Eugenio Elia Levi
Eugenio Elia Levi (18 October 1883 – 28 October 1917) was an Italian mathematician, 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.
Eugenio Elia Levi  

Born  Torino, Italy  18 October 1883
Died  28 October 1917 34) Cormons, Italy  (aged
Nationality  Italian 
Alma mater  Scuola Normale Superiore 
Known for  
Awards 

Scientific career  
Fields  
Institutions  
Academic advisors 
Work
Research activity
He wrote 33 papers, classified by his colleague and friend Mauro Picone[1] according to the scheme reproduced in this section.
Differential geometry
Group theory
He wrote only three papers in group theory: in the first one, Levi (1905) 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[2] 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 (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 in Levi 1959–1960, pp. 101–126, volume I. A a wellknown 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.
 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 the AllRussian 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 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.
 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 4dimensional 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 the theory 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 by Unione 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 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: Levi, Eugenio Elia, Mathematical papers, Los Angeles: UCLA, p. 782, archived from the original on 20160809.
Notes
 This section is mainly based on the survey article by Picone (1959) included in Levi's "Opere (Collected works)", describing his researches briefly but comprehensively; occasionally, also the comments of Guido Fubini in (Fubini & Loria 1917) are taken into account.
 See the two well known papers (Levi 1910) and (Levi 1910): note that Levi deals with functions of two complex variables, but his calculations can be extended to functions with any finite number of variables, as he explicitly states. Note also that Levi, following a then well established practice, does not uses Wirtinger derivatives.
References
Biographical and general references
 Celli, Andrea; Mattaliano, Maurizio, eds. (2015), Eugenio Elia Levi, le speranze perdute della Matematica italiana (in Italian), Milano: EGEA, pp. LIV+326, ISBN 9788823844612. Wide source of unpublished manuscript documents of and about E.E. Levi. A short presentation could be found on EGEA website
 Coen, Salvatore (1999), "Beppo Levi: una biografia", in Levi, Beppo (ed.), Opere. Volume I: 1897–1926 [Collected works. Volume I:1897–1926] (in Italian), Bologna: Edizioni Cremonese (distributed by Unione 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 as Coen, 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 from the original on 20110927. The relation of Enrico D'Ovidio, Luigi Bianchi and Vito Volterra motivating their decision for the awarding of the golden medal of the Accademia 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 known biographical reference on Eugenio Elia Levi, written by Guido Fubini and Gino 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 a Hero. Commemoration of Prof. Eugenio Elia Levi pronounced in the auditorium of the University of Genoa on the 29th of May 1918] (in Italian), Sestri Ponente: S. L. A. G. / Bruzzone e C., p. 31. A pamphlet containing the commemoration of Eugenio Elia Levi pronounced by Gino Loria at the University 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 by Unione Matematica Italiana), pp. V–XV, MR 0123464, Zbl 0091.00108. A commemorative paper on Eugenio Elia Levi written by Mauro Picone and included in the first volume of his "Opere (Levi 1959–1960).
 Roero, C.S. (31 January 2005), Eugenio Elia Levi (18/10/1883–28/10/1917) (in Italian), archived from the original on 27 September 2011, retrieved 12 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), I, p. 120, Zbl 0132.24405. Available from the website of the 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.
 Vitali, Giuseppe (December 1917), "Eugenio Elia Levi", Bollettino della Mathesis (in Italian), 9: 86–89.
External links
 Comoglio, Mauro (November 4, 2010), La guerra del '15–'18 termina il 4 Novembre. Un anno prima era morto Eugenio Elia Levi (in Italian), MATEpristem, retrieved June 13, 2011.
 Dell'Aglio, L. (2005), "LEVI, Eugenio Elia", Enciclopedia Treccani, Dizionario Biografico degli Italiani (in Italian), LXIV, retrieved September 15, 2011. The biographical entry about Eugenio Elia Levi in the "Dizionario Biografico degli Italiani (Biographical Dictionary of Italians)" section of the Enciclopedia Treccani.
 Guerraggio, Angelo; Nastasi, Pietro; Tricomi, Francesco (2008–2010), "Eugenio Elia Levi (1883–1917)", Edizione Nazionale Mathematica Italiana (in Italian), Scuola Normale Superiore, retrieved January 18, 2011. Available from the Edizione Nazionale Mathematica Italiana.
 Eugenio Elia Levi at the Mathematics Genealogy Project