Eugenio Elia Levi (18 October 1883 – 28 October 1917) was an

Italia Italy ( it, Italia ), officially the Italian Republic, ) or the Republic of Italy, is a country in Southern Europe. It is located in the middle of the Mediterranean Sea, and its territory largely coincides with the Italy (geographical region) ...

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...

, known for his fundamental contributions in

group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...

, in the theory of partial differential operators and in the theory of functions of several complex variables. He was a younger brother of

Beppo Levi Beppo Levi (14 May 1875 – 28 August 1961) was an Italian mathematician. He published high-level academic articles and books, not only on mathematics, but also on physics, history, philosophy, and pedagogy. Levi was a member of the Bologna Aca ...

and was killed in action during

First World War World War I (28 July 1914 11 November 1918), often abbreviated as WWI, was one of the deadliest global conflicts in history. Belligerents included much of Europe, the Russian Empire, the United States, and the Ottoman Empire, with fightin ...

Work

Research activity

He wrote 33 papers, classified by his colleague and friend Mauro Picone according to the scheme reproduced in this section.

Differential geometry

Group theory

He wrote only three papers in

: in the first one, discovered what is now called Levi decomposition, which was conjectured by

Wilhelm Killing Wilhelm Karl Joseph Killing (10 May 1847 – 11 February 1923) was a German mathematician who made important contributions to the theories of Lie algebras, Lie groups, and non-Euclidean geometry. Life Killing studied at the University of Mü ...

and proved by

Élie Cartan Élie Joseph Cartan (; 9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups, differential systems (coordinate-free geometric formulation of PDEs), and differential geometr ...

in a special case.

Function theory

In the theory of functions of several complex variables he introduced the concept of

pseudoconvexity In mathematics, more precisely in the theory of functions of several complex variables, a pseudoconvex set is a special type of open set in the ''n''-dimensional complex space C''n''. Pseudoconvex sets are important, as they allow for classificat ...

during his investigations on the domain of existence of such functions: it turned out to be one of the key concepts of the theory.

Cauchy and Goursat problems

Boundary value problems

His researches in the theory of partial differential operators lead to the method of the parametrix, which is basically a way to construct

fundamental solution In mathematics, a fundamental solution for a linear partial differential operator is a formulation in the language of distribution theory of the older idea of a Green's function (although unlike Green's functions, fundamental solutions do not a ...

s for

elliptic In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of ellipse in ...

partial differential operator In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and retur ...

s with variable coefficients: the parametrix is widely used in the theory of

pseudodifferential operator In mathematical analysis a pseudo-differential operator is an extension of the concept of differential operator. Pseudo-differential operators are used extensively in the theory of partial differential equations and quantum field theory, e.g. in m ...

Calculus of variations

Publications

The full scientific production of Eugenio Elia Levi is collected in reference . *, reprinted also in , volume I. A a well-known

memoir A memoir (; , ) is any nonfiction narrative writing based in the author's personal memories. The assertions made in the work are thus understood to be factual. While memoir has historically been defined as a subcategory of biography or autobiog ...

Group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...

: it was presented to the members of the Accademia delle Scienze di Torino during the session of April 2, 1905, by

Luigi Bianchi Luigi Bianchi (18 January 1856 – 6 June 1928) was an Italian mathematician. He was born in Parma, Emilia-Romagna, and died in Pisa. He was a leading member of the vigorous geometric school which flourished in Italy during the later years of th ...

. *. A short note announcing the results of paper . *. An important paper whose results were previously announced in the short note with the same title. It was also translated in Russian by N. D. Ajzenstat, currently available from the

All-Russian Mathematical Portal The All-Russian Mathematical Portal (better known as Math-Net.Ru) is a web portal that provides extensive access to all aspects of Russian mathematics, including journals, organizations, conferences, articles, videos, libraries, software, and people ...

: . *. An important paper in the theory of functions of several complex variables, where the problem of determining what kind of

hypersurface In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface. A hypersurface is a manifold or an algebraic variety of dimension , which is embedded in an ambient space of dimension , generally a Euclidea ...

can be the

boundary Boundary or Boundaries may refer to: * Border, in political geography Entertainment * ''Boundaries'' (2016 film), a 2016 Canadian film * ''Boundaries'' (2018 film), a 2018 American-Canadian road trip film *Boundary (cricket), the edge of the pla ...

of a

domain of holomorphy In mathematics, in the theory of functions of several complex variables, a domain of holomorphy is a domain which is maximal in the sense that there exists a holomorphic function on this domain which cannot be extended to a bigger domain. For ...

