Eugenio Elia Levi
Born	(1883-10-18)18 October 1883 Torino, Italy
Died	28 October 1917(1917-10-28) (aged 34) Cormons, Italy
Nationality	Italian
Alma mater	Scuola Normale Superiore
Known for	Levi decomposition Levi form Parametrix Pseudoconvexity
Awards	Lavagna prize (1904) Golden medal of theAccademia Nazionale delle Scienze detta dei XL (1911)
Scientific career
Fields	Group theory Several complex variables Partial differential operators
Institutions	Scuola Normale Superiore University of Genoa

Eugenio Elia Levi (18 October 1883 – 28 October 1917) was anItalianmathematician, known for his fundamental contributions ingroup theory, in thetheory of partial differential operators and in thetheory of functions of several complex variables. He was a younger brother ofBeppo Levi and was killed in action duringFirst World War.

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He wrote 33 papers, classified by his colleague and friendMauro Picone^[a] according to the scheme reproduced in this section.

He wrote only three papers ingroup theory: in the first one,Levi (1905) discovered what is now calledLevi decomposition, which wasconjectured byWilhelm Killing and proved byÉlie Cartan in a special case.

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

His researches in the theory of partial differential operators lead to the method of theparametrix, which is basically a way to constructfundamental solutions forellipticpartial differential operators with variable coefficients: the parametrix is widely used in the theory ofpseudodifferential operators.

The full scientific production of Eugenio Elia Levi is collected in reference (Levi 1959–1960).

• Levi, Eugenio Elia (2 April 1905),"Sulla struttura dei gruppi finiti e continui" [On the structure of finite simple groups],Atti della Reale Accademia delle Scienze di Torino. (in Italian),XL:551–565,JFM 36.0217.02, reprinted also inLevi 1959–1960, pp. 101–126, volume I. A a well-knownmemoir inGroup theory: it was presented to the members of theAccademia delle Scienze di Torino during the session of April 2, 1905, byLuigi Bianchi.
• Levi, Eugenio Elia (1907a),"Sulle equazioni lineari alle derivate parziali totalmente ellittiche" [On linear totally elliptic partial differential equations],Rendiconti della Reale Accademia dei Lincei, Classe di Scienze Fisiche, Matematiche, Naturali, Serie V (in Italian),16 (1):932–938,JFM 38.0403.01. A short note announcing the results of paper (Levi 1907b).
• Levi, Eugenio Elia (1907b),"Sulle equazioni lineari totalmente ellittiche alle derivate parziali" [On linear totally elliptic partial differential equations],Rendiconti del Circolo Matematico di Palermo (in Italian),24 (1):275–317,doi:10.1007/BF03015067,JFM 38.0402.01. An important paper whose results were previously announced in the short note (Levi 1907a) with the same title. It was also translated in Russian by N. D. Ajzenstat, currently available from theAll-Russian Mathematical Portal:Levi, E. E. (1941),"On linear elliptic partial differential equations",Uspekhi Matematicheskikh Nauk (in Russian) (8):249–292.
• Levi, Eugenio Elia (1910),"Studii sui punti singolari essenziali delle funzioni analitiche di due o più variabili complesse" [Studies on essential singular points of analytic functions of two or more complex variables],Annali di Matematica Pura ed Applicata, s. III (in Italian),XVII (1):61–87,doi:10.1007/BF02419336,JFM 41.0487.01. An important paper in thetheory of functions of several complex variables, where the problem of determining what kind ofhypersurface can be theboundary of adomain of holomorphy.
• Levi, Eugenio Elia (1911), "Sulle ipersuperficie dello spazio a 4 dimensioni che possono essere frontiera del campo di esistenza di una funzione analitica di due variabili complesse" [On the hypersurfaces of the 4-dimensional space that can be the boundary of the domain of existence of an analytic function of two complex variables],Annali di Matematica Pura ed Applicata, s. III (in Italian),XVIII (1):69–79,doi:10.1007/BF02420535,JFM 42.0449.02. Another important paper in thetheory of functions of several complex variables, investigating further the theory started in (Levi 1910).
• Levi, Eugenio Elia (1959–1960),Opere [Collected works] (in Italian), Roma: Edizioni Cremonese (distributed byUnione Matematica Italiana), pp. XX + 418 (Volume I), 448 (Volume II),MR 0123464,Zbl 0091.00108. His "Collected works" in two volumes, collecting all the mathematical papers of Eugenio Elia Levi in a revised typographical form, both amendingtypographical 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 theInternet Archive:Levi, Eugenio Elia,Mathematical papers, Los Angeles:UCLA, p. 782, archived fromthe original on 9 August 2016.

