Giulio Giacomo Levi (1834-1898) practiced as a lawyer, but was also the author of several books on political and social issues, such as 'Labour and freedom' and 'The failure of socialism', which reveal his preoccupation with solving social problems, without abandoning the liberal ideas challenged by nascent socialism.Eugenio attended the Azeglio High School in Turin where he showed that he was an extremely gifted pupil. He graduated in 1900, and entered the Scuola Normale Superiore of Pisa. There he was taught by Luigi Bianchi who had been promoted to professor of analytic geometry at the Scuola Normale Superiore in 1890. Two aspects of Bianchi's work would be particularly significant in his influences on Levi. He had published two important works before Levi began to study with him, namely Lectures on differential geometry (1894) and Lectures on the theory of groups of substitutions (1900). The other mathematician in Pisa who had a major influence on Levi's research was Ulisse Dini who held two chairs at the University of Pisa, the chair of analysis and higher geometry, and the chair of infinitesimal analysis.
In 1904 Levi graduated from the Scuola Normale Superiore of Pisa with the award of his laurea. He remained at Pisa where he became an assistant to Ulisse Dini. Levi published Sulla struttura dei gruppi finiti e continui Ⓣ in 1905, the paper having been presented to Academy of Science of Turin on 2 April of that year by Bianchi. In 1906 Levi submitted his habilitation thesis Sulle equazioni lineari totalmente ellittiche alle derivate parziali to the Scuola Normale Superiore. This important work studied partial differential equations of order 2n, linear in two variables fully elliptic in a certain region of the plane. He published a short note in the journal of the Reale Accademia dei Lincei in 1907 announcing the results which were fully presented in a paper in the journal of the Circolo Matematico di Palermo in the same year. In this work he looked at existence and uniqueness theorems using techniques similar to those introduced by David Hilbert a couple of years later.
Levi became interested in the Cauchy problem. Since the problem could not be solved in general, he examined special cases starting with linear equations in two variables and then extending the ideas to the non-linear case. He published his work in this area in the paper Sul problema di Cauchy per le equazioni lineari in due variabili a caratteristiche reali Ⓣ (1908). Also in 1908 he entered the competition for the chair of infinitesimal analysis at the University of Messina. Although he was not successful, he came second in the ranked list of candidates showing that he would soon gain a chair. Indeed he did just that one year later when he was successful in the competition for the chair of infinitesimal analysis in the Faculty of Science of the University of Genoa. He published important papers on the theory of functions of several complex variables: Studii sui punti singolari essenziali delle funzioni analitiche di due o più variabili complesse Ⓣ (1910) and Sulle ipersuperficie dello spazio a 4 dimensioni che possono essere frontiera del campo di esistenza di una funzione analitica di due variabili complesse Ⓣ (1911). Also in 1911 he published Sopra un teorema di esistenza per le equazioni alle derivate parziali del secondo ordine Ⓣ which made similar contributions to the Goursat problem that he had made earlier to the Cauchy problem. In 1912 he was promoted to full professor. In 1912 Enrico D'Ovidio, Luigi Bianchi and Vito Volterra formed a committee for awarding the Gold Medal of the National Academy of Sciences of Italy (the "Academy of Forty"). The medal was awarded to Levi, and the article  details the reasons of the committee.
The authors of  give the following summary of Levi's mathematical achievements:-
In complex analysis, Eugenio Elia's research focused on the singular point sets of a holomorphic function of several variables. However, he also wrote on issues relating to: differential geometry, Lie groups, partial differential equations and the minimum principle. E E Levi would also demonstrate the falsity of Weierstrass's conjecture according to which given an open A of C2, a meromorphic function will always exist in A which has essential singularities in each point of the boundary of A, providing further evidence in favour of the differentiation between the theory of the single complex variable and the theory of more than one complex variable.Levi's remarkable mathematical career came to an end with the outbreak of World War I. Italy did not enter the war when it broke out in 1914 although at this stage most people in the country would have supported Germany, but had a long-standing rivalry with Austria-Hungary. Only in April 1915 did Italy enter the war, persuaded to come in on the side of the Allies, Russia, France and Britain. It was over one year later, in October 1916, that Levi entered military service in, as a lieutenant in the 1st Regiment of Engineers stationed at their headquarters in Pavia. In December 1916 he returned to teaching. In fact he taught a fundamental discipline that would have given him the right, if he wished, to be exempted from military service. Levi, however, was highly patriotic wished to return to military service. He was an exceptionally good organiser and a very hard worker so he was rapidly promoted to captain. He also showed exceptional bravery for which he was awarded a decorative bronze medal.
In August 1917 he was transferred to the region of Gorizia. A few weeks later, news reached him that his brother Decio had been killed in action on Monte Santo. In May 1917, The Times newspaper had reported, "Fighting is raging furiously for Monte Santo. The Austrian batteries sighted the Italians advancing along the river road and began belching forth shrapnel. ... fighting reached the heights of heroism and tenacity." On 24 August 1917 the Italians captured Monte Santo but the Austrians continued to put up fierce resistance. Decio Levi, who was a major in the Italian army, was killed on 15 September as fighting continued in the Monte Santo area. Eugenio Levi was a captain in charge of a company of engineers in the same general area. Although these victories by the Italians against Austrian troops in northern Italy were not decisive, nevertheless, the Germans sent reinforcements to strengthen the Austrian army. The German troops decided that the weakest point in the Italian lines was at Caporetto and, on 24 October they attacked. After six hours of continuous pounding, the Italian lines were broken near Gabria and San Daniele del Carso. The retreat of the Italians from Caporetto began - it was the most disastrous defeat in the history of the Italian army. During the retreat, Eugenio Levi was hit in the forehead by a stray bullet. He died immediately.
Let us end this biography by quoting Guido Fubini's appreciation of the depth of Levi's contributions :-
The work of E E Levi almost always deal with issues of fundamental importance: he was not discouraged by the difficulties, even major ones, encountered by other mathematicians, and with a more profound analysis, often very simple and ingenious, was able to clarify and overcome them. He did not approach these problems from only one angle, but did so from every direction.
Article by: J J O'Connor and E F Robertson