View source for Hilbert problems

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At the 1990 International Congress of Mathematicians in Paris, D. Hilbert presented a list of open problems. The published version [[#References|[a18]]] contains 23 problems, though at the meeting Hilbert discussed but <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120080/h1200801.png" /> of them (problems 1, 2, 6, 7, 8, 13, 16, 19, 21, 22). For a translation, see [[#References|[a19]]]. These 23 problems, together with short, mainly bibliographical comments, are briefly listed below, using the short title descriptions from [[#References|[a19]]].

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

===Hilbert's first problem.===
Cantor's problem on the [[Continuum, cardinality of the|cardinal number of the continuum]].

More colloquially also known as the [[Continuum hypothesis|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|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 [[#References|[a1]]], [[#References|[a9]]].

===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; [[#References|[a11]]]) and R. Bricard (1896; [[#References|[a8]]]). The study of this problem led to [[scissors-congruence]] problems, [[#References|[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; [[#References|[a34]]]). See [[Desargues geometry|Desargues geometry]] and [[#References|[a35]]], [[#References|[a47]]]. See also [[Hilbert geometry|Hilbert geometry]]; [[Minkowski 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; [[#References|[a15]]], [[#References|[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|Analytic group]]; [[Topological group|Topological group]]). For a much simplified (but non-standard) treatment, see [[#References|[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 [[#References|[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|Constructive quantum field theory]]; [[Quantum field theory, axioms for|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. [[#References|[a28]]], [[#References|[a41]]], [[#References|[a44]]], [[#References|[a45]]], [[#References|[a50]]], [[#References|[a52]]]. Seeing [[Probability theory|probability theory]] as an important tool in physics, Kolmogorov's axiomatization of probability theory is an important positive contribution (see also [[Probability space|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 method|Gel'fond–Schneider theorem]], 1934; see [[Analytic number theory]]). For the general method, the [[Gel'fond–Baker method]], see e.g. [[#References|[a49]]]. A large part of [[#References|[a14]]] is devoted to Hilbert's seventh problem and related questions.

===Hilbert's eighth problem.===
Problems of prime numbers (cf. also [[Prime number|Prime number]]).

This one is usually known as the Riemann hypothesis (cf. also [[Riemann hypotheses|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|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|Reciprocity laws]]). See also [[Class field theory|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 [[#References|[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 [[#References|[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|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|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|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. [[#References|[a25]]], [[#References|[a27]]]. See also [[#References|[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 [[#References|[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 [[#References|[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|Real algebraic variety]]).

The second part concerns the topology of limit cycles of dynamical systems (see [[Limit cycle|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 [[#References|[a3]]], [[#References|[a22]]], [[#References|[a23]]], [[#References|[a39]]].

===Hilbert's seventeenth problem.===
Expression of definite forms by squares.

Solved by E. Artin (1927, [[#References|[a4]]]; see [[Artin–Schreier theory]]). The study of this problem led to the theory of [[formally real field]]s (see also [[Ordered field|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, [[#References|[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, [[#References|[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, [[#References|[a36]]]), who found an anisohedral tiling in $\mathbb{R}^3$, and H. Heesch (1935, [[#References|[a17]]]), who found a non-convex anisohedral polygon in the plane that admits a periodic monohedral tiling,. There also exists convex anisohedral pentagons, [[#References|[a26]]].

