Hilbert's problems

David Hilbert (1862 - 1943) was a famous German mathematician, recognized as one of the most influential and universal mathematicians in his time.

Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory, calculus of variations, commutative algebra, algebraic number theory, the foundations of geometry, spetral theory of operators and its application to integral equations, mathematical physics, and foundations of mathematics (particularly proof theory).

Hilbert adopted and warmly defended Georg Cantor's set theory and transfinite numbers. A famous example of his leadership in mathematics is his 1900 presentation of a collection of problems that set the course for much of the mathematical research of the 20th century.

https://www.wikiwand.com/en/Hilbert%27s_problems

Hilbert's problems:

1. [✔] The continuum hypothesis. Status: solved in 1940, 1963

2. [✔] Proof that the axioms of arithmetic are consistent. Status: solved in 1931, 1936

3. [✔] Given any two polyhedra of equal volume, is it always possible to cut the first into finitely many polyhedral pieces that can be reassembled to yield the second? Status: solved in 1900

4. [?] Construct all metrics where lines are geodesics. Status: Too vague to be stated resolved or not.

5. [✔] Are continuous groups automatically differential groups? Status:

6. [✔] Mathematical treatment of the axioms of physics. Status:

7. [✔] Is $a^b$ transcendental, for algebraic $a\neq 0,1$ and irrational algebraic $b$? Status:

8. [✔] The Riemann hypothesis Status:

9. [✔] Find the most general law of the reciprocity theorem in any algebraic number field. Status:

10. [✔] Find an algorithm to determine whether a given polynomial Diophantine equation with integer coefficients has an integer solution. Status:

11. [✔] Solving quadratic forms with algebraic numerical coefficients. Status:

12. [✔] Extend the Kronecker–Weber theorem on abelian extensions of the rational numbers to any base number field. Status:

13. [✔] Solve 7-th degree equation using algebraic (variant: continuous) functions of two parameters. Status:

14. [✔] Is the ring of invariants of an algebraic group acting on a polynomial ring always finitely generated? Status:

15. [✔] Rigorous foundation of Schubert's enumerative calculus. Status:

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

17. [✔] Express a nonnegative rational function as quotient of sums of squares. Status:

18. [✔] (a) Is there a polyhedron that admits only an anisohedral tiling in three dimensions? (b) What is the densest sphere packing? Status:

19. [✔] Are the solutions of regular problems in the calculus of variations always necessarily analytic? Status:

20. [✔] Do all variational problems with certain boundary conditions have solutions? Status:

21. [✓] Proof of the existence of linear differential equations having a prescribed monodromic group. Status: Partially resolved.

22. [✔] Uniformization of analytic relations by means of automorphic functions. Status: resolved

23. [?] Further development of the calculus of variations. Status: Too vague to be stated resolved or not.

24. Hilbert's 24th Problem In 1900 Hilbert asked a question which was not given at the Paris conference but which has been recently found in his notes for the list: "Find a criterion of simplicity in mathematics".