Mathematics in Pisa and History of the Department

The father of the Pisan mathematical school

The (recent) history of mathematics in Pisa can be said to begin in 1857, with the appointment of Enrico Betti to the Chair of Algebra at the University. It was in Pisa that Betti—born in Pistoia in 1823—had graduated in 1846 in Applied Mathematics under the guidance of Fabrizio Ottaviano Mossotti, Enrico Doveri, and Carlo Matteucci (future Minister of Public Education of unified Italy). For nearly forty years, Betti was the main architect of a profound scientific and cultural renewal that made Pisa and its University, together with the Scuola Normale (of which he was Director almost continuously from 1865 to 1892), an important center for research and training in mathematics, recognized throughout Europe. In addition to producing significant contributions in a wide range of fields—such as Galois theory, complex analysis, and algebraic topology (he introduced the notion of Betti numbers of a manifold)—Betti succeeded in founding a true school, where mathematicians such as Ulisse Dini, Luigi Bianchi, Gregorio Ricci Curbastro, and Vito Volterra studied and worked.

At the same time, Betti worked to renew the faculty of the Mathematical Sciences. In some cases—such as the proposal to bring Bernhard Riemann, a friend and extraordinary scientific interlocutor, to Pisa—the attempt failed; in other circumstances he was more successful, for instance with Eugenio Beltrami, who taught in Pisa between 1863 and 1866 and later became emeritus professor of our University.

Betti taught continuously at the University of Pisa from 1857 until 1892, shortly before his death in the summer of that year. He held the chairs of Algebra (1857–59), Analysis and Higher Geometry (1859–64), and then Mathematical Physics, Celestial Mechanics, and Astronomy, succeeding Mossotti.

Ulisse Dini and the foundations of Analysis

Among Betti’s students, Ulisse Dini played a particularly significant role in shaping the development of mathematics in Pisa—not only as a teacher and researcher, but also through his intense activities in leading academic institutions. He served as Rector of the University of Pisa, Director of the Scuola Normale, and founder of the “Application School for Engineers” (1913), which later evolved into the Faculty of Engineering.

Dini graduated in 1864 at just nineteen years old. After a short period of study in Paris under Joseph Bertrand and Charles Hermite, he took up the Chair of Higher Analysis in 1866, a position he would hold for more than fifty years. His lectures at our University are extraordinarily documented in several textbooks, such as Fondamenti per la teoria delle funzioni di variabili reali (Pisa 1878), Serie di Fourier e altre rappresentazioni analitiche delle funzioni di variabili reali (Pisa 1880), and the Lezioni di Analisi Infinitesimale, which appeared in various lithographed editions and were finally printed in two volumes (1907–1915). These works contain many important results, including the implicit function theorem that still bears his name, criteria for pointwise convergence of trigonometric series, and expansions in series of Bessel functions, known today as “Dini series”.

Among Dini’s students at the Scuola Normale and the University of Pisa were Gregorio Ricci Curbastro (who served as Dini’s assistant for Higher Analysis) and Guido Fubini.

Bianchi and the Pisan school of geometry

Another charismatic figure who grew under Betti’s guidance was Luigi Bianchi. Bianchi graduated in Pisa in 1877 with a thesis on the geometry of pseudospherical surfaces. After a brief period of specialization in Munich under Felix Klein, he began his teaching career first as an internal lecturer at the Scuola Normale and then, from 1886, as a professor at the University.

He taught analytic and projective geometry, differential geometry, the theory of elliptic functions, and more. His work as a passionate teacher left a large number of treatises, including the famous Lezioni di Geometria differenziale (published in many editions, the last in 1923–24), Teoria dei gruppi di sostituzioni e delle equazioni algebriche secondo Galois (1900), Lezioni sulla teoria delle funzioni di variabile complessa e delle funzioni ellittiche (1901, 2nd ed. 1916), Lezioni di geometria analitica (1904), Lezioni sulla teoria dei gruppi continui finiti di trasformazioni (1918), Lezioni sulla teoria dei numeri algebrici e principi d’aritmetica analitica (1921), and Lezioni sulla teoria delle equazioni differenziali lineari (teoria di Fuchs-Riemann) (1924).

Among Bianchi’s many students were Mario Pieri, Guido Fubini, Giovanni Sansone, Mauro Picone, Giuseppe Vitali, and Eugenio Elia Levi.

Alongside Bianchi, an important representative of the Pisan school of geometry between the 19th and 20th centuries was Eugenio Bertini,who held the Chair of Higher Geometry for nearly thirty years, from 1893 to 1921. Bertini published important works in algebraic geometry and authored influential treatises, including Introduzione alla geometria proiettiva degl’iperspazii (1907). His students in Pisa included Gaetano Scorza and Giacomo Albanese, who later (in 1927) took up the Chair of Analytic Geometry in Pisa.

Volterra and functional analysis

Vito Volterra — also a student of Betti and Dini — was one of the most renowned and internationally respected Italian mathematicians of the late 19th and early 20th centuries. He graduated in 1882 under Betti with a thesis on hydrodynamics, but it was likely Dini who exerted the strongest influence on his intellectual development. While still a student, Volterra published several important works on a problem raised by Dini in his analysis lectures, demonstrating the existence of differentiable functions whose derivatives are not Riemann integrable (according to the 1853 definition).

