Hilbert's Problems: Legacy and 23 Mathematical Challenges

1. Hilbert’s Problems By Sharjeel Khan

2. 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

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

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?

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?

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 vsNP 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 • 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