Renato Caccioppoli

Renato Caccioppoli
Born	(1904-01-20)20 January 1904 Napoli
Died	8 May 1959(1959-05-08) (aged 55) Napoli
Nationality	Italian
Alma mater	University of Naples Federico II
Scientific career
Fields	Functional analysis Geometric measure theory Several complex variables Partial differential equations
Institutions	University of Naples Federico II University of Padua
Doctoral advisor	Mauro Picone
Doctoral students	Federico Cafiero Guido Stampacchia

Renato Caccioppoli (Italian: [reˈnaːto katˈtʃɔppoli]; 20 January 1904 – 8 May 1959) was an Italian mathematician, known for his contributions to mathematical analysis, including the theory of functions of several complex variables, functional analysis, measure theory.

Born in Naples, he was the son of Giuseppe Caccioppoli (1852–1947), a surgeon, and his second wife Sofia Bakunin (1870–1956), daughter of the Russian revolutionary Mikhail Bakunin. After earning his high-school diploma in 1921, he enrolled in the Department of engineering to swap to mathematics in November 1923. Immediately after earning his laurea in 1925, he became the assistant of Mauro Picone, who in that year was called to the University of Naples, where he remained until 1932. Picone immediately discovered Caccioppoli's brilliance and pointed him towards research in mathematical analysis. During the following five years, Caccioppoli published about 30 works on topics developed in the complete autonomy provided by a ministerial award for mathematics in 1931, a competition he won at the age of 27 and the chair of algebraic analysis at the University of Padova. In 1934 he returned to Naples to accept the chair in group theory; later he took the chair of superior analysis, and from 1943 onwards, the chair in mathematical analysis.

In 1931 he became a correspondent member of the Academy of Physical and Mathematical Sciences of Naples, becoming an ordinary member in 1938. In 1944 he became an ordinary member of the Accademia Pontaniana, and in 1947 a correspondent member of the Accademia Nazionale dei Lincei, and a national member in 1958. He was also a correspondent member of the Paduan Academy of Sciences, Letters, and Arts. In the years from 1947 to 1957, he directed, together with Carlo Miranda, the journal Giornale di Matematiche, founded by Giuseppe Battaglini. In 1948 he became a member of the editing committee of Annali di Matematica, and starting in 1952 he was also a member of the editing committee of Ricerche di Matematica. In 1953 the Academia dei Lincei bestowed on him the national prize of physical, mathematical, and natural sciences.

He was an excellent pianist, noted as well for his nonconformist temperament. He tried out the vagrant life, and was arrested for begging. In May 1938 he gave a speech against Adolf Hitler and Benito Mussolini, when the latter was visiting Naples. Together with his companion Sara Mancuso, he had the French national anthem played by an orchestra, after which he began to speak against fascism and Nazism in the presence of OVRA agents. He was again arrested, but his aunt, Maria Bakunin, who at the time was a professor of chemistry at the University of Naples, succeeded in having him released by convincing the authorities that her nephew was non compos mentis. Thus Caccioppoli was interned, but he continued his studies in mathematics, and playing the piano.

In his last years, the disappointments of politics and his wife's desertion, together perhaps with the weakening of his mathematical vein, pushed him into alcoholism. His growing instability had sharpened his "strangenesses", to the point that the news of his suicide on May 8, 1959, by a headshot did not surprise those who knew him. He died at his home in Palazzo Cellamare.

His most important works, out of a total of around eighty publications, relate to functional analysis and the calculus of variations. Beginning in 1930 he dedicated himself to the study of differential equations, the first to use a topological-functional approach. Proceeding in this way, in 1931 he extended the Brouwer fixed point theorem, applying the results obtained both from ordinary differential equations and partial differential equations.

In 1932 he introduced the general concept of inversion of functional correspondence, showing that a transformation between two Banach spaces is invertible only if it is locally invertible and if the only convergent sequences are the compact ones.

Between 1933 and 1938 he applied his results to elliptic equations, establishing the majorizing limits for their solutions, generalizing the two-dimensional case of Felix Bernstein. At the same time he studied analytic functions of several complex variables, that is, analytic functions whose domain belongs to the vector space Cⁿ, proving in 1933 the fundamental theorem on normal families of such functions: if a family is normal with respect to every complex variable, it is also normal with respect to the set of the variables. He also proved a logarithmic residue formula for functions of two complex variables in 1949.