On the other hand, this circle of problems is still is a very lively topic (as of 1998), see [[#References|[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 tiling]]s and [[quasi-crystal]]s. As another example of one of the problems that emerged, it is as yet (1998) unknown which polyominos tile the whole plane, [[#References|[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; [[#References|[a13]]], [[#References|[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 [[#References|[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), [[#References|[a6]]], [[#References|[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|Generalized function]]) and, rather recently (1998) for the non-linear case, generalized function algebras [[#References|[a30]]], [[#References|[a37]]], [[#References|[a38]]]. See also, [[Boundary value problem, complex-variable methods|Boundary value problem, complex-variable methods]]; [[Boundary value problem, elliptic equations|Boundary value problem, elliptic equations]]; [[Boundary value problem, ordinary differential equations|Boundary value problem, ordinary differential equations]]; [[Boundary value problem, partial differential equations|Boundary value problem, partial differential equations]]; [[Boundary value problems in potential theory|Boundary value problems in potential theory]]; [[Plateau problem|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|Fuchsian equation]]; [[#References|[a2]]], [[#References|[a5]]], [[#References|[a12]]]). The problem is also sometimes referred to as the Riemann problem or the Hilbert–Riemann problem (see [[Riemann–Hilbert problem|Riemann–Hilbert problem]]; [[Fuchsian equation|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 [[#References|[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|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|Uniformization]]; [[Discrete group of transformations|Discrete group of transformations]]). For higher (complex) dimension, things are still (1998) largely open and that also holds for a variety of generalizations, [[#References|[a1]]], [[#References|[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|Variational calculus]] for developments in the theory of variational problems as classically understood; see [[Variational calculus in the large|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]]; [[Optimal control, mathematical theory of|Optimal control, mathematical theory of]]; [[Pontryagin maximum principle|Pontryagin maximum principle]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> "Die Hilbertschen Probleme" P.S. Alexandrov (ed.) , Geest&amp;Portig (1979) (In Russian) (New ed.: H. Deutsch, 1998) {{MR|0563859}} {{ZBL|0434.00004}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> D.V. Anosov, A.A. Bolibrukh, "The Riemann–Hilbert problem" , Vieweg (1994) {{MR|1276272}} {{ZBL|0801.34002}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> V.I. Arnol'd, Yu.S. Il'yashenko, "Ordinary differential equations" D.V. Anosov (ed.) V.I. Arnol'd (ed.) , ''Dynamical systems I'' , Springer (1988) pp. 7–148 {{MR|}} {{ZBL|0659.58012}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> E. Artin, "Uber die Zerlegung definiter Funktionen in Quadrate" ''Abh. Math. Sem. Univ. Hamburg'' , '''5''' (1927) pp. 100–115 {{MR|}} {{ZBL|52.0122.01}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> A. Beauville, "Equations différentielles à points singuliers réguliers d'apres Bolybrukh" ''Sem. Bourbaki'' , '''1992/3''' (1993) pp. 103–120</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> S.N. Bernstein, "Sur la nature analytique des solutions des équations aux dérivées parteilles des second ordre" ''Math. Ann.'' , '''59''' (1904) pp. 20–76</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> L. Bieberbach, "Über die Bewegungsgruppen des <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120080/h12008042.png" />-dimensionalen euklidisches Raumes met einem endlichen Fundamentalbereich" ''Gött. Nachr.'' (1910) pp. 75–84</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> R. Bricard, "Sur une question de géométrie relative aux polyèdres" ''Nouv. Ann. Math.'' , '''15''' (1896) pp. 331–334 {{MR|}} {{ZBL|27.0407.02}} </TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> "Mathematical developments arising from Hilbert's problems" F.E. Browder (ed.) , Amer. Math. Soc. (1976) {{MR|419125}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top"> J.H. Conway, N.J.A. Sloane, "Sphere