Always professional, prepared and responsible, those were without a doubt the principles that he acquired from his family and especially from his mother, a mathematician by training, who took charge of his basic instruction in Tulancingo, his birthplace. At the end of that stage, his parents sent him along with his brothers to finish their studies in Lyon, France, where some aunts lived. He also studied there for a bachelor's degree in mathematics and a master's degree in physics. In that period he came across the "May Revolution" of 1968.

... arrived in Athens on the agreed date, where Teresa says: "... I met a conversational friend like never before, nice, interesting, gentlemanly ... that first evening with Jorge is unforgettable for me ... ". The next day they embarked and became engaged on the cruise. At the end of the tour, Jorge returned to New York. They wrote to each other abundantly and saw each other on vacation.

Jorge met Alfonso first, in 1973, in a month long meeting in Montreal which has been quite important since many of us started, at that time, a long friendship. Jorge was still a graduate student, taking careful notes at every talk and proud that his adviser, Louis Nirenberg, had included some of the results of his thesis in his mini-course. Alfonso was with a lively group of Italians, in particular the other two musketeers [Massimo Furi and Mario Martelli], who used to sing, laugh and talk about politics until very late at night. They made a Latin group with the participation of some Germans and even a native Hawaiian.

This paper is an extended version of the author's doctoral thesis written at the Courant Institute of New York University. The author would like to express his gratitude to Professor Louis Nirenberg for his advice, for his insight and his kindness. Thanks are also due to the Consejo Nacional de Ciencia y Tecnologia of Mexico which provided a partial support for the author's doctoral studies and in particular for this research.

... the Institute for Research in Applied Mathematics and Systems (IIMAS) of the UNAM, which became his second home. Every day he arrived around ten and at three without fail he left his office to go to his house in Tizapán where he ate with his family, outdoors under a leafy tree. He would return to his office at half past five where he would stay until nine or ten at night, because "the Institute is quieter and you can work better there." Jorge Ize was always a talented and tireless worker.

After a few casual encounters at meetings, Alfonso spent a month in 1976 at the Universidad Nacional Autonoma de Mexico, where he lectured on the nonlinear spectrum and stayed at Jorge's home. One of Jorge's students wrote his bachelor thesis on this topic.

The study of bifurcation, vaguely understood as "change in morphology", arises in many different areas of mathematics as well as in the natural sciences and various branches of engineering. Each of these fields has given his own imprint to the concept, which ultimately led to its indeterminacy. However, one particular aspect of bifurcation theory, bifurcation of solutions of a parametrised family of equations from a trivial branch of solutions, is one of the oldest and better understood notions of bifurcation in mathematics. Indeed, the earliest example of a specific bifurcation phenomenon of this type can be traced back to Leonard Euler who, in 1757, studied the deviation of a loaded vertical elastic column as it buckled from its position of equilibrium, which he modelled as a boundary value problem parametrised by the load. By increasing the load on the top of the column the vertical equilibrium position becomes unstable and the column changes its configuration, acquiring a new configuration each time the amount of load, considered as a parameter, crosses some critical values. The boundary value problem always has a solution corresponding to the vertical configuration, and new solutions appear at critical values of the load. Euler proved that those critical values, when suitably normalised, coincide with the eigenvalues of the linearisation at the trivial solution of the boundary value problem.

In Nice, Ize had been working on the relationship between extensions of maps and the non-triviality of bifurcation invariants for several parameters. Alfonso Vignoli, Ivar Massabò and Jacobo Pejsachowicz had been playing with the idea of "complementing maps" applied to different classes of maps. The idea of complementing maps consists in "filling up" the dimension of the range, in rather the same way as the implicit function theorem may be proved by means of the inverse function theorem. They had given a very nice argument, based on Zorn's lemma, to simplify the Alexander-Antman proof of the dimension of "bifurcating surfaces". This combination of ideas and previous work led them to begin a collaboration ... In Cosenza the whole "gang of four"' worked very intensely, in the house on campus. ... This first step was followed by a whole year that Alfonso spent in Mexico [working with Ize] and where he fell in love with the country.

This paper arose from an attempt to unify, clarify and extend some recent work on the topological dimension of global branches of solutions appearing in different problems of Nonlinear Analysis.

Work performed while the fourth named author [Alfonso Vignoli] was visiting the Department of Mathematics and Mechanics, IIMAS-UNAM, Mexico.

The book under review is the first of its kind on equivariant degree theory and should be considered as an important contribution to modern nonlinear analysis. The importance of symmetry in every aspect of life cannot be underestimated. Unfortunately, studying nonlinear problems with symmetries is not simple and involves advanced mathematical methods and techniques that make this area difficult to access by non-specialists. The book by Ize and Vignoli is an honest and substantial effort to open the area of equivariant analysis to applied mathematicians interested in nonlinear equations with symmetries.

Almost 40 years after beginning his scientific career at UNAM, Dr Jorge Andrés Ize Lamache passed away on 16 August 2012, sheltered by the love and respect of his loved ones. Committed to the University until the last moments of his life, his integrity, dedication, loyalty and honesty will be an example of the academic and human being who enhances the environment in which he lives.

We were all shocked and devastated by the news of Jorge's passing. His mathematical ideas and personal relationship have permanently influenced our lives. Jorge constructively affected our professional attitudes, gave us a strong sense of responsibility as researchers and professionals, and was a close personal friend. We believe that nobody is better suited to express the feelings confronting Jorge's premature departure than Alfonso Vignoli, his dearest friend and close collaborator for forty years.

I met Jorge in June 1973 at a conference in Canada. Our encounter could have not been more explosive. I found myself talking to a person very different from me: reserved and of few, albeit essential words. After each conference I was astounded by the depth in analysis of the results submitted. He often put me in a difficult spot by asking my opinion on a newly announced theorem. He was not, I underscore, a mathematics 'crank'. He adored art, food, travelling and knowing the history of nations. He was abreast of politics, even Italian, and his readings were always lucid. I remember our visits to the church of Santa Maria Maggiore in Rome, with Jorge equipped with small, powerful binoculars to probe the mosaics, looking for precious particulars. I think some of the best results we achieved upon return in our Roman home with a good glass of wine that was never missing. Coming to Italy relieved him from the headaches that assailed him after a glass of wine in Mexico City, probably, due to altitude. I believe one of the reasons our friendship grew stronger over the years is tied to (quite) a few hearty drinks and laughter, and to his subtle irony that let transpire a deep affection for me and my family. Of his extended scientific production I only want to highlight the book 'Equivariant Degree Theory' a book Jorge did not want to write. I had to insist manifold, and almost court him, to convince him. The result was a much cleaner version of the theory and a text filled with his ideas that I believe will give several cues to the coming schools of researchers. Many were the research projects upon which we fantasised, dreamed and ironized of. Many were the meetings we had planned for the coming months and years, even 'just' to drink our glass of wine, to laugh and see him raise his eyebrow to my free and odd stories.

Alfonso Vignoli.