Hilbert’s problems

Hilbert’s problems

On the morning of the

8

t

⁢

h

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

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 (

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

23.

Calculus of variations

See also:

•

David Hilbert,

http://www.mathematik.uni-bielefeld.de/ kersten/hilbert/rede.html

Mathematische Probleme

•

David Hilbert,

http://aleph0.clarku.edu/ djoyce/hilbert/problems.html

Mathematical Problems

•

Wikipedia

,

http://en.wikipedia.org/wiki/Hilbert_problems

Hilbert’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

Canonical name

HilbertsProblems

Date of creation

2013-03-22 16:05:40

Last modified on

2013-03-22 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

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