (http://aleph0.clarku.edu/~djoyce/hilbert/toc.html)
- Details on the solution of the 18th problem (http://www.math.pitt.edu/articles/hilbert.html)
- The Mathematical Gazette, March 2000 (page 2-8) "100 Years On" (http://www.mathsyear2000.org/resources/mathassoc/Maths_Gazette.pdf)
- "On Hilbert's 24th Problem: Report on a New Source and Some Remarks." (http://www.ams.org/amsmtgs/2025_abstracts/962-01-285.pdf)
References
- Rowe, David; Gray, Jeremy J. (2000). The Hilbert Challenge. Oxford University Press. ISBN 0198506511
- Yandell, Benjamin H. (2002). The Honors Class. Hilbert's Problems and Their Solvers. A K Peters. ISBN 1568811411bg:Хилбертови проблеми
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