In 1935 Caccioppoli proved the analyticity of class C² solutions of elliptic equations with analytic coefficients.

The year 1952 saw the publication of his masterwork on the area of a surface and measure theory, the article Measure and integration of dimensionally oriented sets (Misura e integrazione degli insiemi dimensionalmente orientati, Rendiconti dell'Accademia Nazionale dei Lincei, s. VIII, v.12). The article is mainly concerned with the theory of dimensionally oriented sets; that is, an interpretation of surfaces as oriented boundaries of sets in space. Also in this paper, the family of sets approximable by polygonal domains of finite perimeter, known today as Caccioppoli sets or sets of finite perimeter, was introduced and studied.

His last works, produced between 1952 and 1953, deal with a class of pseudoanalytic functions, introduced by him to extend certain properties of analytic functions.

In 1992 his tormented personality inspired the plot of a film directed by Mario Martone, The Death of a Neapolitan Mathematician (Morte di un matematico napoletano), in which he was portrayed by Carlo Cecchi.

An asteroid, 9934 Caccioppoli, has been named after him.

• Caccioppoli, Renato (1963), Opere scelte, Roma: Edizioni Cremonese (distributed by Unione Matematica Italiana), Zbl 0112.28201 ISBN 88-7083-505-7 (Volume 1) AND ISBN 88-7083-506-5 (Volume 2). His "Selected works", a selection from Caccioppoli's scientific works with a biography and a commentary.

• Contraction principle
• Caccioppoli set
• Weyl's inequality

This article is based largely on material from the equivalent article on Italian Wikipedia, accessed 4 March 2006, and also on the following biographical works:

• De Angelis, P. L.; Sbordone, C., eds. (1999), "Renato Caccioppoli", Matematici all'Istituto Universitario Navale (1926 – 1976) [Mathematicians at the Istituto Universitario Navale (1926 – 1976)] (in Italian), Napoli: Istituto Universitario Navale/RCE Edizioni, pp. 15–19. the chapter on Caccioppoli in a book collecting brief biographical sketches and bibliographies of the scientific works produced by the mathematicians who taught at the Parthenope University of Naples during their stay.
• Fichera, Gaetano (1991), "Ricordo di Renato Caccioppoli" [Recollections of Renato Caccioppoli], Ricerche di Matematica (in Italian), 40 (supplement): 11–15, Zbl 0788.01051. The recollections on him by one of his colleagues and close friend.
• Sbordone, Carlo (2004), "Renato Caccioppoli, nel centenario della nascita" [Renato Caccioppoli, on the centenary of his birth] (PDF), Bollettino della Unione Matematica Italiana, Sezione A, la Matematica nella Società e nella Cultura, Serie VIII (in Italian), 7 (2): 193–214, MR 2097985, Zbl 1192.01026. An ample biographical paper on him written by Carlo Sbordone, pupil of Federico Cafifiero.
• UMI (1959), "Renato Caccioppoli", Bollettino dell'Unione Matematica Italiana, Serie III (in Italian), 14 (2): 294. A brief obituary, basically announcing the commemoration of his scientific work published in the following issue 4 of the same Bulletin.
• UMI (1959), "L'opera matematica di Renato Caccioppoli" [The mathematical work of Renato Caccioppoli], Bollettino dell'Unione Matematica Italiana, Serie III (in Italian), 14 (4): 548–551. A survey on his research work published in the UMI Bulletin: even if no author is stated, Sbordone (2004, reference [21], p. 212) attributes the article to Gianfranco Cimmino.

