One of the distinguishing characteristics of the twentieth century is that as its algebraic methods became increasingly introspective as they invaded first all of mathematics and then they went on advancing to the rest of the sciences.

When I say that abstract algebra has an introspective character I mean to say that while in the classical algebra of persian civilization (al-Khwarizmi, Omar Khayyam) is a symbolic art where a letter like x is in essence a representation of an unknown number and a particular equation represents in turn an infinite class of problems, in modern abstract algebra it is the very structure of the methods of solution what concerns us, the next obvious step is to consider the structure of all of mathematics.

There is a second important contrast between classical and abstract algebras, namely, while classical algebra delivers explicit solutions to specific problems (as in the word problems of school), in abstract algebra often what is achieved is an explanation of why a problem (an equation perhaps) is impossible to solve. As is well known, this goes back to the origins of the discipline with Galois who managed to prove that it is impossible to solve the fifth degree polynomial equation using solely radicals by an astonishing use of abstract ideas of symmetry founding hence abstract algebra.

It was René Descartes who revolutionized all of science by the realization that algebra and geometry are indivisibly related: modern mathematicians find both disciplines indistinguishable but this is only after the Cartesian revolution. Later Georg Cantor transformed all of mathematics in the twentieth century with his nearly metaphysical meditations and their result: set theory. Set theory reclaims all of mathematics and gives additional prominence to abstract algebra (above classical algebra) in the methodology of the twentieth and twenty-first centuries. Point set topology is a direct descendant of that tradition. Today it is hard to remember that set theory had its early formidable adversaries as was the French mathematician Henri Poincaré,Footnote 1 specially regarding the infinitary aspects.

It is somewhat ironic that Poincaré, living in the tradition of infinitesimal calculus and mechanics, is who found modern topology as it is understood today in his monumental Analysis situs (1895), nevertheless it is Noether who managed to produce an authentic synthesis between the opposite traditions of Cantor-Hilbert and Poincaré with the foundation of homological algebra.

The next chapter in the historical development of algebraic topology is akin to the much better known development of quantum mechanics, where dozens of brilliant individuals in various parts of the world made fundamental contributions to the astonishing intellectual edifice in which it was to become. The shadow of the edifice of algebraic topology would eventually cover all of contemporary mathematics in one way or another, reason that led Dieudonne to name the twentieth century as the “century of algebraic topology in mathematics” (J. Dieudonné, A History of Algebraic and Differential Topology, 2009).

Two Mexican mathematicians are part of the selected group of individuals whose names are forever linked to this scientific epic: José Ádem and Samuel Gitler, both founding members of Cinvestav,Footnote 2 both members of the Colegio Nacional.Footnote 3

The above mentions complementarity between the classical and abstract versions of algebra saw its reflection on the different emphasis in the activity of (algebraic) geometry in the nineteenth and twentieth centuries. In the later period the methods of abstract algebra became fundamental for the so-called obstruction theory that is the main method for proving the non-existence of solutions for many geometric problems in mathematics (and from there with devilish cleverness this can be used to prove the existence of solutions to some other problems just as in the famous cases of the fundamental theorem of algebra or the fixed point theorem of Brouwer).

Samuel Gitler was a twentieth century master in the use of obstruction theory for the solution of forbidding geometrical problems, his technical mastery was incomparable.

One of the problems that Gitler attacked with lucidly (together with Ádem and other collaborators) was the celebrated immersion problem. This problem asks whether projective geometries with points at infinity in any dimension (Brunelleschi (1404–1472), Kepler, Desargues (1591–1661), Monge, Chasles, Poncelet, etc.) can be immersed in ordinary euclidean geometries. David Hilbert asked his student Werner Boy to show that it was impossible to find an immersion of the real projective space of dimension two \({\mathbb R}{\mathbb P}^2\) into the classical euclidean space \({\mathbb R}^3\) in 1901; Hilbert was very surprised as Boy found such an immersion. Another important result was Whitney’s embedding theorem which states that every n-manifold admits an embedding into \({\mathbb R}^{2n-1}\), together with Boy’s result, at the beginning of Gitler’s career, no additional explicit immersions were known. It was known, due to the independent and simultaneous work of Stiefel and Whitney studying characteristic classes from the 30s, was that in the case of projective spaces of dimensions that are a power of two, Whitney’s immersion is optimal (and so is Boy’s immersion): there is no immersion of \({\mathbb R}{\mathbb P}^{2^k}\) into \({\mathbb R}^{2^{k+1}-2}\). There was an crescendo of contributions to the problem by Ádem, Gitler and Mahowald in the 1960s. Today, only seven families of optimal immersions are known (additionally there are two special optimal immersions: \({\mathbb R}{\mathbb P}^{31}\looparrowright {\mathbb R}^{53}\), and \({\mathbb R}{\mathbb P}^{63}\looparrowright {\mathbb R}^{115})\). Gitler’s work was essential to provide three of such seven families (and one of the two special immersions above mentioned). Ádem and Gitler established the third optimal embedding family (in chronological order): the first family was obtained by Whitney, and the second was recently offered by Brian Sanderson. Gitler published one paper per year devoted to the problem from 1963 to 1969. Three of this papers contain optimal families, one of the optimalities was proved by Neeta Singh in 2004.

The opus magnum of Gitler was undoubtedly A Spectrum whose Cohomology is a certain Cyclic Module over the Steenrod Algebra (Topology 1972), in collaboration with Ed Brown; one of the most relevant works in algebraic topology of the twentieth century. It became a fundamental building block of modern homotopy theory going well beyond its original motivation.

The Brown–Gitler spectra played a fundamental role in various remarkable developments in algebraic topology. For example, there is the immersion conjecture stating that every n-manifold admits en euclidean immersion in \({\mathbb R}^{2n-\alpha (n)}\), where \(\alpha (n)\) is the number on ones in the binary expansion of n. The solution to this problem by R. Cohen published in the Annals of Mathematics uses the Brown–Gitler spectrum in a essential step of the proof. The name of many more famous applications can be given: the Sullivan conjecture, Serre’s conjecture regarding torsion in the homotopy groups of cell complexes, the finiteness of elements with unitary Hopf invariants (Adams), the counterexample of Mahowald to Joel Cohen’s conjecture and, in 2009, the solution of Hill, Hopkins and Ravenel of the Kervaire invariant problem, among many others.

The influence of Samuel Gitler goes far beyond his published oeuvre (all of which has been collected in [1,2,3]), for the best Socratic tradition his oral influence was even bigger. In the Seventh Letter, Plato compares the true process of education to the flame (the flame of love to philosophical knowledge) that ignites the pupil (unlike the simple sata transmission from teacher to student). We knew Gitler as pupils and that was precisely the way he transmitted his love of knowledge during his entire life: as a flame of love for mathematics that never blew out.

The arduous path to the foundation of Cinvestav and the Higher School for Physics and Mathematics of the National Polytechnic Institute in Mexico City (ESFM-IPN) ensure alone a historical place for Gitler in Mexican mathematics.

All his students and pupils and his large mathematical progeny are a testimony to the loving generosity and mathematical influence of Samuel Gitler.