Eugenio Elia Levi

Eugenio Elia Levi (18 October 1883 – 28 October 1917) was an Italian mathematician. He is known for his fundamental contributions in group theory, 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 and was killed in action during First World War.

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 group theory: in the first one, discovered what is now called Levi decomposition, which was conjectured by Wilhelm Killing and proved by Élie Cartan 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.

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

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 in Group theory: it was presented to the members of the Accademia delle Scienze di Torino during the session of April 2, 1905, by Luigi Bianchi. . 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: . . An important paper in the theory of functions of several complex variables, where the problem of determining what kind of hypersurface can be the boundary of a domain of holomorphy. . 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 errors 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: .