Renato Caccioppoli

Overview

**Renato Caccioppoli**was an Italian

Italy

Italy , officially the Italian Republic languages]] under the European Charter for Regional or Minority Languages. In each of these, Italy's official name is as follows:;;;;;;;;), is a unitary parliamentary republic in South-Central Europe. To the north it borders France, Switzerland, Austria and...

mathematician

Mathematician

A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

.

Born in Naples

Naples

Naples is a city in Southern Italy, situated on the country's west coast by the Gulf of Naples. Lying between two notable volcanic regions, Mount Vesuvius and the Phlegraean Fields, it is the capital of the region of Campania and of the province of Naples...

, Campania

Campania

Campania is a region in southern Italy. The region has a population of around 5.8 million people, making it the second-most-populous region of Italy; its total area of 13,590 km² makes it the most densely populated region in the country...

, he was the son of Giuseppe Caccioppoli (1852–1947), a surgeon

Surgeon

In medicine, a surgeon is a specialist in surgery. Surgery is a broad category of invasive medical treatment that involves the cutting of a body, whether human or animal, for a specific reason such as the removal of diseased tissue or to repair a tear or breakage...

, and his second wife Sofia Bakunin (1870–1956), daughter of the Russian revolutionary Mikhail Bakunin

Mikhail Bakunin

Mikhail Alexandrovich Bakunin was a well-known Russian revolutionary and theorist of collectivist anarchism. He has also often been called the father of anarchist theory in general. Bakunin grew up near Moscow, where he moved to study philosophy and began to read the French Encyclopedists,...

. After earning his diploma in 1921, he enrolled in the department of engineering, but in November, 1923 changed to mathematics. Immediately after earning his laurea

Laurea

In Italy, the laurea is the main post-secondary academic degree.-Reforms due to the Bologna process:Spurred by the Bologna process, a major reform was instituted in 1999 to introduce easier university degrees comparable to the bachelors...

, in 1925, he became the assistant of Mauro Picone

Mauro Picone

Mauro Picone was an Italian mathematician. He is known for the Picone identity, for the Sturm-Picone comparison theorem and for being the founder of the Istituto Nazionale per le Applicazioni del Calcolo, presently named after him...

, who in that year was called to the University of Naples, where he remained until 1932.

Unanswered Questions

Encyclopedia

**Renato Caccioppoli**was an Italian

Italy

Italy , officially the Italian Republic languages]] under the European Charter for Regional or Minority Languages. In each of these, Italy's official name is as follows:;;;;;;;;), is a unitary parliamentary republic in South-Central Europe. To the north it borders France, Switzerland, Austria and...

mathematician

Mathematician

A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

.

## Biography

Born in NaplesNaples

Naples is a city in Southern Italy, situated on the country's west coast by the Gulf of Naples. Lying between two notable volcanic regions, Mount Vesuvius and the Phlegraean Fields, it is the capital of the region of Campania and of the province of Naples...

, Campania

Campania

Campania is a region in southern Italy. The region has a population of around 5.8 million people, making it the second-most-populous region of Italy; its total area of 13,590 km² makes it the most densely populated region in the country...

, he was the son of Giuseppe Caccioppoli (1852–1947), a surgeon

Surgeon

In medicine, a surgeon is a specialist in surgery. Surgery is a broad category of invasive medical treatment that involves the cutting of a body, whether human or animal, for a specific reason such as the removal of diseased tissue or to repair a tear or breakage...

, and his second wife Sofia Bakunin (1870–1956), daughter of the Russian revolutionary Mikhail Bakunin

Mikhail Bakunin

Mikhail Alexandrovich Bakunin was a well-known Russian revolutionary and theorist of collectivist anarchism. He has also often been called the father of anarchist theory in general. Bakunin grew up near Moscow, where he moved to study philosophy and began to read the French Encyclopedists,...

