Hilbert problems

2020 Mathematics Subject Classification: Primary: 00-XX [MSN][ZBL]

At the 1900 International Congress of Mathematicians in Paris, D. Hilbert presented a list of open problems. The published version [a18] contains 23 problems, though at the meeting Hilbert discussed but ten of them (problems 1, 2, 6, 7, 8, 13, 16, 19, 21, 22). For a translation, see [a19]. These 23 problems, together with short, mainly bibliographical comments, are briefly listed below, using the short title descriptions from [a19].

Three general references are [a1] (all 23 problems), [a9] (all problems except 1, 3, 16), [a24] (all problems except 4, 9, 14; with special emphasis on developments from 1975–1992).

Hilbert's first problem.

Cantor's problem on the cardinal number of the continuum.

More colloquially also known as the Continuum Hypothesis. Solved by K. Gödel and P.J. Cohen in the (unexpected) sense that the continuum hypothesis is independent of the Zermelo–Frankel axioms. See also Set theory.

Hilbert's second problem.

The compatibility of the arithmetical axioms.

Solved (in a negative sense) by K. Gödel (see Gödel incompleteness theorem). Positive results (using techniques that Hilbert would not have allowed) are due to G. Gentzen (1936) and P.S. Novikov (1941), see [a1], [a9]. See also Formalism, Formal system, Axiomatic method, Consistency, Proof theory, and Unsolvability. For the history of mathematics associated with this problem, see Hilbert 2nd problem and Hilbert program.

Hilbert's third problem.

The equality of the volumes of two tetrahedra of equal bases and equal altitudes.

Solved in the negative sense by Hilbert's student M. Dehn (actually before Hilbert's lecture was delivered, in 1900; [a11]) and R. Bricard (1896; [a8]). The study of this problem led to scissors-congruence problems, [a40], and scissors-congruence invariants, of which the Dehn invariant is one example. See also Equal content and equal shape, figures of.

Hilbert's fourth problem.

The problem of the straight line as the shortest distance between two points.

This problem asks for the construction of all metrics in which the usual lines of projective space (or pieces of them) are geodesics. Final solution by A.V. Pogorelov (1973; [a34]). See Desargues geometry and [a35], [a47]. See also Hilbert geometry; Minkowski geometry.

Hilbert's fifth problem.

Lie's concept of a continuous group of transformations without the assumption of the differentiability of the functions defining the group.

Solved by A.M. Gleason and D. Montgomery and L. Zippin, (1952; [a15], [a29]), in the form of the following theorem: Every locally Euclidean topological group is a Lie group and even a real-analytic group (see also Analytic group; Topological group). For a much simplified (but non-standard) treatment, see [a20].

Hilbert's sixth problem.

mathematical treatment of the axioms of physics.

Very far from solved in any way (1998), though there are (many bits and pieces of) axiom systems that have been investigated in depth. See [a51] for an extensive discussion of Hilbert's own ideas, von Neumann's work and much more. For the Wightman axioms (also called Gårding–Wightman axioms) and the Osterwalder–Schrader axioms of quantum field theory see Constructive quantum field theory; Quantum field theory, axioms for. Currently (1998) there is a great deal of interest and activity in (the axiomatic approach represented by) topological quantum field theory and conformal quantum field theory; see e.g. [a28], [a41], [a44], [a45], [a50], [a52]. Seeing probability theory as an important tool in physics, Kolmogorov's axiomatization of probability theory is an important positive contribution (see also Probability space).

Hilbert's seventh problem.

Irrationality and transcendence of certain numbers.

The numbers in question are of the form $\alpha^\beta$ with $\alpha$ an algebraic number and $\beta$ an algebraic irrational number. For instance, $2^{\sqrt 2}$ and $e^\pi = i^{-2i}$. Solved by A.O. Gel'fond and Th. Schneider (the Gel'fond–Schneider theorem, 1934; see Analytic number theory). For the general method, the Gel'fond–Baker method, see e.g. [a49]. A large part of [a14] is devoted to Hilbert's seventh problem and related questions.

Hilbert's eighth problem.

Problems of prime numbers (cf. also Prime number).

This one is usually known as the Riemann hypothesis (cf. also Riemann hypotheses) and is the most famous and important of the yet (1998) unsolved conjectures in mathematics. Its algebraic-geometric analogue, the Weil conjectures, were settled by P. Deligne (1973). See Zeta-function.

Hilbert's ninth problem.

Proof of the most general law of reciprocity in any number field

Solved by E. Artin (1927; see Reciprocity laws). See also Class field theory, which also is relevant for the 12th problem. The analogous question for function fields was settled by I.R. Shafarevich (the Shafarevich reciprocity law, 1948); see [a46]. All this concerns Abelian field extensions. The matter of reciprocity laws and symbols for non-Abelian field extensions more properly fits into non-Abelian class field theory and the Langlands program, see also below.