packings, lattices and groups" , Springer (1988) {{MR|0920369}} {{ZBL|0634.52002}} </TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top"> M. Dehn, "Uber den Rauminhalt" ''Math. Ann.'' , '''55''' (1901) pp. 465–478 {{MR|}} {{ZBL|32.0486.01}} </TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top"> P. Deligne, "Equations différentielles à points singuliers réguliers" , Springer (1970) {{MR|0417174}} {{ZBL|0244.14004}} </TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top"> L. Fejes Tóth, "Uber einem geometrischen Satz" ''Math. Z.'' , '''46''' (1940) pp. 79–83</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top"> N.I. Fel'dman, Yu.V. Nestorenko, "Transcendental numbers" , Springer (1998) {{MR|}} {{ZBL|0885.11004}} </TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top"> A.M. Gleason, "Groups without small subgroups" ''Ann. of Math.'' , '''56''' (1952) pp. 193–212 {{MR|0049203}} {{ZBL|0049.30105}} </TD></TR><TR><TD valign="top">[a16]</TD> <TD valign="top"> S.W. Golomb, "Tiling rectangles with polyominoes" ''Math. Intelligencer'' , '''18''' : 2 (1996) pp. 38–47 {{MR|1395089}} {{ZBL|0883.01025}} </TD></TR><TR><TD valign="top">[a17]</TD> <TD valign="top"> H. Heesch, "Aufbau der Ebene aus kongruenten Bereiche" ''Nachr. Ges. Wiss. Göttingen, Neue Ser. 1'' (1935) pp. 115–117</TD></TR><TR><TD valign="top">[a18]</TD> <TD valign="top"> D. Hilbert, "Mathematische Probleme" ''Nachr. K. Ges. Wiss. Göttingen, Math.-Phys. Klasse (Göttinger Nachrichten)'' , '''3''' (1900) pp. 253–297 (Reprint: Archiv Math. Physik 3:1 (1901), 44-63; 213-237; also: Gesammelte Abh., dritter Band, Chelsea, 1965, pp. 290-329) {{MR|}} {{ZBL|31.0068.03}} </TD></TR><TR><TD valign="top">[a19]</TD> <TD valign="top"> D. Hilbert, "Mathematical problems" ''Bull. Amer. Math. Soc.'' , '''8''' (1902) pp. 437–479 (Text on the web: http://aleph0.clarku.edu/djoyce/hilbert) {{MR|1557926}} {{ZBL|33.0976.07}} </TD></TR><TR><TD valign="top">[a20]</TD> <TD valign="top"> J. Hirschfeld, "The nonstandard treatment of Hilbert's fifth problem" ''Trans. Amer. Math. Soc.'' , '''321''' (1990) pp. 379–400 {{MR|967314}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a21]</TD> <TD valign="top"> R.-P. Holzapfel, "The ball and some Hilbert problems" , Birkhäuser (1995) {{MR|1350073}} {{ZBL|0905.14013}} </TD></TR><TR><TD valign="top">[a22]</TD> <TD valign="top"> Yu. Il'yashenko, "Finiteness theorems for limit cycles" , Amer. Math. Soc. (1991) {{MR|}} {{ZBL|0753.34015}} {{ZBL|0743.34036}} {{ZBL|0768.34017}} </TD></TR><TR><TD valign="top">[a23]</TD> <TD valign="top"> "Concerning the Hilbert 16th problem" Yu. Ilyashenko (ed.) S. Yakovenko (ed.) , Amer. Math. Soc. (1995) {{MR|1334338}} {{ZBL|0819.00005}} </TD></TR><TR><TD valign="top">[a24]</TD> <TD valign="top"> J.-M. Kantor, "Hilbert's problems and their sequels" ''Math. Intelligencer'' , '''18''' : 1 (1996) pp. 21–34 {{MR|1381576}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a25]</TD> <TD valign="top"> M.M. Kapranov, "Analogies between the Langlands correspondence and topological quantum field theory" S. Gindikhin (ed.) , ''Funct. Anal. on the Eve of the 21st Centuryy'' , '''I''' , Birkhäuser (1995) pp. 119–151 {{MR|1373001}} {{ZBL|0858.11062}} </TD></TR><TR><TD valign="top">[a26]</TD> <TD valign="top"> R.B. Kershner, "On paving the plane" ''Amer. Math. Monthly'' , '''75''' (1968) pp. 839–844 {{MR|0236822}} {{ZBL|0165.23801}} </TD></TR><TR><TD valign="top">[a27]</TD> <TD valign="top"> A.W. Knapp, "Introduction to the Langlands program" T.N. Bailey (ed.) et al. (ed.) , ''Representation theory and automorphic forms'' , Amer. Math. Soc. (1997) pp. 245–302 {{MR|1476501}} {{ZBL|0903.11015}} </TD></TR><TR><TD valign="top">[a28]</TD> <TD valign="top"> R.J. Lawrence, "An introduction to topological field theory" L.H. Kaufmann (ed.) , ''The interface of knots and physics'' , Amer. Math. Soc. (1996) pp. 89–128 {{MR|1372766}} {{ZBL|0844.57021}} </TD></TR><TR><TD valign="top">[a29]</TD> <TD valign="top"> D. Montgomery, L. Zippin, "Small subgroups of finite dimensional groups" ''Ann. of Math.'' , '''56''' (1952) pp. 213–241 {{MR|0049204}} {{MR|0047059}} {{ZBL|0049.30107}} {{ZBL|0049.30106}} </TD></TR><TR><TD valign="top">[a30]</TD> <TD valign="top"> M. Oberguggenberger, E.E. Rosinger, "Solutions of continuous nonlinear PDE's through order completion. Part