Hilbert's problems
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Hilbert’s problems
On the morning of the of August 1900 at the second International Congress of Mathematicians in Paris, David Hilbert gave a talk on ‘The Problems of Mathematics in the Future’ (‘Sur les problèmes futures des mathématiques’).[GGI] He was invited to give a lecture and gave 10 problems (from the 23 known Hilbert’s problems) they were (1,2,6,8,12,13,16,19,21,22).[GGI] The entire 23 problems where published after the conference in Archiv der Mathematik und Physik. Hermann Weyl, one of Hilbert’s students, later on stated that any one who solved one of the 23 problems would be part of the honours class of mathematicians.[GJ]
The 23 problems:
Hilbert’s problem
short description of problem
status
1.
Cantor’s continuum hypothesis
?
2.
Consistency of arithmetic axioms
3.
Polyhedral assembly from polyhedron of equal volume
Solved
4.
Constructibility of metrics by geodesics
5.
Existence of topological groups as manifolds that are not differential groups
Solved
6.
Axiomatization of physics
In progress–AQFT*,TQFT
7.
Genfold-Schneider theorem
8.
Riemann hypothesis
9.
Algebraic number field reciprocity theorem
10.
Matiyasevich’s theorem
Solved
11.
Quadratic form solution with algebraic numerical coefficients
12.
Extension of Kronecker’s theorem to other number fields
13.
Solution of 7th degree equations with 2-parameter functions
14.
Proof of finiteness of complete systems of functions
15.
Schubert’s enumerative calculus
16.
Problem of the topology of algebraic curves and surfaces
Open
17.
Problem related to quadratic forms
Solved
18.
Existence of space-filling polyhedron and densest sphere packing
19.
Existence of Lagrangian solution that is not analytic
20.
Solvability of variational problems with boundary conditions
21.
Existence of linear differential equations with monodromic group
22.
Uniformization of analytic relations
23.
Calculus of variations
See also:
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David Hilbert, Mathematische Probleme
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David Hilbert, Mathematical Problems
References
- GGI Ivor Grattan-Guinness, A Sideways Look at Hilbert’s Twenty-three Problems of 1900, Notices of the AMS, Vol 47, 7, 2000.
- GJ Jeremy Gray, The Hilbert problems, European Mathematical Society, Newsletter 36, 10-12, 2000.
- BF Felix E. Browder (ed.), Mathematical Problems Arising from Hilbert problems, Proceedings of Symposia in Pure Mathematics Vol. XXVII, Part I and Part II, American Mathematical Society, 1976.
- YB Benjamin H. Yandell, The Honors Class: Hilbert’s problems and their solvers, A K Peters, 2002.
Notes:
This entry is under construction please feel free to add information as it editable by anyone who is a member. Please reference what is added, thank you.
The idea, is maybe:
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have a good introduction,
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have a small discription of each problem, and as attached entry have more details on each problem separately,
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have a good bibliography.
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*AQFT = Algebraic, or Axiomatic Quantum Field Theory
Also I think we should not CC wikipedia. This note can be removed once the entry is complete.
Mathematics Subject Classification
01A67 Future prospectives01A60 20th century