• de Lucia, Paolo (1988), "Analisi reale e teoria della misura a Napoli: R. Caccioppoli, C. Miranda e F. Cafiero", in Società Nazionale di Scienze, Lettere ed Arti in Napoli (ed.), Seduta inaugurale dell'anno accademico 1988 (in Italian), Napoli: Francesco Giannini e Figli, pp. 23–33. English: "Real analysis and measure theory in Naples: R. Caccioppoli, C. Miranda and F. Cafiero" is the opening address of the 1988 academic year of the Società Nazionale di Scienze, Lettere ed Arti in Napoli: it describes the contributions of Caccioppoli, Miranda and Cafiero to real analysis and measure theory during their stay in Naples.
• de Lucia, Paolo (2004) [1999], "Teoria della Misura a Napoli: Renato Caccioppoli", in Alvino, A.; Carbone, L.; Sbordone, C.; Trombetti, G. (eds.), In ricordo di Renato Caccioppoli [In memoriam Renato Caccioppoli] (in Italian) (2nd printing ed.), Napoli: Giannini, p. 124, MR 1306300, Zbl 0793.01019 (reviews of the symposium paper, see below). This paper, English: "Measure theory in Naples: Renato Caccioppoli", is a reprint of the contribution of Paulo de Lucia from the "International Symposium Renato Caccioppoli" held in Napoli on 20–22 September 1989 and describes Caccioppoli's and Cafiero's contributions to the development of Measure Theory. The collection includes other papers detailing Caccioppoli's personality and his research, the introduction to his "Opere scelte" (Selected works), a conference held by Caccioppoli himself and related letters by Carlo Miranda, Giovanni Prodi and Francesco Severi.
• Cafiero, Federico (1953), Funzioni additive d'insieme e integrazione negli spazi astratti [Additive set functions and integration in abstract spaces] (in Italian), Napoli: Libreria Editrice Liguori, p. 178, MR 0215954, Zbl 0050.27801. A prize winning monograph where Cafiero first states and proves his convergence theorem.
• Cafiero, Federico (1959), Misura e integrazione [Measure and integration], Monografie matematiche del Consiglio Nazionale delle Ricerche (in Italian), vol. 5, Roma: Edizioni Cremonese, pp. VII+451, MR 0215954, Zbl 0171.01503. A Definitive monograph on integration and measure theory: the treatment of the limiting behavior of the integral of various kind of sequences of measure-related structures (measurable functions, measurable sets, measures and their combinations) is somewhat conclusive.
• Cesari, Lamberto (1956), Surface Area, Annals of Mathematics Studies, vol. 35, Princeton, New Jersey: Princeton University Press, pp. x+595, ISBN 0-691-09585-X, MR 0074500, Zbl 0073.04101. The work of Cesari summarizing the theory of surface area, including his own contributions.
• Miranda, Carlo (1955), Equazioni alle derivate parziali di tipo ellittico, Ergebnisse der Mathematik und ihrer Grenzgebiete – Neue Folge (in Italian), vol. Heft 2 (1st ed.), Berlin – Göttingen – New York: Springer Verlag, pp. VIII+222, MR 0087853, Zbl 0065.08503.
• Miranda, Carlo (1970) [1955], Partial Differential Equations of Elliptic Type, Ergebnisse der Mathematik und ihrer Grenzgebiete – 2 Folge, vol. Band 2, translated by Motteler, Zane C. (2nd Revised ed.), Berlin – Heidelberg – New York: Springer Verlag, pp. XII+370, doi:10.1007/978-3-642-87773-5, ISBN 978-3-540-04804-6, MR 0284700, Zbl 0198.14101.

• Alvino, A.; Carbone, L.; Sbordone, C.; Trombetti, G., eds. (2004) [1999], In ricordo di Renato Caccioppoli [In memoriam Renato Caccioppoli] (in Italian) (2nd printing ed.), Napoli: Giannini, p. 124, Zbl 0928.00071. This is a collection of papers detailing his personality and his research, which includes the introduction to his "Opere scelte" (Selected works), a list of contributions from the "International Symposium Renato Caccioppoli" held in Napoli on September 20–22, 1989, a conference held by Caccioppoli himself and related letters by Carlo Miranda, Giovanni Prodi and Francesco Severi.

• Faber Fabbris, Renato (January 2000), "Renato Caccioppoli", in O'Connor, John J.; Robertson, Edmund F. (eds.), MacTutor History of Mathematics Archive, University of St Andrews
• The Caccioppoli Family (1 July 1997), Renato Caccioppoli, retrieved April 9, 2011: biographical sketch from the Caccioppoli family web site.