. After earning his diploma in 1921, he enrolled in the department of engineering, but in November, 1923 changed to mathematics. Immediately after earning his laurea

Laurea

In Italy, the laurea is the main post-secondary academic degree.-Reforms due to the Bologna process:Spurred by the Bologna process, a major reform was instituted in 1999 to introduce easier university degrees comparable to the bachelors...

, in 1925, he became the assistant of Mauro Picone

Mauro Picone

Mauro Picone was an Italian mathematician. He is known for the Picone identity, for the Sturm-Picone comparison theorem and for being the founder of the Istituto Nazionale per le Applicazioni del Calcolo, presently named after him...

, who in that year was called to the University of Naples, where he remained until 1932. Picone immediately discovered Caccioppoli's gifts and pointed him towards research in mathematical analysis

Mathematical analysis

Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...

. In the course of the next five years, Caccioppoli published about thirty 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, the chair of analisi-algebrica at the University of Padova. In 1934 he returned to Naples to accept the chair in group theory

Group theory

In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...

; 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 Pontanian Academy, and in 1947 a correspondent member of the Accademia dei Lincei

Accademia dei Lincei

The Accademia dei Lincei, , is an Italian science academy, located at the Palazzo Corsini on the Via della Lungara in Rome, Italy....

, 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 Academy of Lincei bestowed on him the national prize of physical, mathematical, and natural sciences.

He was an excellent pianist

Pianist

A pianist is a musician who plays the piano. A professional pianist can perform solo pieces, play with an ensemble or orchestra, or accompany one or more singers, solo instrumentalists, or other performers.-Choice of genres:...

, 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

Adolf Hitler

Adolf Hitler was an Austrian-born German politician and the leader of the National Socialist German Workers Party , commonly referred to as the Nazi Party). He was Chancellor of Germany from 1933 to 1945, and head of state from 1934 to 1945...

and Benito Mussolini

Benito Mussolini

Benito Amilcare Andrea Mussolini was an Italian politician who led the National Fascist Party and is credited with being one of the key figures in the creation of Fascism....

, when the latter was visiting Naples. Together with his companion Sara Mancuso, he had the French national anthem

Anthem

The term anthem means either a specific form of Anglican church music , or more generally, a song of celebration, usually acting as a symbol for a distinct group of people, as in the term "national anthem" or "sports anthem".-Etymology:The word is derived from the Greek via Old English , a word...

played by an orchestra, after which he began to speak against fascism

Fascism

Fascism is a radical authoritarian nationalist political ideology. Fascists seek to rejuvenate their nation based on commitment to the national community as an organic entity, in which individuals are bound together in national identity by suprapersonal connections of ancestry, culture, and blood...

and Nazism

Nazism

Nazism, the common short form name of National Socialism was the ideology and practice of the Nazi Party and of Nazi Germany...

in the presence of OVRA

OVRA

The Organizzazione per la Vigilanza e la Repressione dell'Antifascismo was the secret police of the Kingdom of Italy, founded in 1927 under the regime of Fascist dictator Benito Mussolini and during the reign of King Victor Emmanuel III. The German Gestapo were the equivalent of the 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.

Non compos mentis

Non compos mentis is a term meaning 'not of sound mind'. Non compos mentis derives from the Latin non meaning "not", compos meaning "having ", and mentis , meaning "mind"...

His most important works, out of a total of around eighty publications, relate to functional analysis

Functional analysis

Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...

and the calculus of variations

Calculus of variations

Calculus of variations is a field of mathematics that deals with extremizing functionals, as opposed to ordinary calculus which deals with functions. A functional is usually a mapping from a set of functions to the real numbers. Functionals are often formed as definite integrals involving unknown...

. Beginning in 1930 he dedicated himself to the study of differential equation

Differential equation

A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...

s, the first to use a topological-functional approach. Proceeding in this way, in 1931 he extended the Brouwer fixed point theorem

Brouwer fixed point theorem

Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after Luitzen Brouwer. It states that for any continuous function f with certain properties there is a point x0 such that f = x0. The simplest form of Brouwer's theorem is for continuous functions f from a disk D to...