• Pseudoconvexity
• Levi decomposition
• Parametrix
• Several complex variables

• Celli, Andrea; Mattaliano, Maurizio, eds. (2015),Eugenio Elia Levi, le speranze perdute della Matematica italiana (in Italian), Milano: EGEA, pp. LIV+326,ISBN 978-88-2384-461-2. Wide source of unpublished manuscript documents of and about E.E. Levi. A short presentation could be found onEGEA website
• Coen, Salvatore (1999), "Beppo Levi: una biografia", inLevi, Beppo (ed.),Opere. Volume I: 1897–1926 [Collected works. Volume I:1897–1926] (in Italian), Bologna: Edizioni Cremonese (distributed byUnione Matematica Italiana), pp. XIII–LIV,Zbl 1054.01520. An ample biographical paper (nearly 40 pages) on Beppo Levi: an earlier version of it was published asCoen, Salvatore (1994), "Beppo Levi: la vita", in Coen, Salvatore (ed.),Seminari di geometria, Università di Bologna, Italia, 1991–1993 (in Italian), Bologna: Università degli Studi di Bologna, Dipartimento di Matematica, pp. 193–232,MR 1265762,Zbl 0795.01022. It gives many useful information on the family and on the relationship between the two Levi brothers.
• D'Ovidio, E.;Bianchi, L.;Volterra, V. (1912),"Relazione sul premio per la Matematica relativo all'anno 1912, presentata dalla Commissione composta dei Soci: D'Ovidio, Bianchi, Volterra" [Report on the prize for Mathematics for the year 1912, presented by the Commission consisting of Members: D'Ovidio, Bianchi, Volterra],Rendiconti della Accademia Nazionale delle Scienze Detta dei XL, Memorie di Matematica e Applicazioni, Serie 3 (in Italian),XVII (62):XXIV–XXV, archived fromthe original on 27 September 2011. The relation ofEnrico D'Ovidio,Luigi Bianchi andVito Volterra motivating their decision for the awarding of the golden medal of theAccademia Nazionale delle Scienze detta dei XL to Eugenio Elia Levi.
• Fubini, Guido;Loria, Gino (1918),"Eugenio Elia Levi (1883–1917)",Bollettino di Bibliografia e Storia delle Scienze Matematiche (in Italian),20:38–45,JFM 46.0019.07. The most widely knownbiographical reference on Eugenio Elia Levi, written byGuido Fubini andGino Loria.
• Loria, Gino (1918),Panegirico di un Eroe. Commemorazione del Prof. Eugenio Elia Levi pronunziata nell'aula magna dell'Università di Genova addì XXIX maggio MCMXVIII [Panegyric of aHero. Commemoration of Prof. Eugenio Elia Levi pronounced in the auditorium of the University of Genoa on 29 May 1918] (in Italian),Sestri Ponente: S. L. A. G. / Bruzzone e C., p. 31. Apamphlet containing thecommemoration of Eugenio Elia Levi pronounced byGino Loria at theUniversity of Genova on May 29, 1918.
• Padoa, Alessandro (December 1917), "Eugenio Elia Levi",Bollettino della Mathesis (in Italian),9:89–92.
• Picone, Mauro (1959), "Sulla Vita e sulle Opere di Eugenio Elia Levi", in Levi, Eugenio Elia (ed.),Opere. Volume I [Collected Works] (in Italian), Roma: Edizioni Cremonese (distributed byUnione Matematica Italiana), pp. V–XV,MR 0123464,Zbl 0091.00108. A commemorative paper on Eugenio Elia Levi written byMauro Picone and included in the first volume of hisOpere (Levi 1959–1960).
• Roero, C.S. (31 January 2005),Eugenio Elia Levi (18/10/1883–28/10/1917) (in Italian), archived fromthe original on 27 September 2011, retrieved12 January 2010.
• Tricomi, G. F. (1962),"Eugenio Elia Levi",Matematici italiani del primo secolo dello stato unitario [Italian mathematicians of the first century of the unitary state], Memorie dell'Accademia delle Scienze di Torino. Classe di Scienze fisiche matematiche e naturali, series IV (in Italian), vol. I, p. 120,Zbl 0132.24405, archived fromthe original on 29 December 2016, retrieved28 December 2016. Available from the website of theSocietà 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 theItalian State.
• Vitali, Giuseppe (December 1917), "Eugenio Elia Levi",Bollettino della Mathesis (in Italian),9:86–89.

• Comoglio, Mauro (4 November 2010),La guerra del '15–'18 termina il 4 Novembre. Un anno prima era morto Eugenio Elia Levi (in Italian),MATEpristem, retrieved13 June 2011.
• Dell'Aglio, L. (2005),"LEVI, Eugenio Elia",Enciclopedia Treccani, Dizionario Biografico degli Italiani (in Italian), vol. LXIV, retrieved15 September 2011. The biographical entry about Eugenio Elia Levi in the "Dizionario Biografico degli Italiani (Biographical Dictionary of Italians)" section of theEnciclopedia Treccani.
• Guerraggio, Angelo; Nastasi, Pietro;Tricomi, Francesco (2008–2010),"Eugenio Elia Levi (1883–1917)",Edizione Nazionale Mathematica Italiana (in Italian),Scuola Normale Superiore, retrieved18 January 2011. Available from theEdizione Nazionale Mathematica Italiana.
• Eugenio Elia Levi at theMathematics Genealogy Project

• 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

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