Hilbert’s problems
On the morning of the ${8}^{th}$ 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^{} (http://planetmath.org/ContinuumHypothesis)
?
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 (http://planetmath.org/LieGroup)
Solved
6.
Axiomatization of physics
In progress–AQFT*,TQFT
7.
GenfoldSchneider 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 2parameter 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 (http://planetmath.org/HilbertsSixteenthProblem)
Open
17.
Problem related to quadratic forms (http://planetmath.org/TheoremsOnSumsOfSquares)
Solved
18.
Existence of spacefilling 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:

•
David Hilbert, http://www.mathematik.unibielefeld.de/ kersten/hilbert/rede.htmlMathematische Probleme

•
David Hilbert, http://aleph0.clarku.edu/ djoyce/hilbert/problems.htmlMathematical Problems

•
Wikipedia, http://en.wikipedia.org/wiki/Hilbert_problemsHilbert’s problems
References
 GGI Ivor GrattanGuinness, A Sideways Look at Hilbert’s Twentythree Problems of 1900, Notices of the AMS, Vol 47, 7, 2000.
 GJ Jeremy Gray, The Hilbert problems, European Mathematical Society, Newsletter 36, 1012, 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:

•
have a good introduction,

•
have a small discription of each problem, and as attached entry have more details on each problem separately,

•
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^{}.
Title  Hilbert’s problems 

Canonical name  HilbertsProblems 
Date of creation  20130322 16:05:40 
Last modified on  20130322 16:05:40 
Owner  Daume (40) 
Last modified by  Daume (40) 
Numerical id  18 
Author  Daume (40) 
Entry type  Feature 
Classification  msc 01A67 
Classification  msc 01A60 
Related topic  DehnsTheorem 