At the
1990
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
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].
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
with
algebraic and
algebraic and irrational (cf. also
Algebraic number;
Irrational number).
For instance,
and
.
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
,
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
-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
,
which
remains unsolved, and a
"superposition problem" :
Can every continuous
function in
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
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
be a
field
in between a field
and the field of rational functions
in
variables over
:
.
Is it true that
is finitely generated over
?
The
motivation came from positive answers in a number of important cases where
there is a group,
,
acting on
and
is the field of
-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 precisize
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],
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
,
the
Pfister theorem
says that no more than
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
of isometries of
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
.
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
,
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
have not as yet
(1998)
been classified, even for the plane.
One important theory that emerged is that of Penrose tilings and
quasi-crystals,
see
Penrose tiling.
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
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
,
the face-centred
cubic. The
Leech lattice
is conjecturally the densest packing in
dimensions. The densest lattice packing in dimensions
–
are known. In dimensions
,
,
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
th
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.