Hilbert’s problems

On the morning of the $8^{th}$ of August 1900 at the second International Congress of Mathematicians in Paris, David Hilbertgave a talk on ‘The Problems of Mathematics in the Future’ (‘Sur les problèmes futures des mathématiques’).[GGI] He was invited to give a lecture andgave 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 problemswould be part of the honours class of mathematicians.[GJ]

The 23 problems:
Hilbert’s problemshort description of problemstatus1.Cantor’s continuum hypothesis (http://planetmath.org/ContinuumHypothesis)?2.Consistency of arithmetic axioms3.Polyhedral assembly from polyhedron of equal volumeSolved4.Constructibility of metrics by geodesics5.Existence of topological groups as manifolds that are not differential groups (http://planetmath.org/LieGroup)Solved6.Axiomatization of physicsIn progress–AQFT*,TQFT7.Genfold-Schneider theorem8.Riemann hypothesis9.Algebraic number field reciprocity theorem10.Matiyasevich’s theoremSolved11.Quadratic form solution with algebraic numerical coefficients12.Extension of Kronecker’s theorem to other number fields13.Solution of 7th degree equations with 2-parameter functions14.Proof of finiteness of complete systems of functions15.Schubert’s enumerative calculus16.Problem of the topology of algebraic curves and surfaces (http://planetmath.org/HilbertsSixteenthProblem)Open17.Problem related to quadratic forms (http://planetmath.org/TheoremsOnSumsOfSquares)Solved18.Existence of space-filling polyhedron and densest sphere packing19.Existence of Lagrangian solution that is not analytic20.Solvability of variational problems with boundary conditions21.Existence of linear differential equations with monodromic group22.Uniformization of analytic relations23.Calculus of variations

References

GGI Ivor Grattan-Guinness, A Sideways Look at Hilbert’sTwenty-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 oncethe entry is complete.

单词	HilbertsProblems
释义	Hilbert’s problems On the morning of the $8^{th}$ of August 1900 at the second International Congress of Mathematicians in Paris, David Hilbertgave a talk on ‘The Problems of Mathematics in the Future’ (‘Sur les problèmes futures des mathématiques’).[GGI] He was invited to give a lecture andgave 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 problemswould be part of the honours class of mathematicians.[GJ] The 23 problems: Hilbert’s problemshort description of problemstatus1.Cantor’s continuum hypothesis (http://planetmath.org/ContinuumHypothesis)?2.Consistency of arithmetic axioms3.Polyhedral assembly from polyhedron of equal volumeSolved4.Constructibility of metrics by geodesics5.Existence of topological groups as manifolds that are not differential groups (http://planetmath.org/LieGroup)Solved6.Axiomatization of physicsIn progress–AQFT,TQFT7.Genfold-Schneider theorem8.Riemann hypothesis9.Algebraic number field reciprocity theorem10.Matiyasevich’s theoremSolved11.Quadratic form solution with algebraic numerical coefficients12.Extension of Kronecker’s theorem to other number fields13.Solution of 7th degree equations with 2-parameter functions14.Proof of finiteness of complete systems of functions15.Schubert’s enumerative calculus16.Problem of the topology of algebraic curves and surfaces (http://planetmath.org/HilbertsSixteenthProblem)Open17.Problem related to quadratic forms (http://planetmath.org/TheoremsOnSumsOfSquares)Solved18.Existence of space-filling polyhedron and densest sphere packing19.Existence of Lagrangian solution that is not analytic20.Solvability of variational problems with boundary conditions21.Existence of linear differential equations with monodromic group22.Uniformization of analytic relations23.Calculus of variations See also: • 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’sTwenty-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 oncethe entry is complete.
随便看	7.4 Colimits of n-types 75Connectedness 7.5 Connectedness 76OrthogonalFactorization 7.6 Orthogonal factorization 77Modalities 7.7 Modalities 7HomotopyNtypes 7. Homotopy n-types 810AdditionalResults 8.10 Additional Results 811GettingStarted stochastic integral StochasticIntegrationAsALimitOfRiemannSums stochastic integration as a limit of Riemann sums StochasticIntegrationByParts stochastic integration by parts Stochastic maps StochasticMatrix stochastic matrix StochasticProcess stochastic process StolzCesaroTheorem Stolz-Cesaro theorem StoneSpace