Zoghman Mebkhout (born 1949 ) (زغمان مبخوت) is a

French French may refer to: * Something of, from, or related to France ** French language, which originated in France ** French people, a nation and ethnic group ** French cuisine, cooking traditions and practices Arts and media * The French (band), ...

Algeria Algeria, officially the People's Democratic Republic of Algeria, is a country in the Maghreb region of North Africa. It is bordered to Algeria–Tunisia border, the northeast by Tunisia; to Algeria–Libya border, the east by Libya; to Alger ...

n mathematician. He is known for his work in

algebraic analysis Algebraic analysis is an area of mathematics that deals with systems of linear partial differential equations by using sheaf theory and complex analysis to study properties and generalizations of functions such as hyperfunctions and microfuncti ...

geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

and

representation theory Representation theory is a branch of mathematics that studies abstract algebra, abstract algebraic structures by ''representing'' their element (set theory), elements as linear transformations of vector spaces, and studies Module (mathematics), ...

, more precisely on the theory of ''D''-modules.

Career

Mebkhout is currently a research director at the

French National Centre for Scientific Research The French National Centre for Scientific Research (, , CNRS) is the French state research organisation and is the largest fundamental science agency in Europe. In 2016, it employed 31,637 staff, including 11,137 tenured researchers, 13,415 engi ...

and in 2002 Zoghman received the Servant Medal from the CNRS a prize given every two years with an amount of €10,000.

Notable works

In September 1979 Mebkhout presented the

Riemann–Hilbert correspondence In mathematics, the term Riemann–Hilbert correspondence refers to the correspondence between regular singular flat connections on algebraic vector bundles and representations of the fundamental group, and more generally to one of several generali ...

, which is a generalization of

Hilbert's twenty-first problem The twenty-first problem of the 23 Hilbert problems, from the celebrated list put forth in 1900 by David Hilbert, concerns the existence of a certain class of linear differential equations with specified singular points and monodromic group. S ...

to higher dimensions. The original setting was for

Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...

s, where it was about the existence of regular differential equations with prescribed

monodromy group In mathematics, monodromy is the study of how objects from mathematical analysis, algebraic topology, algebraic geometry and differential geometry behave as they "run round" a singularity. As the name implies, the fundamental meaning of ''mono ...

s. In higher dimensions, Riemann surfaces are replaced by

complex manifold In differential geometry and complex geometry, a complex manifold is a manifold with a ''complex structure'', that is an atlas (topology), atlas of chart (topology), charts to the open unit disc in the complex coordinate space \mathbb^n, such th ...

s of dimension > 1. Certain systems of

partial differential equation In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives. The function is often thought of as an "unknown" that solves the equation, similar to ho ...

s (linear and having very special properties for their solutions) and possible monodromies of their solutions correspond. An independent proof of this result was presented by

Masaki Kashiwara is a Japanese mathematician and professor at the Kyoto University Institute for Advanced Study (KUIAS). He is known for his contributions to algebraic analysis, microlocal analysis, ''D''-module theory, Hodge theory, sheaf theory and represent ...

in April 1980. Zoghman is now largely known as a specialist in ''D''-modules theory.Alexander Grothendieck
"''Récoltes et sémailles'', ''Réflexions et témoignage sur un passé de mathématicien.''"
Zoghman Mebkhout

Recognition

Zoghman is one of the first modern international-caliber North-African mathematicians. A symposium in Spain was held on his sixtieth birthday. He was invited to the

Institute for Advanced Study The Institute for Advanced Study (IAS) is an independent center for theoretical research and intellectual inquiry located in Princeton, New Jersey. It has served as the academic home of internationally preeminent scholars, including Albert Ein ...

and gave a recent talk at Institut Fourier. In his quasi-autobiographical text Récoltes et semailles

Alexander Grothendieck Alexander Grothendieck, later Alexandre Grothendieck in French (; ; ; 28 March 1928 – 13 November 2014), was a German-born French mathematician who became the leading figure in the creation of modern algebraic geometry. His research ext ...

wrote extensively about what he for a time thought of as gross mistreatment of Mebkhout, in particular in the context of attribution of credit for the formulation and proof of the Riemann-Hilbert correspondence. However, in May 1986, after being contacted by a number of mathematicians involved in the matter, Grothendieck retracted his former viewpoints (that had been based on direct testimony of Mebkhout) in a number of additions to the manuscript. They were not included in the first edition of the book by

Éditions Gallimard Éditions Gallimard (), formerly Éditions de la Nouvelle Revue Française (1911–1919) and Librairie Gallimard (1919–1961), is one of the leading French book publishers. In 2003, it and its subsidiaries published 1,418 titles. Founded by G ...

, but were added in the later "augmented" edition.

References

{{DEFAULTSORT:Mebkhout, Zoghman Differential equations Representation theory 1949 births Algerian mathematicians Living people People from Naâma Province 21st-century Algerian people