Pia Maria Nalli (10 February 1886 – 27 September 1964) was an Italian mathematician known for her work on the

summability In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a finite limit. If a series converges, the individual terms of the series must ...

Fourier series A Fourier series () is an Series expansion, expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series. By expressing a function as a sum of sines and cosines, many problems ...

, on

Morera's theorem In complex analysis, a branch of mathematics, Morera's theorem, named after Giacinto Morera, gives a criterion for proving that a function is holomorphic. Morera's theorem states that a continuous, complex-valued function ''f'' defined on an ...

for analytic functions of several variables and for finding the solution to the

Fredholm integral equation In mathematics, the Fredholm integral equation is an integral equation whose solution gives rise to Fredholm theory, the study of Fredholm kernels and Fredholm operators. The integral equation was studied by Ivar Fredholm. A useful method to ...

of the third kind for the first time. Her research interests ranged from

algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...

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 (for example, Inner product space#Definition, inner product, Norm (mathematics ...

and

tensor analysis In mathematics and physics, a tensor field is a function (mathematics), function assigning a tensor to each point of a region (mathematics), region of a mathematical space (typically a Euclidean space or manifold) or of the physical space. Tens ...

; she was a speaker at the 1928

International Congress of Mathematicians The International Congress of Mathematicians (ICM) is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union (IMU). The Fields Medals, the IMU Abacus Medal (known before ...

. She is also remembered for her struggles against discrimination against women in the Italian university hiring system. A street in Rome is named after her.

Life and academic career

Early life and education

Nalli was born on February 10, 1886, in

Palermo Palermo ( ; ; , locally also or ) is a city in southern Italy, the capital (political), capital of both the autonomous area, autonomous region of Sicily and the Metropolitan City of Palermo, the city's surrounding metropolitan province. The ...

, to a middle-class family with seven children.. She studied at the

University of Palermo The University of Palermo () is a public university, public research university in Palermo, Italy. It was founded in 1806, and is currently organized in 12 Faculties. History The University of Palermo was officially founded in 1806, although it ...

, where she obtained a

laurea In Italy, the ''laurea'' is the main post-secondary academic degree. The name originally referred literally to the laurel wreath, since ancient times a sign of honor and now worn by Italian students right after their official graduation ceremo ...

in 1910 under the supervision of Giuseppe Bagnera, with a thesis concerning

, and in the same year joined the

Circolo Matematico di Palermo The Circolo Matematico di Palermo (Mathematical Circle of Palermo) is an Italian mathematical society, founded in Palermo by Sicilian geometer Giovanni B. Guccia in 1884.

. After finishing her studies, Nalli assisted Bagnera in Palermo in 1911 and then began working as a school teacher. She completed a

habilitation Habilitation is the highest university degree, or the procedure by which it is achieved, in Germany, France, Italy, Poland and some other European and non-English-speaking countries. The candidate fulfills a university's set criteria of excelle ...

thesis in 1914 on the theory of

integral In mathematics, an integral is the continuous analog of a Summation, sum, which is used to calculate area, areas, volume, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental oper ...

s, and continued to work on

Fourier analysis In mathematics, Fourier analysis () is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Joseph Fo ...

and

Dirichlet series In mathematics, a Dirichlet series is any series of the form \sum_^\infty \frac, where ''s'' is complex, and a_n is a complex sequence. It is a special case of general Dirichlet series. Dirichlet series play a variety of important roles in anal ...

for the next several years.

Academic career

Nalli served as assistant to Giuseppe Bagnera at the University of Palermo from 1 April 1911 to 16 November 1911. She then taught at a number of secondary schools, first in the girls' school at Avellino, then in Trapani, and from 16 November 1912 in the girls' technical school in Palermo. During this time Nalli continued her research, completing her thesis "''Esposizione e confronto critico delle diverse definizioni proposte per l'integrale definito di una funzione limitata o no''", a study of the theory of the integral based on recent work on the subject by

Émile Borel Félix Édouard Justin Émile Borel (; 7 January 1871 – 3 February 1956) was a French people, French mathematician and politician. As a mathematician, he was known for his founding work in the areas of measure theory and probability. Biograp ...

