In this post I’m going to talk about the last half of Hilbert’s problems. To read the first part, click here.
Description: This problem asks to extend the Kronecker-Weber Theorem on abelian extensions of the rational numbers to any base number field. The current Kronecker-Weber Theorem does this for the case of any imaginary quadratic field, showing that every algebraic integer whose Galois group is abelian can be expressed as a sum of roots of unity with rational coefficients.
Status: The work done by Shimura and Taniyama in the complex multiplication of abelian varieties gave rise to abelian extensions of CM-fields. Additionally, Stark’s conjecture resulted in a great conjectural development of L-functions, and is capable of producing concrete, numerical results. However, this problem still remains unresolved.
Description: One of the simplest problems to understand, it asks to show whether or not a 7th degree equation can be solved using (continuous) algebraic functions of two parameters (e.g. x and y).
The ‘continuous’ part of the question was added as a later version of this problem.
Status: This problem was solved in 1957 by Vladimir Arnold, based on the work of Andrey Kolmogorov, who showed that several variables can be constructed with a finite number of three-variable functions.
Description: Are certain alegbras finitely generated?
Assume that k is a field and let K be a subfield of the field of rational functions in n variables. Now consider the k-algebra R, which is defined by the intersection:
Hilbert conjectured that all algebras of this type are finitely generated over k.
Status: Although there were results that confirmed his conjecture was true in special cases and for certain classes of rings, in 1959 Masayoshi Nagata found a counterexample, hence disproving the conjecture.
Description: Justify Schubert’s enumerative geometry and calculus by putting it on rigorous foundations. Enumerative geometry has recently emerged as a central element of string theory, making this a very important problem in modern day physics.
Status: This problem is currently only partially resolved.
Description: Describe the relative positions of ovals from a real algebraic curve and as limit cycles of a polynomial vector field on the plane. Limit cycles are isolated closed trajectories. This problem was split into two separate parts (for more details, click here).
Status: In 1991-1992, Yulii Ilyashenko and Jean Écalle showed that every polynomial vector field in the plane has only finitely many limit cycles. However, this problem has still not been solved.
Description: Concerns the expression of definite positive rational functions as the sums of quotients of squares. The problem can be worded as:
“Given a multivariate polynomial that takes only non-negative values over the reals, can it be represented as a sum of squares of rational functions?”
Status: In 1888, Hilbert had already shown that every polynomial in n variables and degree 2d can only be expressed as a sum of squares of other polynomials if and only if n = 2 and d= 1 or n = 3 and 2d = 4. In 1927, Emil Artin, proved that it was true for positive definite functions over the reals. In 1984, an algorithmic solution was found by Charles Delzell.
Description: The 18th problem can be split up into two separate questions:
- Is there a polyhedron that can only be tiled in a anisohedral way in three dimensions?
- What is the densest sphere packing?
Status: The first question was answered by Karl Reinhardt, who found an example of such tile in three-dimensions in 1928. The second part of the problem, which is generally taken to be equivalent to the Kepler conjecture, was proved by Thomas Callister Hales using a computer-aided proof.
Description: Are the solutions of regular problems in the calculus of variations always necessarily analytic? An analytic function is one that can be given by a convergent power series.
Status: This was proved in the affirmative by Ennio de Giorgi, and John Forbes Nash independently and using different methods.
Description: This problems asks whether boundary value problems can be solved. A boundary value problem is a differential equation with a set of additional constraints.
Status: This has been resolved due to a significant amount of research throughout the 20th century, which culminated in solutions for the non-linear case.
Description: Show the existence of a certain class of differential equations which have a specified monodromic group. Monodromy is the study of groups as they move around a singularity.
Status: The answer to this problem was proven to be yes or no, depending on which of the more exact forms of the problem you look at.
Description: This involves the uniformisation of analytic relations using automorphic functions.
Status: Although this problem is currently open, a partial solution has been offered by Koebe. Furthermore, some progress has also been made by Griffith and Bers.
Description: This was not a specific problem, but rather was an encouragement towards the further development of the calculus of variations. The calculus of variations entails maximizing or minimizing functionals, which are mappings from a set of functions to the real numbers.
Status: Mathematicians such as Noether, Tonelli, Lebesgue, Hadamard and Hilbert himself made significant contributions to the calculus of variations following the statement of the problem. Marston Morse applied it to Morse theory and Pontryagin, Rockafellar and Clarke developed new tools for the calculus of variations in optimal control theory. As this is such an open ended question it is hard to tell whether or not it has been resolved.
Hoped you liked this mini two-part series! M x