Francesco Gerbaldi was appointed to the chair of analytic and projective geometry at the University of Palermo in 1890. In 1893 Bagnera, who was now studying mathematics, was appointed as Gerbaldi's assistant. His first contributions to mathematics studied Fuchsian and Kleinian functions and linear systems of algebraic curves :-
Bagnera began his scientific work with a series of works marked by the elegant simplicity which was the essential character of all his work.In 1895 Bagnera was awarded his laurea in mathematics by the University of Palermo and, in the following year, he published two papers Sul teorema di esistenza delle funzioni fuchsiane , which further developed ideas which had been introduced by Henri Poincaré, and Sul luogo dei contatti tripunti delle curve di un fascio con le curve di una rete . The second of these papers was published in the journal of the Mathematical Circle of Palermo. It was around this time that he became interested in the theory of finite groups, particularly the study of finite collineation groups. On this new topic, he published Sopra la costruzione del gruppo dell'icosaedro (1897) and a number of papers on group theory in the following year: Sopra i divisori normali d'indice primo di un gruppo finito ; Un teorema relativo agli invarianti delle sostituzioni di un gruppo kleiniano ; and La composizione dei gruppi finiti il cui grado è la quinta potenza di un numero primo . Two important papers by Bagnera contain the list of the groups of order p5, where p is a prime.
He was named professor at the R Educatorio Maria Adelaide di Palermo in 1897. This school had been built as a nunnery in 1697 but was turned into the Educatorio Carolino boarding school in 1735 before being renamed the Educatorio Maria Adelaide in 1860. In 1899 Bagnera was appointed as a teaching assistant in algebraic analysis at the University of Palermo and Gerbaldi encouraged him to compete for chairs. In 1901 he entered the competition for the extraordinary professorship of infinitesimal calculus at the University of Messina and was appointed to the post. He was appointed as an ordinary professor of algebra and analytic geometry at the University of Messina in 1905.
From 1906 to 1909 Bagnera worked in collaboration with Michele de Franchis on the study of irregular surfaces, obtaining fundamental results for the classification of hyperelliptic surfaces. During these years Michele de Franchis, who was also a Sicilian, was professor of projective and descriptive geometry at the University of Parma, and Bagnera and de Franchis collaborated on an outstanding series of papers on the theory of hyperelliptic surfaces. They won the Paris Academy of Sciences' Bordin prize in 1909 for their classification of hyperelliptic surfaces. A hyperelliptic surface is an algebraic surface which can be rationally covered by an abelian surface. If not a ruled surface, it is a quotient of an abelian surface by a finite group of automorphisms. There is, however, a rather strange story attached to the 1909 Bordin prize since the same prize had been awarded in 1907 to Federigo Enriques and Francesco Severi, also for classifying hyperelliptic surfaces :-
Strange as it may seem that two couples get two prizes for the same theorem, instead of sharing one, this story is even more complicated, since the first version of the paper by Enriques and Severi was withdrawn after a conversation of Severi with de Franchis, and soon corrected. Bagnera and de Franchis were only a little later, since they had to admit a restriction; their proof however was simpler ...The remarkable papers Bagnera wrote with de Franchis over this period are: Sopra le superficie algebriche che hanno le coordinate del punto generico esprimibili con funzioni meromorfe quadruplamente periodiche di due parametri (1907); Sur les surfaces hyperelliptiques (1907); Le superficie algebriche le quali ammettono una rappresentazione parametrica mediante funzioni iperellittiche di due argomenti (1908); Sopra le funzioni algebriche che si lasciano risolvere con X,Y,Z funzioni quadruplamente periodiche di due parametri (1909); and Intorno alle superficie regolari di genere zero che ammettono una rappresentazione parametrica mediante funzioni iperellittiche di due argomenti (1909).
Bagnera was extremely fortunate to escape with his life when Sicily was hit with a 7.1 magnitude earthquake on 28 December 1908. He was still holding the chair of infinitesimal calculus at the University of Messina at this time but he had travelled to Palermo to spend the Christmas vacation there. On the evening of 27 December, he went to the railway station in Palermo to catch a train back to Messina but he was late and missed the train. Therefore, he had to spend the night of 27/28 December in Palermo waiting to catch a train the next morning. However, at 5.20 in the morning of 28th, the earthquake struck, its epicentre being very close to Messina. Shortly after the earthquake, Messina was struck by a tsunami. The devastation produced by the earthquake and the tsunami was horrific with 91% of buildings in the city being destroyed and 70,000 of the inhabitants being killed. Palermo was 200 km from the epicentre of the earthquake and suffered very much less damage. In fact many of those who survived the Messina destruction, fled to Palermo in the days following the natural disaster. Most of the buildings of the University of Messina had been destroyed in the earthquake, including their famous library building. The university was not able to reopen properly until 1914 although the law school did manage to reopen in 1909. Bagnera had no home and no job to return to in Messina.
In 1909, already living in Palermo, he was appointed to the chair of infinitesimal and higher analysis at the University there. Later he served as dean of the faculty of sciences at the university. He also taught financial mathematics at the Higher Institute of Commerce in Palermo and served as director of that Institute. Among his pupils at Palermo we should mention Michele Cipolla and Pia Nalli. On 14 April 1914 the thirtieth anniversary of the founding of the Mathematical Circle of Palermo was celebrated at the University of Palermo. Bagnera was one of the members of the organising committee of this international event. A gold medal was presented to Giovanni Battista Guccia who founded both the Mathematical Circle of Palermo and its journal the Rendiconti. Bagnera was appointed to the chair of infinitesimal and higher analysis at the University of Rome in 1921, taking up the appointment at the beginning of the following year.
After the 1908 earthquake, Bagnera published little in the way of research papers but did publish his lecture notes (some in lithographed form). These are: Lezioni di calcolo delle variazioni (1913), Lezioni di matematica finanziaria (1914), Lezioni di calcolo bancario e commerciale (1914), Corso di Analisi Infinitesimale (1915), Teoria dei numeri reali (1919), Lezioni di calcolo infinitesimale per gli allievi ingegneri (1924), Lezioni di analisi algebrica (1926), Elementi di algebra (1926), and Lezioni sulla teoria delle funzioni analitiche (1927). These are quality works written by an excellent lecturer who polished his material to the highest possible standards :-
Bagnera's higher level courses were characterized by having great clarity and the highest scientific rigour.In addition to the Bordin prize awarded by the Paris Académie des Sciences, which we mentioned above, Bagnera was honoured with election to the Accademia Nazionale dei Lincei (which awarded him their mathematical sciences prize in 1901). In addition, he was elected to the Accademia di scienze, lettere e belle arti of Palermo, the Accademia Peloritana of Messina, the National Academy of Sciences of Italy (the "Academy of Forty"), and the Società italiana per il progresso delle scienze. He was a member and served on the board of directors of the Mathematical Circle of Palermo. He was made an honorary professor of the University of Washington. We should also mention that he is remembered with the via Giuseppe Bagnera in his home town of Bagheria and the Scuole Primarie e Secondarie, Scuola Media Statale Bagnera, on the via Giuseppe Bagnera in Rome. The Associazione Culturale "Giuseppe Bagnera", based in Bagheria, is named after Giuseppe Bagnera. The Honorary President of the Association is the engineer Giuseppe Bagnera, who is the grandson of the subject of this biography.
Article by: J J O'Connor and E F Robertson