ON MAURICIO PEIXOTO, ALBERTO PINTO AND STEVE SMALE

Este es el hermoso prólogo que escribió Steve Smale, matemático americano y ganador de la Medalla Field en  1966, al libro 'Dynamics, Games and Science I', Springer Proceeding in Mathematics, Springer, Berlin, 2011, Mauricio M. Peixoto y Alberto A. Pinto (Editores). Cabe resaltar la profunda influencia de la escuela brasilera de sistemas dinámicos. 
Los matemáticos están conectados por un enlace bello, profundo e interno que hace que todos sean parte de un hermoso árbol. Ese árbol no tiene fronteras ni banderas.

Mauricio Peixoto

Alberto Pinto has asked me to write about Mauricio Peixoto in this book that honors him as well as David Rand. I am happy to do so. Mauricio is among my oldest friends in mathematics, having met him more than fifty years ago. Moreover he was instrumental in my entry into the field of dynamical systems. So important is this part of my life that my collected works contain four articles that bear on Mauricio in one way or another. That is fortunate since I wrote that material when events were fresher in my mind than they are now. Thus I will borrow freely from these references.

A most important period in my relationship with Mauricio is the summer of 1958 to June of 1960. This is discussed in an article titled “On how I got started in dynamical systems” appearing in the “Mathematics of Time”, based on a talk given at a Berkeley seminar circa 1976. There I wrote how I met Mauricio in the summer of 1958 through a mutual friend, Elon Lima, who was a student from Brazil finishing his PhD at Chicago in topology. Through Lefshetz, Peixoto had become interested in structural stability and he explained to me that subject and described his own work in that area. I became immediately enthusiastic, and started making some early conjectures on how to pass from two to higher dimension. Shortly thereafter, Peixoto and Lima invited me and Clara to Rio for a visit to IMPA, or Instituto de Matemática, Pura e Aplicada.

It was during the next six months (January–June, 1960) that I did some of my most well known work, firstly the introduction of the horseshoe dynamical system and its consequences and secondly the proof of Poincare’s conjecture in dimensions five or more. I sometimes described these works as having been done on the beaches of Rio; this part of the story is told in two articles in the Mathematics Intelligencer in the 1980s.

Thus we may see here what a big influence Mauricio had on my career. Another impact was his “sending” me a student to write a PhD thesis at Berkeley. That student in fact finished such a thesis and went on to become a world leader in dynamical systems. Jacob Palis’ contributions in science go well beyond that. He is a main figure in developing third world science, and mathematics in Brazil in particular. In the article “What is Global Analysis”, based on a talk I gave before the Mathematical Association of America, 1968, I gave a focus to one result as an excellent theorem in global analysis. That result was Peixoto’s theorem that structurally stable differential equations on a two dimensional manifold form an open and dense set.

Another example of the influence of Mauricio!

I will end on a final note that reinforces all that I have said here. Over the last fifty years I have made fifteen visits to IMPA, the institute founded by Mauricio Peixoto (and Leopoldo Nachbin).

Steve Smale