Hilbert's tenth problem.

Determination of the solvability of a Diophantine equation.

Solved (in the negative sense) by Yu. Matiyasevich (1970; see Diophantine set; Algorithmic problem). For a discussion of various refinements and extensions, see [a33].

For the ring of algebraic integers there is, contrary to the case of the integers $\mathbb{Z}$, a positive solution to Hilbert's tenth problem; cf. Local-global principles for the ring of algebraic integers.

Hilbert's eleventh problem.

Quadratic forms with any algebraic numerical coefficients.

This asks for the classification of quadratic forms over algebraic number fields. Partially solved. The Hasse–Minkowski theorem (see Quadratic form) reduces the classification of quadratic forms over a global field to that over local fields. This represents the historically first instance of the Hasse principle.

Hilbert's twelfth problem.

Extension of the Kronecker theorem on Abelian fields to any algebraic realm of rationality.

For Abelian extensions of number fields (more generally, global fields and also local fields) this is (more or less) the issue of class field theory. For non-Abelian extensions, i.e. non-Abelian class field theory and the much therewith intertwined Langlands program (Langlands correspondence, Langlands–Weil conjectures, Deligne–Langlands conjecture), see e.g. [a25], [a27]. See also [a21] for two complex variable functions for the explicit generation of class fields.

Hilbert's thirteenth problem.

Impossibility of the solution of the general equation of the $7$-th degree by means of functions of only two variables.

This problem is nowadays (1998) seen as a mixture of two parts: a specific algebraic (or analytic) one concerning equations of degree $7$, which remains unsolved, and a "superposition problem" : Can every continuous function in $n$ variables be written as a superposition of continuous functions of two variables? The latter problem was solved by V.I. Arnol'd and A.N. Kolmogorov (1956–1957; see Composite function): Each continuous function of $n$ variables can be written as a composite (superposition) of continuous functions of two variables. The picture changes drastically if differentiability or analyticity conditions are imposed.

Hilbert's fourteenth problem.

Proof of the finiteness of certain complete systems of functions.

The precise form of the problem is as follows: Let $K$ be a field in between a field $k$ and the field of rational functions $k(x_1,\ldots,x_n)$ in $n$ variables over $k$: $k \subset K \subset k(x_1,\ldots,x_n)$. Is it true that $K \cap k[x_1,\ldots,x_n]$ is finitely generated over $k$? The motivation came from positive answers in a number of important cases where there is a group$G$ acting on $k^n$ and $K$ is the field of $G$-invariant rational functions. A counterexample, precisely in this setting of rings of invariants, was given by M. Nagata (1959). See Invariants, theory of; see also Mumford hypothesis for a large class of invariant-theoretic cases where finite generation is true.

Hilbert's fifteenth problem.

Rigorous foundation of Schubert's enumerative calculus.

The problem is to justify and make precise Schubert's "principle of preservation of numbers" under suitable continuous deformations. It mostly concerns intersection numbers. For instance, to prove rigorously that there are indeed, see [a42], $666841048$ quadric surfaces tangent to nine given quadric surfaces in space. There are a great number of such principles of conservation of numbers in intersection theory and cohomology and differential topology. Indeed, one version of another such idea is often the basis of definitions in singular cases. In spite of a great deal of progress (see [a42]) there remains much to be done to obtain a true enumerative geometry such as Schubert dreamt of.

Hilbert's sixteenth problem.

Problem of the topology of algebraic curves and surfaces.

Even in its original formulation, this problem splits into two parts.

First, the topology of real algebraic varieties. For instance, an algebraic real curve in the projective plane splits up in a number of ovals (topological circles) and the question is which configurations are possible. For degree six this was finally solved by D.A. Gudkov (1970; see Real algebraic variety).

The second part concerns the topology of limit cycles of dynamical systems (see Limit cycle). A first problem here is the Dulac conjecture on the finiteness of the number of limit cycles of vector fields in the plane. For polynomial vector fields this was settled in the positive sense by Yu.S. Il'yashenko (1970). See [a3], [a22], [a23], [a39].

Hilbert's seventeenth problem.

Expression of definite forms by squares.

Solved by E. Artin (1927, [a4]; see Artin–Schreier theory). The study of this problem led to the theory of formally real fields (see also Ordered field). For a definite function on a real irreducible algebraic variety of dimension $d$, the Pfister theorem says that no more than $2^d$ terms are needed to express it as a sum of squares, [a32].

Hilbert's eighteenth problem.

Building up of space from congruent polyhedra.

