Hilbert’s Problems
By Sharjeel Khan
David Hilbert
• Born in Königsberg, Russia
• Went to University of
Königsberg
• Went on to teach at University
of Königsberg
• Left Königsberg and went to
University of Göttingen
David Hilbert’s Legacy
• One of the most influential mathematicians of the
19th and 20th century
• His famous Basis Theorem in Invariant Theory
• His famous 21 axioms in the axiomation of
geometry
• The Hilbert’s Space
• Hilbert’s Program
• 23 mathematical problems
David Hilbert’s Legacy
• One of the most influential mathematicians of the
19th and 20th century
• His famous Basis Theorem in Invariant Theory
• His famous 21 axioms in the axiomation of
geometry
• The Hilbert’s Space
• 23 mathematical problems
The 23 Mathematical Problems of Hilbert
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
The continuum hypothesis
Prove that the axioms of arithmetic are consistent.
Given any two polyhedra of equal volume, is it always possible to cut the first into finitely many polyhedral pieces
which can be reassembled to yield the second?
Construct all metrics where lines are geodesics.
Are continuous groups automatically differential groups?
Mathematical treatment of the axioms of physics
Is a b transcendental, for algebraic a ≠ 0,1 and irrational algebraic b ?
The Riemann hypothesis ("the real part of any non-trivial zero of the Riemann zeta function is ½") and other prime
number problems, among them Goldbach's conjecture and the twin prime conjecture
Find the most general law of the reciprocity theorem in any algebraic number field.
Find an algorithm to determine whether a given polynomial Diophantine equation with integer coefficients has an
integer solution.
Solving quadratic forms with algebraic numerical coefficients.
Extend the Kronecker–Weber theorem on abelian extensions of the rational numbers to any base number field.
Solve 7-th degree equation using continuous functions of two parameters.
Is the ring of invariants of an algebraic group acting on a polynomial ring always finitely generated?
Rigorous foundation of Schubert's enumerative calculus.
Describe relative positions of ovals originating from a real algebraic curve and as limit cycles of a polynomial vector
field on the plane.
Express a nonnegative rational function as quotient of sums of squares.
(a) Is there a polyhedron which admits only an anisohedral tiling in three dimensions?
(b) What is the densest sphere packing?
Are the solutions of regular problems in the calculus of variations always necessarily analytic?
Do all variational problems with certain boundary conditions have solutions?
Proof of the existence of linear differential equations having a prescribed monodromic group
Uniformization of analytic relations by means of automorphic functions
Further development of the calculus of variations
Consequences of Hilbert’s 23 problems
Famous Hilbert Talk in 1900
• Presented the 23 problems for research
• Talked about mathematics history
• Showed expectations for the future of
mathematics
• Showed the necessity why to solve these
problems
• Gave a challenge and hope to mathematician
through the 23 problems
• Became the most influential speech in the study
of mathematics
Famous Hilbert Talk in 1900
• Presented the 23 problems for research
Why
• Talked about mathematics history
give this
• Showed expectations for the future of
talk?
mathematics
• Showed the necessity why to solve these
problems
• Gave a challenge and hope to mathematician
through the 23 problems
• Became the most influential speech in the study
of mathematics
Mathematics right before the Talk
• Mathematics in 19th century was growing
• New theories were being found and many
contributions were being made
• The limits of mathematics were being
explored
• Foundations of mathematics became a
problem with the rise of mathematical logic
• Hilbert’s questions gave more incentive for
people to explore these limits
Hilbert’s 1st Problem (The Continuum
Hypothesis)
• It was created by the creator of Set Theory, Georg
Cantor in 1878
• Hilbert used this hypothesis and agreed that it makes
sense so he wanted someone to prove it right
• Cardinality is same for two sets if they have a bijection
• Cardinality of integers and natural numbers is
countable and it is aleph zero
• Cardinality of real numbers is uncountable and it is
2^(aleph zero)
• Question was is there another set of infinite between
integers and real numbers such that aleph_0 < |S| <
2^aleph_0
Hilbert’s tenth Problem (Diophantine
Equation)
• Given a Diopantine equation, is it solvable in
rational integers. Example: 15x^2 + 23z +45y = 0
• It took 21 years and 4 people to help solve this
question where it was solved in 1970
• It was Yuri Matiyasevich who came up with
solution.
• The answer ended up being that it was impossible.
Famous Hilbert Talk in 1900
• Presented the 23 problems for research
Were
• Talked about mathematics history
they all
solved?
• Showed expectations for the future of
mathematics
• Showed the necessity why to solve these
problems
• Gave a challenge and hope to mathematician
through the 23 problems
• Became the most influential speech in the study
of mathematics
The state of the 23 problems
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
The continuum hypothesis (1963)
Prove that the axioms of arithmetic are consistent.
Given any two polyhedra of equal volume, is it always possible to cut the first into finitely many polyhedral pieces which can be reassembled to
yield the second? (1900)
Construct all metrics where lines are geodesics.
Are continuous groups automatically differential groups?
Mathematical treatment of the axioms of physics
Is a b transcendental, for algebraic a ≠ 0,1 and irrational algebraic b ? (1935)
The Riemann hypothesis ("the real part of any non-trivial zero of the Riemann zeta function is ½") and other prime number problems, among them
Goldbach's conjecture and the twin prime conjecture
Find the most general law of the reciprocity theorem in any algebraic number field.
Find an algorithm to determine whether a given polynomial Diophantine equation with integer coefficients has an integer solution. (1970)
Solving quadratic forms with algebraic numerical coefficients.
Extend the Kronecker–Weber theorem on abelian extensions of the rational numbers to any base number field.
Solve 7-th degree equation using continuous functions of two parameters.
Is the ring of invariants of an algebraic group acting on a polynomial ring always finitely generated? (1959)
Rigorous foundation of Schubert's enumerative calculus.
Describe relative positions of ovals originating from a real algebraic curve and as limit cycles of a polynomial vector field on the plane.
Express a nonnegative rational function as quotient of sums of squares. (1927)
(a) Is there a polyhedron which admits only an anisohedral tiling in three dimensions? (1928)
(b) What is the densest sphere packing? (1928)
Are the solutions of regular problems in the calculus of variations always necessarily analytic? (1957)
Do all variational problems with certain boundary conditions have solutions?
Proof of the existence of linear differential equations having a prescribed monodromic group
Uniformization of analytic relations by means of automorphic functions
Further development of the calculus of variations
Too Vague
Effects of the 23 problems
• Each solved problem became a huge thing as it
further the mathematics field.
• Motivated all mathematicians
• Most questions were solved in the 20th century
(less than 100 years after the talk)
• Inspired people to challenge the general public to
solve problems like Millennium Problems (which
includes P vs NP problem)
Secret Hilbert’s 24th Problem
• This problem was not included in the original
list but was in Hilbert’s notes
• It was discovered by German historian Rudiger
Thiele in 2000, 100 years after the talk.
• Asks for criterion of simplicity in mathematical
proofs and development of proof theory
References
• http://aleph0.clarku.edu/~djoyce/hilbert/proble
ms.html
• http://mathworld.wolfram.com/HilbertsProble
ms.html
• http://www-history.mcs.standrews.ac.uk/Biographies/Hilbert.html
• http://en.wikipedia.org/wiki/Hilbert%27s_prob
lems