, 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 space

Banach space

In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...

s 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

Felix Bernstein

Felix Bernstein was a German Jewish mathematician known for developing a theorem of the equivalence of sets in 1897, and less well known for demonstrating the correct blood group inheritance pattern of multiple alleles at one locus in 1924 through statistical analysis...

. At the same time he studied analytic function

Analytic function

In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions, categories that are similar in some ways, but different in others...

s of several complex variables

Several complex variables

The theory of functions of several complex variables is the branch of mathematics dealing with functionson the space Cn of n-tuples of complex numbers...

, i.e. analytic functions whose domain

Domain (mathematics)

In mathematics, the domain of definition or simply the domain of a function is the set of "input" or argument values for which the function is defined...

belongs to the vector space

**C**

^{}, 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

Residue (complex analysis)

In mathematics, more specifically complex analysis, the residue is a complex number proportional to the contour integral of a meromorphic function along a path enclosing one of its singularities...

formula for functions of two complex variables in 1949.

In 1935 Caccioppoli proved the analyticity of class 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 set

Caccioppoli set

In mathematics, a Caccioppoli set is a set whose boundary is measurable and has a finite measure. A synonym is set of finite perimeter...

s or sets of finite perimeter

Caccioppoli set

In mathematics, a Caccioppoli set is a set whose boundary is measurable and has a finite measure. A synonym is set of finite perimeter...

, was introduced and studied.

His last works, produced between 1952 and 1953, deal about a class of pseudoanalytic function

Pseudoanalytic function

In mathematics, pseudoanalytic functions are functions introduced by that generalize analytic functions and satisfy a weakened form of the Cauchy–Riemann equations....

s, introduced by him to extend certain properties of analytic function

Analytic function

In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions, categories that are similar in some ways, but different in others...

s.

In his last years, the disappointments of politics

Politics

Politics is a process by which groups of people make collective decisions. The term is generally applied to the art or science of running governmental or state affairs, including behavior within civil governments, but also applies to institutions, fields, and special interest groups such as the...

and his wife's desertion, together perhaps with the weakening of his mathematical vein, pushed him into alcoholism

Alcoholism

Alcoholism is a broad term for problems with alcohol, and is generally used to mean compulsive and uncontrolled consumption of alcoholic beverages, usually to the detriment of the drinker's health, personal relationships, and social standing...

. His growing instability had sharpened his "strangenesses", to the point that the news of his suicide on May 8, 1959 by a gunshot to the head did not surprise those who knew him. He died at his home in Palazzo Cellammare

In 1992 his tormented personality was remembered in a film directed by Mario Martone

Mario Martone

Mario Martone is an Italian film director and screenwriter. He has directed 15 films since 1985. His film L'amore molesto was entered into the 1995 Cannes Film Festival...

,

*The Death of a Neapolitan Mathematician*(

*Morte di un matematico napoletano*

), in which he was portrayed by Carlo Cecchi

Morte di un matematico napoletano

Death of a Neapolitan Mathematician is a 1992 Italian drama film, written and directed by Mario Martone....

Carlo Cecchi

Carlo Cecchi is an Italian actor.Born in Florence, Cecchi studied under the Living Theatre and with the Workshop of Eduardo De Filippo. In 1968 he made his debut for cinema in La sua giornata di gloria...

. He names also an asteroid

Asteroid

Asteroids are a class of small Solar System bodies in orbit around the Sun. They have also been called planetoids, especially the larger ones...

, 9934 Caccioppoli

9934 Caccioppoli

9934 Caccioppoli is a C-type main belt asteroid of the solar system. It orbits the Sun once every 4.14 years.Discovered on October 20, 1985 by Ted Bowell it was given the provisional designation "1985 UC"...

.