Died: 18 August 1961 in Rosario, Santa Fe, Argentina

Beppo Levi's parents were Giulio Giacomo Levi and Diamantina Pugliese. The family was Jewish and Beppo was the fourth of his parents' ten children. The ninth of these ten children was Eugenio Elia Levi (born 1883) who also became a mathematician and has a biography in this archive. The youngest member of the family was Decio Levi (1885-1917), who became an engineer. Their father [1]:-

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.

Beppo Levi studied mathematics at the University of Turin, beginning his studies in 1892. He attended courses given by Corrado Segre, Enrico D'Ovidio and Giuseppe Peano, and these mathematicians had a major influence on him [23]:-

For all three he retained a deep affection and reverence for the rest of his life.

He was also taught by Vito Volterra and Mario Pieri. Corrado Segre, who had himself studied at Turin with D'Ovidio, had been appointed to the Chair of Higher Geometry there in 1888. He became Levi's thesis advisor, and Levi's thesis Sulla varietá delle cordi di una curva algebraica was a brilliant piece of work adding to the outstanding work of the Italian school of geometry. It studied [20]:-

... the variety of secants of algebraic curves, with a view to studying singularities of space curves.

For the last three years of study at the University of Turin, Levi was supported by a scholarship which he had won. After graduating in July 1896, Levi was appointed as an assistant to Luigi Berzolari at Turin and he held this position until 1899. His father had died in 1898 and this gave Levi the responsibility of becoming head of the family (he was the eldest surviving son). To support himself and other family members he took on various teaching positions at middle schools. He taught first at Sassari in Sardinia, next at Bari in Apulia, moving on to Vercelli in Piedmont before teaching at two towns, first Bobbio then Piacenza, in Emilia-Romagna. Some of these jobs took him far from his family which distressed him and he tried to find jobs nearer to his home town. He returned to Turin where he taught at the Technical Institute until 1906 when his position there was made permanent.

During these seven years as a school teacher, Levi had tried to obtain a number of different university appointments but he was not successful. For example, in 1901 he entered the competition for the chair at the University of Turin which had been held by Luigi Berzolari. He came third in this competition, the post going to Gino Fano [2]:-

The referees generally regarded Levi as outstanding in intellect and culture but inferior as an expositor.

In 1906 he was successful in a competition when he was appointed professor of descriptive and projective geometry at the University of Cagliari in Sardinia. Cagliari was situated on the south coast of the island of Sardinia and was an ancient town whose university had been founded in 1606. While in Cagliari, Levi did some outstanding work on the arithmetic of elliptic curves which he published in four papers entitled Saggio per una teoria aritmetica delle forme cubiche ternarie (one paper in 1906, and three in 1908). He reported on this work in the invited lecture Sull'equazione indeterminata del 3^o ordine to the International Congress of Mathematicians in Rome in 1908. He remained at Cagliari, teaching analytic geometry, for four years until he was called to the chair of algebraic analysis at the University of Parma in 1910. While in Cagliari, Levi had married Albina Bachi from the town of Torre Pelice in Piemonte. Albina, like Levi, was Jewish; they had three children, Giulio, Laura and Emilia. Although, for Albina [2]:-

... Cagliari was an exotic locale; for him, it was too far from his family.

The year Levi left Cagliari he had been promoted to an ordinary professorship there, but he was so keen to leave that he was prepared to accept a lower rank at Parma at a university which did not offer the laureate in mathematics. However, Mario Pieri, who had been one of Levi's teachers at Turin, was at Parma and was keen to have his former student join him there. Once in Parma, Pieri became Levi's closest friend. He spent eighteen years at Parma putting considerable effort into the scientific development of the university with a number of new policies which produced excellent results. As well as the chair of algebraic analysis, Levi also held the chair of analytic geometry and, for a year, the chair of mathematical physics too. This meant that his workload was extremely heavy. However, he made strenuous efforts to have the laureate in mathematics set up in Parma and gained the approval of the rector for such a move. However, the outbreak of World War I and Italy's entry into the conflict in April 1915, prevented Levi's plans being carried through. The war saw tragedy strike the Levi family, for his two brothers Decio Levi and Eugenio Levi were both killed in action in 1917.

After the war ended, Levi renewed his efforts to introduce the laureate in mathematics in Parma. His hand was strengthen when he became president of the Faculty of Science. However, in the 1920s the political situation in Italy began to make his work increasingly difficult, having a serious impact on his attempts to improve the status of Parma. Giovanni Gentile (1875-1944) was a philosopher, with extreme Fascist views, who became professor of the history of philosophy at the University of Rome in 1917. However he became minister of education in the Fascist Italian government in 1922 and over the following two years carried out major reforms of education. Gentile organised the first Congress of Fascist Institutions of Culture in Bologna in March 1925 and this led to the 'manifesto Gentile' in April which sought the support of intellectuals for Fascism. Two mathematicians, Corrado Gini and Salvatore Pincherle, supported the manifesto while others drew up a counter-manifesto arguing for the independence of intellectuals from political interference. Levi signed the counter-manifesto, as did Leonida Tonelli, Vito Volterra, Guido Castelnuovo, Tullio Levi-Civita and Francesco Severi. However, Fascist reforms continued, leading to the closure of the mathematics programme at the University of Parma. All the mathematicians left except Levi who became professor of special mathematics and president of the school of chemistry. However, in 1928 the school of chemistry at Parma was also closed as part of the Fascist reforms.

