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 Hilbert1. The continuum hypothesis

2. Prove that the axioms of arithmetic are consistent.

3. 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?

4. Construct all metrics where lines are geodesics.

5. Are continuous groups automatically differential groups?

6. Mathematical treatment of the axioms of physics

7. Is a b transcendental, for algebraic a ≠ 0,1 and irrational algebraic b ?

8. 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

9. Find the most general law of the reciprocity theorem in any algebraic number field.

10. Find an algorithm to determine whether a given polynomial Diophantine equation with integer coefficients has an integer solution.

11. Solving quadratic forms with algebraic numerical coefficients.

12. Extend the Kronecker–Weber theorem on abelian extensions of the rational numbers to any base number field.

13. Solve 7-th degree equation using continuous functions of two parameters.

14. Is the ring of invariants of an algebraic group acting on a polynomial ring always finitely generated?

15. Rigorous foundation of Schubert's enumerative calculus.

16. Describe relative positions of ovals originating from a real algebraic curve and as limit cycles of a polynomial vector field on the plane.

17. Express a nonnegative rational function as quotient of sums of squares.

18. (a) Is there a polyhedron which admits only an anisohedral tiling in three dimensions?(b) What is the densest sphere packing?

19. Are the solutions of regular problems in the calculus of variations always necessarily analytic?

20. Do all variational problems with certain boundary conditions have solutions?

21. Proof of the existence of linear differential equations having a prescribed monodromic group

22. Uniformization of analytic relations by means of automorphic functions

23. 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• 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

Why give this

talk?

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• 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

Were they all solved?

The state of the 23 problems1. The continuum hypothesis (1963)

2. Prove that the axioms of arithmetic are consistent.

3. 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)

4. Construct all metrics where lines are geodesics.

5. Are continuous groups automatically differential groups?

6. Mathematical treatment of the axioms of physics

7. Is a b transcendental, for algebraic a ≠ 0,1 and irrational algebraic b ? (1935)

8. 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

9. Find the most general law of the reciprocity theorem in any algebraic number field.

10. Find an algorithm to determine whether a given polynomial Diophantine equation with integer coefficients has an integer solution. (1970)

11. Solving quadratic forms with algebraic numerical coefficients.

12. Extend the Kronecker–Weber theorem on abelian extensions of the rational numbers to any base number field.

13. Solve 7-th degree equation using continuous functions of two parameters.

14. Is the ring of invariants of an algebraic group acting on a polynomial ring always finitely generated? (1959)

15. Rigorous foundation of Schubert's enumerative calculus.

16. Describe relative positions of ovals originating from a real algebraic curve and as limit cycles of a polynomial vector field on the plane.

17. Express a nonnegative rational function as quotient of sums of squares. (1927)

18. (a) Is there a polyhedron which admits only an anisohedral tiling in three dimensions? (1928)(b) What is the densest sphere packing? (1928)

19. Are the solutions of regular problems in the calculus of variations always necessarily analytic? (1957)

20. Do all variational problems with certain boundary conditions have solutions?

21. Proof of the existence of linear differential equations having a prescribed monodromic group

22. Uniformization of analytic relations by means of automorphic functions

23. 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/problems.html

• http://mathworld.wolfram.com/HilbertsProblems.html

• http://www-history.mcs.st-andrews.ac.uk/Biographies/Hilbert.html

• http://en.wikipedia.org/wiki/Hilbert%27s_problems