. *. Another important paper in the theory of functions of several complex variables, investigating further the theory started in . *. His "''Collected works''" in two volumes, collecting all the mathematical papers of Eugenio Elia Levi in a revised typographical form, both amending

typographical error A typographical error (often shortened to typo), also called a misprint, is a mistake (such as a spelling mistake) made in the typing of printed (or electronic) material. Historically, this referred to mistakes in manual type-setting (typography) ...

s and author's oversights. A collection of all his published papers (in their original typographical form), probably an unordered uncorrected collection of offprints, is available online at the

Internet Archive The Internet Archive is an American digital library with the stated mission of "universal access to all knowledge". It provides free public access to collections of digitized materials, including websites, software applications/games, music, ...

: .

Notes

References

Biographical and general references

*. Wide source of unpublished manuscript documents of and about E.E. Levi. A short presentation could be found o
EGEA website
*. An ample biographical paper (nearly 40 pages) on Beppo Levi: an earlier version of it was published as . It gives many useful information on the family and on the relationship between the two Levi brothers. *. The relation of Enrico D'Ovidio,

and

Vito Volterra Vito Volterra (, ; 3 May 1860 – 11 October 1940) was an Italian mathematician and physicist, known for his contributions to mathematical biology and integral equations, being one of the founders of functional analysis. Biography Born in An ...

motivating their decision for the awarding of the golden medal of the Accademia Nazionale delle Scienze detta dei XL to Eugenio Elia Levi. *. The most widely known

biographical A biography, or simply bio, is a detailed description of a person's life. It involves more than just the basic facts like education, work, relationships, and death; it portrays a person's experience of these life events. Unlike a profile or c ...

reference on Eugenio Elia Levi, written by

Guido Fubini Guido Fubini (19 January 1879 – 6 June 1943) was an Italian mathematician, known for Fubini's theorem and the Fubini–Study metric. Life Born in Venice, he was steered towards mathematics at an early age by his teachers and his father, wh ...

and

Gino Loria Gino Benedetto Loria (19 May 1862, Mantua – 30 January 1954, Genoa) was a Jewish-Italian mathematician and historian of mathematics. Loria studied mathematics in Mantua, Turin, and Pavia and received his doctorate in 1883 from the University ...

. *. A pamphlet containing the

commemoration Commemoration may refer to: *Commemoration (Anglicanism), a religious observance in Churches of the Anglican Communion *Commemoration (liturgy) In the Latin liturgical rites of the Catholic Church, a commemoration is the recital, within the Li ...

of Eugenio Elia Levi pronounced by

at the

University of Genova The University of Genoa, known also with the acronym UniGe ( it, Università di Genova), is one of the largest universities in Italy. It is located in the city of Genoa and regional Metropolitan City of Genoa, on the Italian Riviera in the Liguri ...

on May 29, 1918. *. *. A commemorative paper on Eugenio Elia Levi written by Mauro Picone and included in the first volume of his ''Opere'' . *. *. Available from the website of th
Società Italiana di Storia delle Matematiche
An ample historical paper written by Francesco Tricomi to commemorate all Italian mathematicians who worked during the first century of the Italian State. *.

External links

*. *. The biographical entry about Eugenio Elia Levi in the "''Dizionario Biografico degli Italiani'' (''Biographical Dictionary of Italians'')" section of the

Enciclopedia Treccani The ''Enciclopedia Italiana di Scienze, Lettere e Arti'' (Italian for "Italian Encyclopedia of Science, Letters, and Arts"), best known as ''Treccani'' for its developer Giovanni Treccani or ''Enciclopedia Italiana'', is an Italian-language en ...

. *. Available from th
Edizione Nazionale Mathematica Italiana
* {{DEFAULTSORT:Levi, Eugenio Elia 1883 births 1917 deaths 20th-century Italian mathematicians University of Pisa alumni Academic staff of the University of Genoa 20th-century Italian Jews Complex analysts Mathematical analysts PDE theorists