However, despite these extreme difficulties, Levi's years in Parma had been ones in which he had greatly extended the already broad range of research topics he had studied. Before going to Parma he had already published over forty articles on topics ranging from algebraic geometry to logic, particularly working on the axiom of choice. He had also studied the theory of integration, partial differential equations and the Dirichlet Principle, producing the famous "Beppo Levi theorem" and spaces now called "Beppo Levi spaces". To this already broad range of work, he added contributions to subjects such as number theory, electrical engineering, the theory of physical measurements, and theoretical physics. In 1928 he left Parma and transferred to the chair of the theory of functions at the University of Bologna. He had spent over a year trying to negotiate this move which proved very difficult because in 1925 he had signed the counter-manifesto opposing Fascist policies.

At Bologna, Levi had a heavy teaching and administrative load, yet he continued to undertake research with the same passions as he had done throughout his life. He wrote articles on logic, differential equations, complex variable, as well as on the border between analysis and physics. He also played a significant role in the Italian Mathematical Union as the editor of its journal the Bolletino dell'Unione Matematica Italiana and being director of the Union from 1931 to 1938. In many ways things went well for Levi at Bologna: his daughter Laura (the author of [1]) began doctoral studies in physics there, he had an excellent relationship with Salvatore Pincherle who was retired but still active, and he was elected to the Reale Accademia dei Lincei in 1935.

Let us look now at Analisi Matematica Algebrica ed Infinitesimale which Levi published in 1937. We do this by quoting from the start of a detailed review by E S Allen [3]:-

In 1916 Beppo Levi published volume I of an 'Introduzione alla Analisi Matematica', an excellent treatise on algebra with the subtitle 'Teorie Formali'. In this earlier work all questions of continuity and infinite processes were avoided - or, rather, postponed to later sections of the treatise, which have not been printed. Rather than complete the work previously begun, Levi offers us the present book, which can probably be regarded as a somewhat briefer version of both the written and the unwritten parts of the 'Introduzione'. In content it has much in common with American works on "advanced calculus," differing from them in the greater attention given to algebra and in the commendably comprehensive points of view. For instance, the "numbers" used are of a very general type, except where the contrary is stated. Definite integrals are defined, from the start, so as to include those of both Lebesgue and Riemann. Simpson's rule comes in as a special case of a much more general method. Levi intended, it seems, to write a text requiring previous knowledge only of "algebra through quadratics" and of the elements of trigonometry. He takes no acquaintance with analytic geometry for granted. Of course it is a freshman of extraordinarily tough mind to whom the work is ostensibly addressed; and even this prodigy is forgotten when the author later assumes acquaintance with cross-ratios. There is a hint of an upper bound for the topics to be considered when Levi declines to give a proof of Stirling's formula, "which would be beyond the theoretical limits we have set ourselves."

Allen ends the review by writing:-

In conclusion, we would warmly commend this work of Beppo Levi, which brings to important topics already frequently treated deep insight and a stimulating wealth of fresh and fertile ideas.

Despite his hatred of Fascism, Levi had signed the "oath to Fascism" in 1931 along with around 1200 other mathematicians (only eleven refused to sign). Perhaps because of this, he was able to carry out his duties in Bologna with little political interference for several years. However, this changed dramatically in July 1938 when, under pressure from Hitler, Mussolini brought in the Manifesto of Race. This law was totally anti-Semitic, removing Italian citizenship from Jews and banning them from jobs in education, government and banking. This resulted in Levi being dismissed from his position in Bologna in 1938. He had made contacts with several mathematicians from Argentina through his work as the editor of the Bollettino and, although he was sixty-three years old, he immediately began to try to negotiate a move to Argentina. Cortés Plá invited Levi to be the director of the recently founded mathematical institute at the Universidad del Litoral in Rosario. Plá [20]:-

... was an engineer who taught physics and had an active interest in the history of science. He was a friend and admirer of Rey Pastor, the undisputed leading mathematician of Argentina at the time.

In October 1939 Levi, with his wife and two daughters, left Italy and emigrated to Argentina. His son Giulio, who was a biologist, emigrated to Palestine at this time. Remarkably, although Levi was 64 when he took up the positions of professor and director of the Institute in Rosario, he was able to continue to teach, undertake research and undertake administrative duties for a further 20 years. As well as teaching courses on analysis, geometry and rational mechanics, he was very active in research, publishing around one third of all his work in Spanish. He founded the journal Mathematicae Notae, the series Publicaciones del Instituto de matemáticas, and the series of books Monografias. He published Sistemas de ecuaciones analiticas en terminos finitos, diferenciales y en derivadas partiales (Systems of Analytic Equations: Equations in Finite Terms, Ordinary and Partial Differential Equations) (1944) as the first volume in theMonografias series. Levi writes in the Introduction:-

The problems related to the resolution of equations, whether finite or differential, assume fundamentally different aspects according to whether we postulate only the existence of such properties of continuity and differentiability of the given functions and the unknowns as are strictly necessary if a particular problem is to have meaning, or admit additional hypotheses relative to the existence of a certain number of successive derivatives, or finally grant at once the existence of all derivatives.

