In 1858, together with Betti from Pisa and his own student Casorati, Brioschi visited Göttingen, Berlin and Paris. In Göttingen they met with Riemann and quickly realised the significance of the contributions he was making. This visit is often taken as the point where Italian mathematics joined the mainstream of European mathematics. When they returned to Italy they began to publish Italian translations of Riemann's works and Brioschi began to lecture on these ideas. Casorati was not the only student that Brioschi had in Pavia, for he also advised the doctoral students Cremona (doctorate in 1853) and Beltrami (doctorate in 1856).
Brioschi was living through a very dramatic period in Italian politics as the country moved towards unification. Camillo Benso, conte di Cavour, was one of the main leaders in this bid for unification. In 1859 he appointed Brioschi to a committee which had as its remit the reform of secondary schools. It is worth noting that Giuseppe Verdi, the leading Italian composer of operas, was also a member of the same committee. The Kingdom of Italy came into existence in 1860 and was officially proclaimed on 17 March 1861, by a parliament assembled in Turin. The reforms put forward by the committee, which were to a certain extent modelled on German reforms, were implemented after the Kingdom of Italy was created.
From 1861 to 1862 Brioschi was secretary of the Italian Ministry of Education. In 1863 he founded the Politecnico, the Technical University of Milan, serving as its director and professor of mathematics and hydraulics for the rest of his life. The Politecnico made a highly significant contribution to Italian life as it led the way in the development of electricity. Northern Italy had a large potential for generating electricity by hydroelectric power and, thanks to research and development done in the Politecnico, in 1883 Italy became the first European nation with an hydroelectric power station. Brioschi also continued to work more generally for improving Italian education and, from 1870 to 1882, he was on the Executive Council of the Ministry of Education.
L Pepe, in a review of  writes:-
Francesco Brioschi was an important mathematician in the European context owing to his contributions to the theory of algebraic equations and to the applications of mathematics to hydraulics. He also devoted himself to providing Italy with a scientific culture which was comparable with the new role of Italy as an emerging European nation.Brioschi studied the theory and application of determinants and published a major work Teoria dei determinanti Ⓣ in 1854. His four part work La teorie dei covarianti e degli invarianti delle forme binarie, e le sue principali applicazioni Ⓣ was published in 1858, 1859, 1860 and 1861. Hermite said that this work was:-
... one of the major mathematical achievements of our time.Brioschi also obtained new results in the theory of transformation of elliptic and abelian functions. One of his most important results was his application of elliptical modular functions to the solution of equations of the fifth degree in 1858. His method uses ideas due to Hermite and Betti who had both worked on the problem. This problem was also solved by Kronecker at almost exactly the same time. Brioschi however later went on to solve sixth degree equations using similar techniques. In 1888, Maschke proved that a particular sixth-degree equation could be solved by using hyperelliptic functions and Brioschi then showed that any sixth-degree algebraic equation could be reduced to Maschke's equation and therefore solved using hyperelliptic functions.
We mentioned above that he was professor of hydraulics yet we have not yet looked at the applied mathematical aspect of his research :-
In mechanics Brioschi dealt with problems of statics, proving Möbius's results by analytic means; with the integration of equations in dynamics, according to Jacobi's method; with hydrostatics; and with hydrodynamics. His work as a hydraulic engineer was significant, although it is reflected comparatively little in his publications. Brioschi used the findings of a series of major projects or participated in the projects' development - for example, in the regulation of the Po and Tiber ...Brioschi highly valued the significance of pure mathematics in applications and greatly influenced the direction of mathematics in Italy. He made fundamental contributions in 1857 to changing the Annali di Scienze Matematiche e Fisiche into a journal of international standing, in 1886 to the new series of the journal Politecnico, to an Italian edition of Euclid's Elements for secondary education, and he edited Leonardo da Vinci's Codice Atlantico Ⓣ which was a major contribution to the understanding of the history of science and technology.
Brioschi, however, was not one to lay the foundations of new areas of mathematics but rather one who grasped the power of new ideas he met rapidly and saw much further than others how these ideas could be used to make major advances. He said modestly of himself:-
I am only a calculator.This is, of course, a vast understatement but it does show that he understood where his strengths lay.
Article by: J J O'Connor and E F Robertson