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

Algeria ) , image_map = Algeria (centered orthographic projection).svg , map_caption = , image_map2 = , capital = Algiers , coordinates = , largest_city = capital , relig ...

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 microfunctio ...

geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

and

representation theory Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...

, 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 (french: link=no, Centre national de la recherche scientifique, CNRS) is the French state research organisation and is the largest fundamental science agency in Europe. In 2016, it employed 31,637 ...

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 generaliz ...

, 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 an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic. The term complex manifold is variously used to mean a com ...

s of dimension > 1. Certain systems of

partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function. The function is often thought of as an "unknown" to be sol ...

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. He was a student of Mikio Sato at the University of Tokyo. Kashiwara made leading contributions towards algebraic analysis, microlocal analysis, D-module, ''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.''"

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), located in Princeton, New Jersey, in the United States, is an independent center for theoretical research and intellectual inquiry. It has served as the academic home of internationally preeminent scholar ...

and gave a recent talk at Institut Fourier. Alexander Grothendieck said that Mebkhout's name was hidden and his role neglected in an operation headed by

Pierre Deligne Pierre René, Viscount Deligne (; born 3 October 1944) is a Belgian mathematician. He is best known for work on the Weil conjectures, leading to a complete proof in 1973. He is the winner of the 2013 Abel Prize, 2008 Wolf Prize, 1988 Crafoord Pr ...

in the Luminy congress in June 1981. He calls it "a great disgrace of the mathematical world of this century" and is one of the reasons of Grothendieck's departure from mathematics.

References

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