**Pia Nalli**'s parents were Giovanni Nalli, who worked as a clerk, and Carmela Fazello. Pia was the fourth of Giovanni and Carmela Nalli's seven children. Among their other children, we mention at this stage that her brother Vitangelo Nalli (1876-1932) entered the medical profession (but was well-known as a passionate proponent of the Esperanto language and an author of books on that topic), and her brother Paolo Nalli (1887-1967) became a writer and librarian holding positions as head of some of the main libraries of Italy, for example at Emilia, Lombardy, Naples, and Verona. Pia was educated in Palermo and after graduating from the local school, entered the University of Palermo.

While Nalli was studying at the University of Palermo, Sicily was hit with a 7.1 magnitude earthquake on 28 December 1908. The epicentre of the earthquake was close to Messina, around 200 km from Palermo, so the city of Palermo escaped with minor destruction. However, one consequence of the earthquake was that Giuseppe Bagnera, who was on the staff at Messina at the time, joined the University of Palermo and was advisor to Nalli during the final stages of writing her thesis. She was awarded her laurea on 10 January 1910 after submitting a thesis on algebraic geometry.

Giovanni Battista Guccia set up the Mathematical Circle of Palermo in 1884 making Palermo an important mathematical centre. The journal for the new society was the *Rendiconti del Circolo matematico di Palermo* which was edited by Guccia. Nalli joined the Mathematical Circle in 1910, after graduating from Palermo. She was joining the largest mathematical society in the world with over 900 members, the majority of whom came from outside Italy. The leading mathematicians of the day were members, for example in the early years of the 20^{th} century Henri Poincaré, Jacques Hadamard, David Hilbert, Vito Volterra, Federigo Enriques, Guido Castelnuovo, Corrado Segre, Giuseppe Peano, Tullio Levi-Civita, Gregorio Ricci-Curbastro and Luigi Bianchi were members. At the time when Nalli joined, the Mathematical Circle had, over the preceding few years, published some of the most important mathematical papers in the world in the *Rendiconti*. Nalli submitted her first paper *Riduzione di un fascio di curve piane di genere uno, corrispondente a se stesso in una trasformazione birazionale involutoria del piano* Ⓣ to the Mathematical Circle in May 1910. It was read at a meeting of the society on 14 August 1910 and the paper was printed on 22 December 1910. Her second paper *Sopra una definizione di dominio piano limitato da una curva continua, senza punti multipli* Ⓣ was submitted to the Mathematical Circle in September 1911, read at a meeting of the society on 12 November 1911 and printed on 25 November 1911.

Nalli was Giuseppe Bagnera's assistant 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. Despite a very heavy teaching load, Nalli was able to carry out high quality research during her time as a school teacher. She worked on her habilitation thesis and, in 1914, she completed an impressive thesis *Esposizione e confronto critico delle diverse definizioni proposte per l'integrale definito di una funzione limitata o no* Ⓣ. In this work she studied the theory of the integral, in particular bringing together recent work on the subject by Émile Borel, Henri Lebesgue, Charles de la Vallée Poussin, Giuseppe Vitali and Arnaud Denjoy.

A function *f* can be expanded as a Fourier series and the coefficients calculated as integrals. If *f* is Riemann integrable, then du Bois-Reymond proved that the Riemann integral can be used to calculate the coefficients. De la Vallee Poussin generalised this by showing that if *f* is Lebesgue integrable, then the Lebesgue integral can be used to calculate the coefficients. Between 1915 and 1918, Nalli concentrated her efforts on extending the theorem of de la Vallee Poussin to functions *f* that are integrable using the restricted Denjoy integral. Here she was working with very new ideas for Arnaud Denjoy had only announced his integral in two short notes in 1912 and gave full details in four long papers published between 1915 and 1917. Nalli published *Sulle serie di Fourier delle funzione non assolutamente integrabili* Ⓣ in 1915. Then, two years later, she proved that if *f* is integrable using the restricted Denjoy integral then this integral can be used to calculate the coefficients. She also proved uniqueness theorems for the trigonometric series expansion for this class of functions. Progress along these lines came to a stop at that point and Nalli's result was the high point in this area for a long time. Also during 1915-1917, she focused her attention on the problem of summation of series, with special reference to Dirichlet series.

In the year 1919 Nalli started to work on issues related to the theory of linear integral equations and the study of integral operators. Her work continued that begun by Ivar Fredholm on this topic. She published her results in a series of four notes entitled *Sulle equazioni integrali* Ⓣ in 1919 and the memoir *Generalizzazione di alcuni punti della teoria delle equazioni integrali di Fredholm* Ⓣ in the same year. In 1922 she published another important paper *Sulle operazioni funzionali linear* Ⓣ on this topic. Already, at this time, Nalli was a university lecturer being appointed as an extraordinary professor of analysis at the University of Cagliari in 1921. She had entered the competition for the chair of infinitesimal analysis at the University of Cagliari, being the only woman among the eight applicants that included Mauro Picone and Giuseppe Vitali. The panel examining the candidates put Picone first and Nalli second. It is now widely believed that Nalli was actually ranked first but pushed down the list because she was a woman. Picone, however, who had been appointed as Professor of Analysis at the University of Catania in 1919, may well have been using his application to Cagliari as a way to improve his position at Catania, for he was quickly offered the position as Head of Mathematics at Catania. Cagliari then gave Nalli an extraordinary professorship of analysis.

