Dynamics, Games and Science II: DYNA 2008, in Honor of Maurício Peixoto and David Rand, University of Minho, Braga, Portugal, September 8-12, 2008

784 Pages·2011·7.63 MB·English

by  D. A. Rand (auth.), Mauricio Matos Peixoto, Alberto Adrego Pinto, David A. Rand (eds.)

Springer Proceedings in Mathematics
Volume 2
For further volumes:
http://www.springer.com/series/8806
Springer Proceedings in Mathematics
The book series will feature volumes of selected contributions from workshops and
conferences in all areas of current research activity in mathematics. Besides an
overall evaluation, at the hands of the publisher, of the interest, scientific quality,
and timeliness of each proposal, every individual contribution will be refereed to
standards comparable to those of leading mathematics journals. It is hoped that
this series will thus propose to the research community well-edited and authori-
tative reports on newest developments in the most interesting and promising areas
of mathematical research today.
Mauricio Matos Peixoto Alberto Adrego Pinto
David A. Rand
Editors
Dynamics, Games
and Science II
DYNA 2008, in Honor of Mauricio Peixoto
and David Rand, University of Minho, Braga,
Portugal, September 8-12, 2008
123
Editors
Mauricio Matos Peixoto Alberto Adrego Pinto
Instituto de Matema´tica Universidade do Porto
Pura e Aplicada (IMPA) Departamento de Matema´tica
Estrada Dona Castorina 110 Faculdade de Cieˆncias
22460-320 Rio de Janeiro Rua do Campo Alegre 687
Brazil 4169-007 Porto
To my wife Alcilea Augusto and my four
children and eight grandchildren:
Marta: Daniel, Thomas and Mariana;
Ricardo: Gabriel;
Marcos: Andre and Livia; and
Elisa: Clara and Bruno.
To Anto´nio Pereira Pinto
To Barbel Finkensta¨dt and the Rand Kids:
Ben,
Tamsin,
Rupert and
Charlotte.
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 dynam-
ical 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 conjec-
tures 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 Matema´tica,
Pura e Aplicada.
It was during the next six months (January through 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 1980’s.
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 stu-
dent 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 Math-
ematical 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
vii
viii Mauricio Peixoto
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
Alberto Adrego Pinto
I met Alberto a few years ago, in the office of Mauricio Peixoto at IMPA, the
Brazilian Institute of Pure and Applied Mathematics. Alberto was on a summer
visit, and he wanted to discuss results he had obtained with his former student Diogo
Pinheiro on the focal decomposition proposed years earlier by Mauricio.
By sheer accident, I had come across an application of the focal decomposition
in finite temperature quantum mechanics. In fact, semi-classical approximations to
the problem practically forced one to make use of the focal decomposition, although
it was only much later that I became aware of its existence. That was why I was part
of the meeting: my mathematician friends were curious about possible applications,
and we were eager to collaborate.
Alberto immediately impressed me by his enthusiasm, his genuine interest in
science, and by his easy-going style, much appreciated by a “carioca” like myself.
Besides, our discussions were lively, and touched upon various conceptual points
that seemed quite natural to a physicist, and eventually proved very useful from a
mathematical point of view. Our collaboration has been going on ever since, and has
already led to a couple of articles.
Alberto has also offered us all with a wonderful event back in 2008, when he
organized a conference in honor of David Rand and Mauricio Peixoto in the pre-
cious city of Braga. The conference made me appreciate, even more, the versatility
and scientific depth of Alberto, as he and his PhD students and postdocs presented
seminars that covered a wide variety of subjects.
As a final word about Alberto, it must be said that he is a marvelous host. He
showed us the finest of the region of Minho, using a well balanced combination of
science, art, good food, good wine, and above all, good humor. That is the reason I
always look forward to our next meeting: whether in Brazil or in Portugal, I am sure
we will have a pleasant and productive time.
Carlos Alberto Araga˜o de Carvalho
ix