This problem has three parts (in its original formulation).

a) Show that there are only finitely many types of subgroups of the group $E(n)$ of isometries of $\mathbb{R}^n$ with compact fundamental domain. Solved by L. Bieberbach, (1910, [a7]). The subgroups in question are now called Bieberbach groups, see (the editorial comments to) Space forms.

b) Tiling of space by a single polyhedron which is not a fundamental domain as in a). More generally, also non-periodic tilings of space are considered. A monohedral tiling is a tiling in which all tiles are congruent to one fixed set $T$. If, moreover, the tiling is not one that comes from a fundamental domain of a group of motions, one speaks of an anisohedral tiling. In one sense, b) was settled by K. Reinhardt (1928, [a36]), who found an anisohedral tiling in $\mathbb{R}^3$, and H. Heesch (1935, [a17]), who found a non-convex anisohedral polygon in the plane that admits a periodic monohedral tiling,. There also exists convex anisohedral pentagons, [a26].

On the other hand, this circle of problems is still is a very lively topic (as of 1998), see [a43] for a recent survey. See also (the editorial comments to) Packing; Geometry of numbers.

For instance, the convex polytopes that can give a monohedral tiling of $\mathbb{R}^d$ have not as yet (1998) been classified, even for the plane.

One important theory that emerged is that of Penrose tilings and quasi-crystals. As another example of one of the problems that emerged, it is as yet (1998) unknown which polyominos tile the whole plane, [a16]. (A polyomino is a connected figure obtained by taking $n$ identical unit squares and connecting them along common edges.)

c) Densest packing of spheres. Still (1998) unsolved in general. The densest packing of circles in the plane is the familiar hexagonal one, as proved by A. Thue (1910, completed by L. Fejes-Tóth in 1940; [a13], [a48]). Conjecturally, the densest packing in three-dimensional space is the lattice packing $A_3$, the face-centred cubic. The Leech lattice is conjecturally the densest packing in $24$ dimensions. The densest lattice packing in dimensions $1$ to $8$ are known. In dimensions $10$, $11$ and $13$ there are packings that are denser than any lattice packing. See the standard reference [a10]. See also Voronoi lattice types; Geometry of numbers.

Hilbert's nineteenth problem.

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

This problem links to the twentieth problem through the Euler–Lagrange equation of the variational calculus, see Euler equation. Positive results on the analyticity for non-linear elliptic partial equations were first obtained by S.N. Bernshtein (1903) and, in more or less definite form, by I.G. Petrovskii (1937), [a6], [a31]. See also Elliptic partial differential equation; Boundary value problem, elliptic equations.

Hilbert's twentieth problem.

The general problem of boundary values.

In 1900, the general matter of boundary value problems and generalized solutions to differential equations, as Hilbert wisely specified, was in its very beginning. The amount of work accomplished since is enormous in achievement and volume and includes generalized solution ideas such as distributions (see Generalized function) and, rather recently (1998) for the non-linear case, generalized function algebras [a30], [a37], [a38]. See also, Boundary value problem, complex-variable methods; Boundary value problem, elliptic equations; Boundary value problem, ordinary differential equations; Boundary value problem, partial differential equations; Boundary value problems in potential theory; Plateau problem.

Hilbert's twenty-first problem.

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

Solved by the work of L. Plemelj, G. Birkhoff, I. Lappo-Danilevskij, P. Deligne, and A. Bolibrukh (see Fuchsian equation; [a2], [a5], [a12]). The problem is also sometimes referred to as the Riemann problem or the Hilbert–Riemann problem (see Riemann–Hilbert problem; Fuchsian equation).

The solution is negative or positive depending on how the problem is understood. If extra "apparent singularities" (where the monodromy is trivial) are allowed or if linear differential equations are understood in the generalized sense of connections on non-trivial vector bundles, the solution is positive. If no apparent singularities are permitted and the underlying vector bundle must be trivial, there are counterexamples; see [a5] for a very clear summing up.

Hilbert's twenty-second problem.

Uniformization of analytic relations by means of automorphic functions.

This is the uniformization problem, i.e representing an algebraic or analytic manifold parametrically by single-valued functions. The dimension-one case was solved by H. Poincaré and P. Koebe (1907) in the form of the Koebe general uniformization theorem: A Riemann surface topologically equivalent to a domain in the extended complex plane is also conformally equivalent to such a domain, and the Poincaré-Koebe theorem or Klein–Poincaré uniformization theorem (see Uniformization; Discrete group of transformations). For higher (complex) dimension, things are still (1998) largely open and that also holds for a variety of generalizations, [a1], [a9].

Hilbert's twenty-third problem.

Further development of the methods of the calculus of variations.

Though there were already in 1900 a great many results in the calculus of variations, very much more has been developed since. See Variational calculus for developments in the theory of variational problems as classically understood; see Variational calculus in the large for the global analysis problems that emerged later. For the much related topic of optimal control, see Optimal control; Optimal control, mathematical theory of; Pontryagin maximum principle.

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