Henri Lebesgue Henri Léon Lebesgue (; ; June 28, 1875 – July 26, 1941) was a French mathematician known for his Lebesgue integration, theory of integration, which was a generalization of the 17th-century concept of integration—summing the area between an ...

Charles de la Vallée Poussin,

Giuseppe Vitali Giuseppe Vitali (26 August 1875 – 29 February 1932) was an Italian mathematician who worked in several branches of mathematical analysis. He gives his name to several entities in mathematics, most notably the Vitali set with which he was the fi ...

and Arnaud Denjoy. In 1921, Nalli became extraordinary professor at the

University of Cagliari The University of Cagliari () is a public research university in Cagliari, Sardinia, Italy. It was founded in 1606 and is organized in 11 faculties. History The ''Studium Generalis Kalaritanum'' was founded in 1606 along the lines of the old ...

. She had been ranked second to

Mauro Picone Mauro Picone (2 May 1885 – 11 April 1977) was an Italian mathematician. He is known for the Picone identity, the Sturm-Picone comparison theorem and being the founder of the Istituto per le Applicazioni del Calcolo, presently named after hi ...

in the competition for the position, possibly in part because she was female, but Picone chose to stay at the

University of Catania The University of Catania () is a university located in Catania, Sicily. Founded in 1434, it is the oldest university in Sicily, the 13th oldest in Italy, and the 29th oldest in the world. With over 38,000 enrolled students, it is the largest uni ...

and become head of mathematics there, so the Cagliari position fell to Nalli. In 1923, she was listed first in a search for a position at the

University of Pavia The University of Pavia (, UNIPV or ''Università di Pavia''; ) is a university located in Pavia, Lombardy, Italy. There was evidence of teaching as early as 1361, making it one of the List of oldest universities in continuous operation, oldest un ...

, but not offered the position. Finally, after similar mistreatment from several other universities, she moved to the University of Catania as a full professor in 1927. At around this time, perhaps encouraged by

Tullio Levi-Civita Tullio Levi-Civita, (; ; 29 March 1873 – 29 December 1941) was an Italian mathematician, most famous for his work on absolute differential calculus ( tensor calculus) and its applications to the theory of relativity, but who also made signifi ...

, she switched her research focus from

tensor calculus In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other ...

. She was an

invited speaker at the International Congress of Mathematicians An invitation system is a method of encouraging people to join an organization, such as a Club (organization), club or a website. In regular society, it refers to any system whereby new members are chosen; they cannot simply apply. In relation to w ...

in 1928.

Death and legacy

Nalli died on 27 September 1964, in Catania. A street in Rome, the Via Pia Nalli, is named after her.

Selected works

Pia Nalli published 61 mathematical works, including the monograph and a textbook.According to the list of her works included in . Her "Selected works" include this monograph plus eleven articles on topics mainly belonging to

functional Functional may refer to: * Movements in architecture: ** Functionalism (architecture) ** Form follows function * Functional group, combination of atoms within molecules * Medical conditions without currently visible organic basis: ** Functional s ...

and mathematical analysis: the following list includes also her doctoral thesis and other works on

*. *. *. *. *. *.

Notes

References

Biographical and general references

*. *. *.

Scientific references

*. A survey paper describing the development of

infinitesimal calculus Calculus is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations. Originally called infinitesimal calculus or "the calculus of ...

during the twentieth century and trying to trace possible scenarios for its future evolution. *. A survey paper describing the development of

mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series ( ...

in Italy during the two world wars, describing the contributions of several Italian scientists who worked during that period.

External links

* {{DEFAULTSORT:Nalli, Pia Maria 1886 births 1964 deaths 20th-century Italian mathematicians Mathematicians from Sicily Italian mathematical analysts Functional analysts University of Palermo alumni Academic staff of the University of Cagliari Academic staff of the University of Catania Scientists from Palermo 20th-century women mathematicians