Hilbert's twenty-first problem
Encyclopedia
The twenty-first problem of the 23 Hilbert problems, from the celebrated list put forth in 1900 by David Hilbert
, was phrased like this (English translation from 1902).
In fact it is more appropriate to speak not about differential equations but about linear systems of differential equations: in order to realise any monodromy by a differential equation one has to admit, in general, the presence of additional apparent singularities, i.e. singularities with trivial local monodromy. In more modern language, the (systems of) differential equations in question are those defined in the complex plane
, less a few points, and with a regular singularity at those. A more strict version of the problem requires these singularities to be Fuchsian, i.e. poles of first order (logarithmic poles). A monodromy group is prescribed, by means of a finite-dimensional complex representation
of the fundamental group
of the complement in the Riemann sphere
of those points, plus the point at infinity, up to equivalence. The fundamental group is actually a free group
, on 'circuits' going once round each missing point, starting and ending at a given base point. The question is whether the mapping from these Fuchsian equations to classes of representations is surjective.
This problem is more commonly called the Riemann–Hilbert problem. There is now a modern (D-module
and derived category
) version, the 'Riemann–Hilbert correspondence
' in all dimensions. The history of proofs involving a single complex variable is complicated. Josip Plemelj
published a solution in 1908. This work was for a long time accepted as a definitive solution; there was work of G. D. Birkhoff in 1913 also, but the whole area, including work of Ludwig Schlesinger
on isomonodromic deformations that would much later be revived in connection with soliton theory, went out of fashion. wrote a monograph summing up his work. A few years later the Soviet mathematician Yuliy S. Il'yashenko and others started raising doubts about Plemelj's work. In fact, Plemelj correctly proves that any monodromy group can be realised by a regular linear system which is Fuchsian at all but one of the singular points. Plemelj's claim that the system can be made Fuchsian at the last point as well is wrong. (Il'yashenko has shown that if one of the monodromy operators is diagonalizable, then Plemelj's claim is true.)
Indeed found a counterexample to Plemelj's statement.
This is commonly viewed as providing a counterexample to the precise question Hilbert had in mind;
Bolibrukh showed that for a given pole configuration certain monodromy groups can be realised by regular, but not by Fuchsian systems. (In 1990 he published the thorough study of the case of regular systems of size 3 exhibiting all situations when such counterexamples exists. In 1978 Dekkers had shown that for systems of size 2 Plemelj's claim is true.) Parallel to this the Grothendieck school of algebraic geometry had become interested in questions of 'integrable connections on algebraic varieties', generalising the theory of linear differential equations on Riemann surface
s. Pierre Deligne
proved a precise Riemann–Hilbert correspondence in this general context (a major point being to say what 'Fuchsian' means). With work by Rohrl, the case in one complex dimension was again covered.
David Hilbert
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...
, was phrased like this (English translation from 1902).
- Proof of the existence of linear differential equations having a prescribed monodromic group
- In the theory of linear differential equationLinear differential equationLinear differential equations are of the formwhere the differential operator L is a linear operator, y is the unknown function , and the right hand side ƒ is a given function of the same nature as y...
s with one independent variable z, I wish to indicate an important problem one which very likely Riemann himself may have had in mind. This problem is as follows: To show that there always exists a linear differential equation of the Fuchsian class, with given singular pointsMathematical singularityIn mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability...
and monodromic group. The problem requires the production of n functions of the variable z, regular throughout the complex z-plane except at the given singular points; at these points the functions may become infinite of only finite order, and when z describes circuits about these points the functions shall undergo the prescribed linear substitutions. The existence of such differential equations has been shown to be probable by counting the constants, but the rigorous proof has been obtained up to this time only in the particular case where the fundamental equations of the given substitutions have roots all of absolute magnitude unity. L. Schlesinger has given this proof, based upon PoincaréHenri PoincaréJules Henri Poincaré was a French mathematician, theoretical physicist, engineer, and a philosopher of science...
