1. Hilbert’s Problems
By Sharjeel Khan

2. David Hilbert Born in Königsberg, Russia Went to University of
Went on to teach at University
of Königsberg
Left Königsberg and went to
University of Göttingen

3. 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

4. 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

5. The 23 Mathematical Problems of Hilbert
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

6. Consequences of Hilbert’s 23 problems

7. 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
International Congress of Mathematicians in Paris in the year 1900

8. 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?
International Congress of Mathematicians in Paris in the year 1900

9. 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

10. 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

11. 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.

12. 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?
International Congress of Mathematicians in Paris in the year 1900

13. The state of the 23 problems
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

15. 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)

16. 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

17. References http://aleph0.clarku.edu/~djoyce/hilbert/problems.html