Shortly after graduation, on Betti’s initiative, Volterra was appointed assistant to the Chair of Rational Mechanics. The following year, he won the competition for the professorship of the same subject at the University of Pisa, where he also taught Graphic Statics and, after Betti’s death, Mathematical Physics. He later served as Dean of the Faculty of Sciences (1892–93).

During the decade he spent teaching in Pisa (before moving first to Turin and then to Rome, where he taught until 1931, when he was removed from his position for refusing to take an oath of loyalty to the Fascist regime), Volterra produced numerous contributions to mathematical analysis, particularly in the theory of integral equations, making him one of the founders of modern functional analysis.

The post–World War I period

If the early 20th century in Pisan mathematics was marked by the figures of Dini and Bianchi, the following generation found its central point of reference in Leonida Tonelli. Tonelli, who graduated in Bologna in 1907 under Salvatore Pincherle and Federigo Enriques (himself a former student of our University), arrived in Pisa only in 1930 at the invitation of Giovanni Gentile, then Royal Commissioner of the Scuola Normale. As in Bologna, Tonelli succeeded in Pisa in creating a prestigious school that quickly became an international point of reference for research on the calculus of variations. Among Tonelli’s many students were Lamberto Cesàri, Alessandro Faedo (Rector of the University of Pisa from 1958 to 1972), and Guido Stampacchia.

Faedo’s students in the immediate postwar years included Gianfranco Capriz—considered one of the modern fathers of electronic computation—Giorgio Dall’Aglio, and later Mario Miranda.

At Faedo’s initiative, Federico Cafiero (1914–1980) was appointed in 1956 to the then Institute of Mathematics of the Faculty of Sciences. His arrival in Pisa marked, as Giorgio Letta wrote, “a kind of ideal union between Tonelli’s Pisan school and Renato Caccioppoli’s Neapolitan school.”[1] This was followed by the appointments of Aldo Andreotti, Iacopo Barsotti, and Edoardo Vesentini, and somewhat later by those of Giovanni Prodi, Sergio Campanato, Guido Stampacchia, Enrico Giusti, and Enrico Bombieri, future Fields Medal recipient (1974).

In much more recent times, our Department welcomed among its students Alessio Figalli, who was awarded the Fields Medal in 2018 for his important contributions to optimal transport theory—an achievement that once again highlights the long-standing tradition of excellence of the Pisan mathematical school.

After the peaceful flight of Leopold II, Grand Duke of Tuscany, in April 1859, the fate of Tuscany was governed for a time by a provisional Government that led the Grand Duchy toward annexation to the newly unified Italian State. One of the first measures of the new political order was the reconstitution of the University of Pisa, which Leopold II had abolished in 1851 after the battalion of the Tuscan universities (Siena and Pisa) joined the First War of Independence. The Grand Duke had planned to merge them into a single “Etruscan University”, which he believed would be easier to control centrally. A decree issued by the provisional Government in July 1859 reorganized academic studies. Article 13 established the Faculty of Mathematical Sciences and two degree programs: Pure Mathematics and Applied Mathematics. The programs lasted four years and followed rather rigid curricula.

Pure Mathematics 1st year: Algebra, Analytic Geometry, Descriptive Geometry, Physics 2nd year: Differential Calculus and Principles of Integral Calculus, Technological Physics and Experimental Mechanics, Chemistry 3rd year: Integral Calculus, Mechanics, Higher Analysis, Geodesy 4th year: Integral Calculus, Mechanics, Higher Analysis, Mathematical Physics and Celestial Mechanics

Applied Mathematics 1st year: Algebra, Analytic Geometry, Descriptive Geometry, Physics 2nd year: Differential Calculus and Principles of Integral Calculus, Technological Physics and Experimental Mechanics, Descriptive Geometry and its Applications to Geometric Drawing, Chemistry 3rd year: Integral Calculus, Mechanics, Civil and Hydraulic Architecture, Mineralogy and Geology, Technological Physics and Experimental Mechanics 4th year: Mechanics, Civil and Hydraulic Architecture, Geodesy, Terrestrial Physics and Physical Geography, Agronomy

The teaching staff of the Faculty of Mathematical Sciences for the academic year 1860–1861 consisted of only ten professors who taught a few dozen students (in 1877–78, just over twenty students enrolled in Mathematical Sciences):

Giovanni Novi (Algebra), Fabio Sbragia (Analytic Geometry), Guglielmo Martolini (Descriptive Geometry, Civil and Hydraulic Architecture), Luigi Pacinotti (Technological Physics and Experimental Mechanics), Gaspero Botto (Differential Calculus and Principles of Integral Calculus), Giovanni Maria Lavagna (Integral Calculus), Enrico Betti (Analysis and Higher Geometry), Giovanni Barsotti (Analytical Mechanics), Ottaviano Mossotti (Mathematical Physics, Celestial Mechanics, and Geodesy), Angelo Nardi Dei (Geometric Drawing).

The Department of Mathematics was established in March 1982, together with the dissolution of the Institute of Mathematics of the Faculty of Mathematical, Physical, and Natural Sciences. From 1996 to 2012, it was joined by the Department of Applied Mathematics “Ulisse Dini”, created following the suppression of the Institute of Applied Mathematics of the Faculty of Engineering.

Below is a list of the Directors of both Departments.