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

Italia Italy, officially the Italian Republic, is a country in Southern Europe, Southern and Western Europe, Western Europe. It consists of Italian Peninsula, a peninsula that extends into the Mediterranean Sea, with the Alps on its northern land b ...

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, mathematical structure, structure, space, Mathematica ...

, 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 on mathematics as well as on physics, history, philosophy, and pedagogy. Levi was a member of the Bologna Academy of S ...

and was killed in action during

First World War World War I or the First World War (28 July 1914 – 11 November 1918), also known as the Great War, was a World war, global conflict between two coalitions: the Allies of World War I, Allies (or Entente) and the Central Powers. Fighting to ...

Work

Research activity

He wrote 33 papers, classified by his colleague and friend

Mauro Picone Mauro Picone (2 May 1885 – 11 April 1977) was an Italian mathematician. He is known for the Picone identity, the Sturm-Picone comparison theorem and being the founder of the Istituto per le Applicazioni del Calcolo, presently named after hi ...

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 In Lie theory and representation theory, the Levi decomposition, conjectured by Wilhelm Killing and Élie Cartan and proved by , states that any finite-dimensional Lie algebra ''g'' over a field of characteristic zero is the semidirect product o ...

, which was

conjecture In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis or Fermat's conjecture (now a theorem, proven in 1995 by Andrew Wiles), ha ...

d 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 M ...

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 geometry. He ...

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 classificati ...

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

, 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 ...

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 return ...

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 on 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 autob ...

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 Accademia (Italian for "academy") often refers to: * The Galleria dell'Accademia, an art museum in Florence * The Gallerie dell'Accademia, an art museum in Venice Accademia may also refer to: Academies of art * The Accademia Carrara di Belle ...

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 peo ...

: . *. 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 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. Forma ...

. *. 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 or transposition error) made in the typing of printed or electronic material. Historically, this referred to mistakes in manual typesettin ...

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 501(c)(3) organization, non-profit organization founded in 1996 by Brewster Kahle that runs a digital library website, archive.org. It provides free access to collections of digitized media including web ...

: .

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 Enrico D'Ovidio (1842–1933) was an Italian mathematician who is known by his works on geometry. Life and work D'Ovidio, son of liberal parents involved in the Italian independence movement, studied at the University of Naples under his uncl ...

and

Vito Volterra Vito Volterra (, ; 3 May 1860 – 11 October 1940) was an Italian mathematician and physicist, known for his contributions to Mathematical and theoretical biology, mathematical biology and Integral equation, integral equations, being one of the ...

motivating their decision for the awarding of the golden medal of the

Accademia Nazionale delle Scienze detta dei XL The Accademia Nazionale delle Scienze (), or more formally L'Accademia Nazionale delle Scienze detta dei XL, and also called the Accademia dei XL (), is Italy's national academy of science. Its offices are located within the Villino Rosso, at the co ...

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 basic facts like education, work, relationships, and death; it portrays a person's experience of these life events. Unlike a profile or curric ...

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, ...

and

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

. *. A

pamphlet A pamphlet is an unbound book (that is, without a Hardcover, hard cover or Bookbinding, binding). Pamphlets may consist of a single sheet of paper that is printed on both sides and folded in half, in thirds, or in fourths, called a ''leaflet'' ...

containing the

commemoration Commemoration may refer to: *Commemoration (Anglicanism), a religious observance in Churches of the Anglican Communion *Commemoration (liturgy), insertion in one liturgy of portions of another *Memorialization *"Commemoration", a song by the 3rd a ...

of Eugenio Elia Levi pronounced by

at the

University of Genova The University of Genoa () is a Public university, public research university. It is one of the largest universities in Italy and it is located in the city of Genoa, on the Italian Riviera in the Liguria region of northwestern Italy. The original ...

on May 29, 1918. *. *. A commemorative paper on Eugenio Elia Levi written by

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 Italian(s) may refer to: * Anything of, from, or related to the people of Italy over the centuries ** Italians, a Romance ethnic group related to or simply a citizen of the Italian Republic or Italian Kingdom ** Italian language, a Romance languag ...

. *.

External links

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

Enciclopedia Treccani Institute Giovanni Treccani for the publication of the Italian Encyclopedia (), also known as Treccani Institute or simply Treccani, is a cultural institution of national interest, active in the publishing field, founded by Giovanni Treccani a ...

. *. 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 Italian mathematical analysts Partial differential equation theorists