# 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. Genfold-Schneider 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 (http://planetmath.org/HilbertsSixteenthProblem) Open 17. Problem related to quadratic forms (http://planetmath.org/TheoremsOnSumsOfSquares) 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

• David Hilbert, http://www.mathematik.uni-bielefeld.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 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:

• 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.

• *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 HilbertsProblems 2013-03-22 16:05:40 2013-03-22 16:05:40 Daume (40) Daume (40) 18 Daume (40) Feature msc 01A67 msc 01A60 DehnsTheorem