This book is dedicated by the author to his beloved daughter Laura

In Germany, at the end of the nineteenth century, relying on his own prestige as an outstanding scientist, Felix Klein promoted a campaign in order to stimulate the diffusion, at all levels of teaching and in all types of schools, of an education solidly based on functional thinking. The concept of function was already almost three centuries old, with differential and integral calculus being not too much younger, and the theory of analytic functions of one complex variable had been fully developed in the course of the current century, yet functional thinking had not yet become the automatic and customary way of thinking for any educated person in whatever professional field.

About 120 years later, functional thinking was relatively well established. In the whole world the basic theory of functions and integral-differential calculus are taught at the level of the final years of high school and are an essential part of most university curricula, at least of those whose character is technical-scientific like physics, chemistry, biology, engineering in all of its declensions, statistics, and economics. The most important aspect of this evolution is that, while teaching and learning such mathematical topics, only the abstract concepts are dealt with: no one doubts what a function f(x) and its derivative ∂xf(x) might represent, what they might be good for, and so on. It is just clear, both to the students and to the teachers, that f(x) might describe anything in whatever field; so all efforts are focused on abstract constructions and manipulations. In this way calculus has finally reached the same status as grammar and syntaxis already did since antiquity. The educational process, elaborated through a long historical evolution that led to the medieval crystallization in the trivium (grammar, logic, and rhetoric; see Figure 1), focused on the formal structure of the language and not on its actual content. Everyone perfectly knew what language was good for and the discussion of the cases in declension was not preceded by any illustration of the practical advantages or of the concrete applications of such abstract notions. Assimilated and examined by each individual, grammar and logic became an automatic habit forging one’s thinking.

Figure 1 Priscian or “The Grammar” relief by Luca della Robbia from Giotto’s Bell Tower in Florence.

The contemporary assimilation and analysis of functional thinking is a great and valuable progress although even now it is not yet a full achievement: indeed there are still countries like Italy where, in the twenty-first century, you can become a medical doctor while ignoring functions and derivatives.

The reader has certainly noted that, while praising the educational virtues of grammar and syntaxis, I utilized the past tense. It was not without a reason. Indeed, while functional thinking was progressing, the centrality of trivium education started losing ground everywhere in the world with a wide spectrum of different velocities, but a uniform negative sign. The earliest disruption of classical education occurred in the Soviet Union with the October Revolution, and the outcome of the complete obliteration of Latin and Greek teaching in a country whose language is inflectional, like Latin is, is visible nowadays after one century of absence. By personal experience I know how difficult it is to argue in grammatical terms about the Russian language with Russian educated people, even academicians. Furthermore, present-day Russia is a country where being gramatny, namely, speaking and writing in the native language without introducing macroscopic mistakes, is a matter of distinction carefully boasted by those few who possess such an ability. The same decline, however, is rapidly occurring also in Europe and the United States through many ill-conceived reforms of the educational systems that have affected all the countries and are generally marked by a tendency to privilege an alluvion of disorganized information with respect to formation. So the balance of abstract thinking, which is proper to mathematics, yet not exclusive to it, and is highly promoted by an early and serious grammatical and logical education, is still shaking.

A non-marginal responsibility of this state of affairs is located in the economical structure of the contemporary world, dominated by the short time horizons of private investors, who naturally and legitimately privilege applied science with respect to fundamental science, this dominance not being sufficiently counterbalanced by public (national and international) long time horizon investment policies.

Nowadays, in the correct balance of the abstract thinking diffusion, one notices a new impellent urgency. Just as more than one century ago Felix Klein envisaged the urge to promote functional thinking, I deem that priority number one is at the present time that of promoting the diffusion of group thinking, which should become an essential part of one’s education forging one’s mental framework, at least among scientists and engineers of all specializations.

Group theory codifies the concept of symmetry and, as such, it is the backbone of mathematics, physics, and chemistry, which, up to a certain degree, is just that branch of physics that focuses on atoms and molecules. The notion of what is a group developed slowly, from the end of the eighteenth century with the contributions of Lagrange and others, which were mainly focused on the group of permutations, named at the time substitutions, through the whole span of the nineteenth century. The first very significant results on group theory were those obtained by Galois before his death in 1832 and published by Liouville in 1846 [1]. Galois groups are the symmetry groups of algebraic equations and, as such, they are finite groups, subgroups of the permutation group Sn, where n is the degree of the considered algebraic equation. In the middle of the nineteenth century the first attempts at a rigorous abstract definition of the group structure were developed by Arthur Cayley [2]. Finite group theory came to ripeness by the end of the nineteenth century and the beginning of the twentieth century through the work of Camille Jordan [3], Enrico Betti [4], Adolf Hurwitz [5], Felix Klein [6], Issai Schür [7], and others. An essential allied topic and ingredient of group theory was linear algebra and matrix theory, which were slowly developed in the central part of the nineteenth century most intensively by Arthur Cayley [8] and James Sylvester [9], [8], [10]. It is mandatory to stress that the key notion in linear algebra, namely, that of vector space, although correctly conceived and explained by Grassmann since 1844 [11], had a hard time to be accepted and metabolized by the mathematical scientific community and became part of shared knowledge only after the admirable lectures by Giuseppe Peano published in 1888 [12].

Along a parallel route, starting from the 1870s, the theory of continuous analytic groups, namely, Lie groups with their associated Lie algebras, was firmly established, before the end of the nineteenth century, through the work of Sophus Lie [13], [14], [15], [16], [17], of Wilhelm Killing [18], [19], and finally of Élie Cartan [20], [21]. In this way modern group theory was essentially accomplished already more than 120 years ago and underwent some important additional refinements through the work of Hermann Weyl [22], Coxeter [23], and Dynkin [24] just by the end of World War II.

The complete merge of geometry with group theory, started by the monumental work of Élie Cartan on symmetric homogeneous spaces in the 1930s and continued in between the two wars by the development of the notion of fiber bundles and their classification by means of characteristic classes (Chern and Weil), was finally achieved in the 1950s by the parallel mathematical work on connections conducted by Ehresmann [25], [26] and by the introduction in physics of Yang–Mills gauge theories [27]. Although physicists already knew groups and used them in elaborating their theories, it took a long while before the complete identity of gauge fields with connections on principal bundles was recognized and a decisive upgrade in the mathematical education of younger generation physicists came into being. I think that the complete inclusion into fundamental physics of the fields of group theory, algebraic topology, and differential and algebraic geometry was accomplished...