R P Boas Jr. writes in a review:-

This monograph is a clearly written exposition of the fundamental existence theorems for systems of analytic partial differential equations, together with necessary preliminary material on implicit functions and ordinary differential equations. ... The theory is illustrated by numerous interesting examples.

R H Bruck writes [7]:-

.... we should like to remark upon the elegance of mathematical expression which is possible in Spanish and to suggest that with the present volume the new series of monographs has made a most satisfactory beginning.

In 1947 Levi published Leyendo a Euclides (Reading Euclid). L M Blumenthal writes in a review:-

This pleasant little book records in informal fashion some thoughts of a mathematician occasioned by a reading of Euclid's 'Elements'. Though the author disclaims any intention of writing a serious historical study or a modern critique of Euclid, there is much of both in the book.

In 1956 Levi received the Antonio Feltrinelli Prize from the Accademia dei Lincei. The authors of [20] note the ironical citation which congratulates the Accademia dei Lincei for the honour that Levi has brought to Italy with his work in Argentina.

Books:

1. L Levi, Beppo Levi: Italia y Argentina en la vida de un matemático (Libros del Zorzal, 2000).
2. E A Marchisotto and J T Smith, The Legacy of Mario Pieri in Geometry and Arithmetic (Springer, 2007).

Articles:

1. E S Allen, Review: Analisi Matematica Algebrica ed Infinitesimale by Beppo Levi, Amer. Math. Monthly 46 (1) (1939), 37-40.
2. Beppo Levi on his 80th anniversary (Spanish), Rev. Un. Mat. Argentina 17 (1955), 7-16.
3. L R Berrone, The correspondence of Beppo Levi and Misha Cotlar. Cotlar's first publications (Spanish), Bol. Asoc. Mat. Venez. 16 (2) (2009), 115-131.
4. L R Berrone, An unpublished manuscript of Beppo Levi (Spanish), LLULL 29 (63) (2006), 19-35.
5. R H Bruck, Review: Sistemas de Ecuaciones Analiticas en Terminos Finitos, Diferenciales y en Derivadas Parciales by Beppo Levi, Amer. Math. Monthly 52 (1) (1945), 39-40.
6. J Cassinet, The approximation principle of Beppo Levi. An attempt at replacing the axiom of choice (1918-1923) (Italian), in Italian mathematics between the two world wars (Italian), Milan/Gargnano, 1986 (Pitagora, Bologna, 1987), 99-105.
7. S Coen, Beppo Levi: una biografia, in Beppo Levi, Opere I: 1897-1906 (Unione Matematica Italiana, Bologna, 1999), 13-54.
8. S Coen, Elenco completo delle opere di Beppo Levi, in Beppo Levi, Opere I: 1897-1906 (Unione Matematica Italiana, Bologna, 1999), 85-122.
9. S Coen, Beppo Levi, Dizionario Biografico degli Italiani 64 (2005).
10. S Coen, Indicazioni bibliografiche ragionate sull'opera di Beppo Levi, in Cuadernos del Instituto de Matemáticas 'Beppo Levi' (Rosario) 30 (2001), 13-30.
11. S Coen, Beppo Levi: la vita, in Salvatore Coen, Seminari di geometria, Università di Bologna, Italia, 1991-1993 (Department of Mathematics, Università degli Studi di Bologna, Bologna, 1994), 193-232.
12. S Coen, Geometry and complex variables in the work of Beppo Levi, in Geometry and complex variables, Bologna, 1988/1990 (Dekker, New York, 1991), 111-139.
13. B Levi, Some reflections on mathematics and philosophy (Spanish), Math. Notae 14 (1955), 133-140.
14. G Lolli, L'opera logica di B. L., in Beppo Levi, Opere I: 1897-1906 (Unione Matematica Italiana, Bologna, 1999), 67-76.
15. B Moss, Beppo Levi and the axiom of choice, Historia Math. 6 (1) (1979), 54-56.
16. C Pla, Beppo Levi en la Argentina, Math. Notae 18 (1962), 13-22.
17. L A Santaló, La obra cient’fica de Beppo Levi, Math. Notae 18 (1962), 23-28.
18. N Schappacher and R Schoof, Beppo Levi and the arithmetic of elliptic curves, The Mathematical Intelligencer 18 (1) (1996), 57-69.
19. S Spagnolo, 1906: un anno di grazia per Beppo Levi, in Beppo Levi, Opere I: 1897-1906 (Unione Matematica Italiana, Bologna, 1999), 67-81.
20. A Terracini, Commemorazione del Corrispondente Beppo Levi, Rendiconti Acc. dei Lincei 34 (1963), 590-606.
21. T Viola, Necrologio di Beppo Levi, Bollettino della Unione Matematica Italiana, Serie 3 (4) 16 (1961), 513-516.