's theory of the Fuchsian zeta-functions. The theory of linear differential equations would evidently have a more finished appearance if the problem here sketched could be disposed of by some perfectly general method. http://aleph0.clarku.edu/~djoyce/hilbert/problems.html
In fact it is more appropriate to speak not about differential equations but about linear systems of differential equations: in order to realise any monodromy by a differential equation one has to admit, in general, the presence of additional apparent singularities, i.e. singularities with trivial local monodromy. In more modern language, the (systems of) differential equations in question are those defined in the complex plane
Complex plane
In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis...
, less a few points, and with a regular singularity at those. A more strict version of the problem requires these singularities to be Fuchsian, i.e. poles of first order (logarithmic poles). A monodromy group is prescribed, by means of a finite-dimensional complex representation
Complex representation
The term complex representation has slightly different meanings in mathematics and physics.In mathematics, a complex representation is a group representationof a group on a complex vector space....
of the fundamental group
Fundamental group
In mathematics, more specifically algebraic topology, the fundamental group is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other...
of the complement in the Riemann sphere
Riemann sphere
In mathematics, the Riemann sphere , named after the 19th century mathematician Bernhard Riemann, is the sphere obtained from the complex plane by adding a point at infinity...
of those points, plus the point at infinity, up to equivalence. The fundamental group is actually a free group
Free group
In mathematics, a group G is called free if there is a subset S of G such that any element of G can be written in one and only one way as a product of finitely many elements of S and their inverses...
, on 'circuits' going once round each missing point, starting and ending at a given base point. The question is whether the mapping from these Fuchsian equations to classes of representations is surjective.
This problem is more commonly called the Riemann–Hilbert problem. There is now a modern (D-module
D-module
In mathematics, a D-module is a module over a ring D of differential operators. The major interest of such D-modules is as an approach to the theory of linear partial differential equations...
and derived category
Derived category
In mathematics, the derived category D of an abelian category C is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on C...
) version, the 'Riemann–Hilbert correspondence
Riemann–Hilbert correspondence
In mathematics, the Riemann–Hilbert correspondence is a generalization of Hilbert's twenty-first problem to higher dimensions. The original setting was for Riemann surfaces, where it was about the existence of regular differential equations with prescribed monodromy groups...
' in all dimensions. The history of proofs involving a single complex variable is complicated. Josip Plemelj
Josip Plemelj
Josip Plemelj was a Slovene mathematician, whose main contributions were to the theory of analytic functions and the application of integral equations to potential theory.- Life :...
published a solution in 1908. This work was for a long time accepted as a definitive solution; there was work of G. D. Birkhoff in 1913 also, but the whole area, including work of Ludwig Schlesinger
Ludwig Schlesinger
Ludwig Schlesinger was a German mathematician known for the research in the field of linear differential equations.-Biography:...
on isomonodromic deformations that would much later be revived in connection with soliton theory, went out of fashion. wrote a monograph summing up his work. A few years later the Soviet mathematician Yuliy S. Il'yashenko and others started raising doubts about Plemelj's work. In fact, Plemelj correctly proves that any monodromy group can be realised by a regular linear system which is Fuchsian at all but one of the singular points. Plemelj's claim that the system can be made Fuchsian at the last point as well is wrong. (Il'yashenko has shown that if one of the monodromy operators is diagonalizable, then Plemelj's claim is true.)
Indeed found a counterexample to Plemelj's statement.
This is commonly viewed as providing a counterexample to the precise question Hilbert had in mind;
Bolibrukh showed that for a given pole configuration certain monodromy groups can be realised by regular, but not by Fuchsian systems. (In 1990 he published the thorough study of the case of regular systems of size 3 exhibiting all situations when such counterexamples exists. In 1978 Dekkers had shown that for systems of size 2 Plemelj's claim is true.) Parallel to this the Grothendieck school of algebraic geometry had become interested in questions of 'integrable connections on algebraic varieties', generalising the theory of linear differential equations on Riemann surface
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as "deformed versions" of the complex plane: locally near every point they look like patches of the...
s. Pierre Deligne
Pierre Deligne
- See also :* Deligne conjecture* Deligne–Mumford moduli space of curves* Deligne–Mumford stacks* Deligne cohomology* Fourier–Deligne transform* Langlands–Deligne local constant- External links :...
proved a precise Riemann–Hilbert correspondence in this general context (a major point being to say what 'Fuchsian' means). With work by Rohrl, the case in one complex dimension was again covered.