Although she was a professor at Cagliari, Nalli continued to enter competitions for professorships at other universities. She was ranked third in the competition for the chair of calculus at the University of Modena in 1922 but then ranked first in the competition for the chair of calculus at the University of Pavia in the following year. However, despite being ranked first, Pavia did not appoint Nalli to the chair and she wrote a strong letter of complaint to the rector of the university as well as writing to Tullio Levi-Civita complaining bitterly about the injustice. She complained to Levi-Civita about (see [4]):-

Other similar events did follow, for she was placed second in the competition for the chair of algebraic analysis at the University of Catania in 1926 and second in the competition for the chair of algebraic analysis at the University of Florence in the same year. We do not know what reply she had from Levi-Civita but he seems to have given her encouragement since, in February 1927, she was took up an appointment as professor at the University of Catania. However, the greatest indication of the reply Nalli must have received is seen from the fact that at this time she changed her research topic and after this worked on tensor calculus, the topic for which Levi-Civita is famed. Her first publications on this new topic were... the shameful persecution to which I have been subjected for a few years, all based on fantastic slander, ridiculous and anonymous(for example, that of my lack of educational quality, finally exploded by two great results in two recent competitions)has happened again with the usual treatment by the Faculty of Pavia. Other similar events will follow: of that I'm absolutely sure, because there is very little danger in persecuting a woman, who would not be allowed even the slightest verbal outburst, under the sentence of being subjected to gossip. I just call it gossip!

*Sul parallelismo di Levi-Civita e sopra certe possibili estensioni*Ⓣ (1928) and

*Parallelismo e coordinate geodetiche*Ⓣ (1929).

Her move to Catania was, like so much of her career, only made after considerable controversy. When the dean of the faculty of sciences told her the news, he noted that the "resolution was subject to official participation by the ministry". The next competition for a mathematics chair was in Florence and she had already entered that competition before hearing from the dean at Catania. Nalli carried on and took part in the competition for the chair at Florence, and she was again ranked second (as we noted above) while the local candidate, who the faculty at Florence had been expecting would be appointed, was not ranked at all. The University of Catania claimed that Nalli had not responded quickly enough to their proposal of 4 August 1926 so withdrew their offer to her. Nalli was not going to accept this sort of treatment, so she wrote to the Minister of Education, Pietro Fedele (1873-1943), who was a modern historian at Turin before being made Education Minister in 1925. In her letter she listed all the competitions she had entered, and those she had won, and asked him to intervene with the University of Catania and insist that they had a moral obligation to vote again on their decision. Despite some time having now elapsed, no appointment had been made to the chair at the University of Catania and, after Pietro Fedele intervened, they offered Nalli a permanent chair of analysis starting on 16 February 1927.

The professorship at Catania did not suit Nalli, partly because she felt that her colleagues resented her being there, having only accepted her at Fedele's insistence, partly because she would have preferred to be in Palermo where she felt much more at home. However, there was resistance from the staff at Palermo to her being appointed there. In particular, Michele La Rosa (1880-1933), who held the chair of physics at Palermo but was also dean of the Faculty of Science before becoming rector of the university, was opposed to her appointment. Nalli wrote in a letter to the Minister of Education on 12 May 1928 (see, for example [4]):-

Let us indicate some of Nalli's publications after she moved to Catania. These include the papers: (with Giulio Andreoli)Professor La Rosa claims that I would not be able to keep discipline, because the pupils of Palermo are large and to keep discipline, according to professor la Rosa, requires a teacher with solid manly qualities. I protest against this assertion: I have always kept discipline, and if anything I can be blamed for being too harsh. To keep discipline there is no need to do battle with the students: you just do not be ridiculous. And I'm not ridiculous either as a woman or as a teacher.

*Sui processi integrali di Stieltjes*Ⓣ (1929);

*Spazi di Riemann di seconda classe*Ⓣ (1932);

*Risoluzione di due problemi classici per mezzo di una equazione di Volterra*Ⓣ (1938);

*Sopra un problema relativo ai trasporti rigidi di vettori negli spazi quadrimensionali*Ⓣ (1947);

*Trasporti rigidi di vettori negli spazi quadridimensionali*Ⓣ (1949); and

*Equazioni indipendenti dalla scelta delle variabili e caratterizzazione di varietà metriche*Ⓣ (1955). She also published the book

*Lezioni de calcolo differenziale assoluto*Ⓣ (1952) which was material from lecture courses she had given over many years.

Gaetano Fichera paints a sad picture of how Nalli was deprived of the recognition she deserved, presumably because she was a woman [2]:-

Despite the lack of honours given to Nalli in her lifetime, we note that she has been honoured with the naming of a street Via Pia Nalli in Rome.Her aspiration to teach in her hometown, Palermo, became increasingly frustrated and was for her a source of great bitterness when she saw mathematicians appointed of very different stature from her. Retiring from teaching from the Faculty of Catania, which she had served for thirty years, she was given the recognition of being proposed for the appointment as professor emeritus. But in the national scene, Pia Nalli was left in complete obscurity. No academy ever thought to welcome her among its members, she was never called to judge a university competition, ... and she never had a position of distinction or prestige. On the other hand, she had the respect of the authentic scientific fraternity, which prevented her from begging for recognition and positions. Tired out with the passing of the years, resentful of those who had been hostile, resentful that her exuberant southern temperament had, at times, given its bright tone, become more and more closed in on itself, in a desperately lonely life, not cheered by family affections. But the few who were her friends knew that behind the outward harshness of her character was concealed a soul who was sensitive to the delicate nuances of feeling.

**Article by:** *J J O'Connor* and *E F Robertson*