I" ''Univ. Pretoria'' (1991)</TD></TR><TR><TD valign="top">[a31]</TD> <TD valign="top"> I.G. Petrovskij, "Sur l'analyticité des solutions des systèmes d'équations différentielles" ''Mat. Sb.'' , '''5''' (1939) pp. 3–70</TD></TR><TR><TD valign="top">[a32]</TD> <TD valign="top"> A. Pfister, "Zur Darstellung definiter Funktionen als Summe von Quadraten" ''Invent. Math.'' , '''4''' (1967) pp. 229–237 {{MR|0222043}} {{ZBL|0222.10022}} </TD></TR><TR><TD valign="top">[a33]</TD> <TD valign="top"> T. Pheidas, "Extensions of Hilbert's tenth problem" ''J. Symbolic Logic'' , '''59''' (1994) pp. 372–397 {{MR|1276621}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a34]</TD> <TD valign="top"> A.V. Pogorelov, "A complete solution of Hilbert's fourth problem" ''Soviet Math. Dokl.'' , '''14''' (1973) pp. 46–49 {{MR|317256}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a35]</TD> <TD valign="top"> A.V. Pogorelov, "Hilbert's fourth problem" , Winston&amp;Wiley (1979) {{MR|550440}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a36]</TD> <TD valign="top"> K. Reinhardt, "Zur Zerlegung Euklische Räume in kongruente Polytope" ''Sitzungsber. Preuss. Akad. Wiss.'' (1928) pp. 150–155</TD></TR><TR><TD valign="top">[a37]</TD> <TD valign="top"> E.E. Rosinger, "Non-linear partial differential equations" , North-Holland (1990) {{MR|1091547}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a38]</TD> <TD valign="top"> E.E. Rosinger, "Parametric Lie group actions on global generalised solutions of nonlinear PDEs" , Kluwer Acad. Publ. (1998) {{MR|1658516}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a39]</TD> <TD valign="top"> R. Roussarie, "Bifurcation of planar vectorfields and Hilbert's sixteenth problem" , Birkhäuser (1998) {{MR|1628014}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a40]</TD> <TD valign="top"> C.-H. Sah, "Hilbert's third problem: scissors congruence" , Pitman (1979) {{MR|554756}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a41]</TD> <TD valign="top"> S. Sawin, "Links, quantum groups, and TQFT's" ''Bull. Amer. Math. Soc.'' , '''33''' : 4 (1996) pp. 413–445 {{MR|1388838}} {{ZBL|0872.57002}} </TD></TR><TR><TD valign="top">[a42]</TD> <TD valign="top"> H.C.H. Schubert, "Kalkül der abzählenden Geometrie" , Teubner (1879) {{MR|0555576}} {{ZBL|0417.51008}} </TD></TR><TR><TD valign="top">[a43]</TD> <TD valign="top"> E. Schulte, "Tilings" P.M. Gruber (ed.) J.M. Wills (ed.) , ''Handbook of convex geometry'' , '''B''' , North-Holland (1993) pp. 899–932 {{MR|1242999}} {{ZBL|0791.52015}} </TD></TR><TR><TD valign="top">[a44]</TD> <TD valign="top"> G. Segal, "The definition of conformal field theory" K. Bleuler (ed.) M. Werner (ed.) , ''Differential Geometrical Methods in Theoretical Physiscs'' , Kluwer Acad. Publ. (1988) pp. 165–171 {{MR|0981378}} {{ZBL|0657.53060}} </TD></TR><TR><TD valign="top">[a45]</TD> <TD valign="top"> G. Segal, "Geometric aspects of quantum field theory" , ''Proc. Internat. Congress Math. Kyoto, 1990'' , '''II''' , Springer (1991) pp. 1387–1396 {{MR|1159323}} {{ZBL|0757.53048}} </TD></TR><TR><TD valign="top">[a46]</TD> <TD valign="top"> I.R. Shafarevich, "General reciprocity laws" ''Mat. Sb.'' , '''26''' (1950) pp. 13–146</TD></TR><TR><TD valign="top">[a47]</TD> <TD valign="top"> Z.I. Szabo, "Hilbert's fourth problem" ''Adv. Math.'' , '''59''' (1986) pp. 185–301</TD></TR><TR><TD valign="top">[a48]</TD> <TD valign="top"> A. Thue, "Uber die dichteste Suzammenstellung von kongruenten Kreisen in einer Ebene" ''Skr. Vidensk–Selsk Christ.'' , '''1''' (1910) pp. 1–9</TD></TR><TR><TD valign="top">[a49]</TD> <TD valign="top"> R. Tijdeman, "The Gel'fond–Baker method" F.E. Browder (ed.) , ''Mathematical developments arising from Hilbert's problems'' , Amer. Math. Soc. (1976) pp. 241–268 {{MR|}} {{ZBL|0341.10026}} {{ZBL|0335.10022}} </TD></TR><TR><TD valign="top">[a50]</TD> <TD valign="top"> V.G. Turaev, "Quantum invariants of knots and 3-manifolds" , W. de Gruyter (1994) pp. Chap. II {{MR|1292673}} {{ZBL|0812.57003}} </TD></TR><TR><TD valign="top">[a51]</TD> <TD valign="top"> A.S. Wightman, "Hilbert's sixth problem" F.E. Browder (ed.) , ''Mathematical developments arising from Hilbert's problems'' , Amer. Math. Soc. (1976) pp. 147–240</TD></TR><TR><TD valign="top">[a52]</TD> <TD valign="top"> E. Witten, "Topological quantum field theory" ''Comm. Math. Phys.'' , '''117''' (1988) pp. 353–386 {{MR|0953828}} {{ZBL|0656.53078}} </TD></TR></table>