Eugenio Elia Levi was an Italian mathematician

Mathematician

A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

, known for his fundamental contributions in group theory

Group theory

In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...

, in the theory of partial differential operators and in the theory of functions of several complex variables

Several complex variables

The theory of functions of several complex variables is the branch of mathematics dealing with functionson the space Cn of n-tuples of complex numbers...

: he was the younger brother of Beppo Levi

Beppo Levi

Beppo Levi was an Italian mathematician. He published high-level academic articles and books not only in mathematics, but also in physics, history, philosophy, and pedagogy. Levi was a member of the Bologna Academy of Sciences and of the Accademia dei Lincei.- Early years :Beppo Levi was born on...

 and died in the First World War.

Research activity

He wrote 33 papers, classified by his colleague and friend Mauro Picone

Mauro Picone

Mauro Picone was an Italian mathematician. He is known for the Picone identity, for the Sturm-Picone comparison theorem and for being the founder of the Istituto Nazionale per le Applicazioni del Calcolo, presently named after him...

 according to the scheme reproduced in this section.

Group theory

He wrote only three papers in group theory

Group theory

In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...

: in the first one, discovered what is now called Levi decomposition

Levi decomposition

In Lie theory and representation theory, the Levi decomposition, conjectured by Killing and Cartan and proved by , states that any finite dimensional real Lie algebra g is the semidirect product of a solvable ideal and a semisimple subalgebra....

, which was conjecture

Conjecture

A conjecture is a proposition that is unproven but is thought to be true and has not been disproven. Karl Popper pioneered the use of the term "conjecture" in scientific philosophy. Conjecture is contrasted by hypothesis , which is a testable statement based on accepted grounds...

d by Wilhelm Killing

Wilhelm Killing

Wilhelm Karl Joseph Killing was a German mathematician who made important contributions to the theories of Lie algebras, Lie groups, and non-Euclidean geometry....

 and proved by Élie Cartan

Élie Cartan

Élie Joseph Cartan was an influential French mathematician, who did fundamental work in the theory of Lie groups and their geometric applications...

 in a special case.

Function theory

In the theory of functions of several complex variables he introduced the concept of pseudoconvexity during his investigations on the domain of existence of such functions: it turned out to be one of the key concepts of the theory.

Boundary value problems

His researches in the theory of partial differential operators lead to the method of the parametrix

Parametrix

In mathematics, and specifically the field of partial differential equations , a parametrix is an approximation to a fundamental solution of a PDE, and is essentially an approximate inverse to a differential operator....

, which is basically a way to construct fundamental solution

Fundamental solution

In mathematics, a fundamental solution for a linear partial differential operator L is a formulation in the language of distribution theory of the older idea of a Green's function...

s for elliptic partial differential operators with variable coefficients: the parametrix is widely used in the theory of pseudodifferential operators.

See also

• Pseudoconvexity
• Levi decomposition
  Levi decomposition
  In Lie theory and representation theory, the Levi decomposition, conjectured by Killing and Cartan and proved by , states that any finite dimensional real Lie algebra g is the semidirect product of a solvable ideal and a semisimple subalgebra....
  
  
• Parametrix
  Parametrix
  In mathematics, and specifically the field of partial differential equations , a parametrix is an approximation to a fundamental solution of a PDE, and is essentially an approximate inverse to a differential operator....
  
  
• Several complex variables
  Several complex variables
  The theory of functions of several complex variables is the branch of mathematics dealing with functionson the space Cn of n-tuples of complex numbers...
  
  

External links

. (in Italian

Italian language

Italian is a Romance language spoken mainly in Europe: Italy, Switzerland, San Marino, Vatican City, by minorities in Malta, Monaco, Croatia, Slovenia, France, Libya, Eritrea, and Somalia, and by immigrant communities in the Americas and Australia...

). The biographical entry about Eugenio Elia Levi in the "Dizionario Biografico degli Italiani (Biographical Dictionary of Italians)" section of the Enciclopedia Treccani.. Available from the Edizione Nazionale Mathematica Italiana.