Some words in memory of Dan Henry These words are dedicated to the memory of Dan Henry. He was, in fact, an inspiration and friend to many of his colleagues, including myself, at the University of São Paulo (USP). I heard about Dan Henry since 1969 when I visited Brown University. He was student of Jack Hale. I came back many times to Brown and I knew him very well. In the end of 1979, as President of the USP, I had the great opportunity of signing the first contract of Dan with the University of São Paulo, brought to me by Chaim Hönig. He came to the “Departamento de Matemática Aplicada" (MAP-department). In 1982, when I returned to the academic research works, Dan and I used to attend together our seminar on Dynamical Systems. It has been a time of strong activities in this field with the participation of good mathematicians as well as of our graduate students. During one of these seminars I was presenting to him details of my work on A -structural stability, later on inserted as the technical report “Relatório Técnico MAP-8301 (Jan. 1983)", and I mentioned the famous figures of Dan Henry, relative to the Chafee-Infante problem, as suggestive examples of Morse Smale (MS) systems (see the next picture).
So, Dan Henry asked me: — Why are you saying that these examples are (MS)? I answered: — It is clear from the pictures, at least in these special cases of low dimensions. I also said: — I am sure that you will prove that the same is true for the general case because you are an expert in parabolic partial differential equations. In fact he did everything in a few months, partly in response to this challenge, and published his famous paper “Some Infinite-Dimensional Morse-Smale Systems defined by Parabolic Partial Differential Equations" (J.of Differential Equations 59, 165-205, 1985). I reproduce here the first page of that paper.
In 1986 appeared also a paper on the same subject, independently written by S.B.Angenent, called “The Morse-Smale Property for a Semi-linear Parabolic Equation" (J.of Differential Equations 62, 427-442, 1986) and it is interesting to reproduce the last page where Angenent comments the contribution above of Dan Henry.
I also have to mention that in the same “Relatório Técnico MAP-8301" appear again the inuence of Dan Henry. In fact, the first two (unpublished) sections were written after a suggestion of Henry who mentioned to me the possibility of including the dissipative properties of the so called a-contractions, specially because these maps have finite dimension unstable manifolds, a crucial hypothesis for the theory of (MS)-maps. At that time the results of these two sections gave an answer to the following question proposed by Hale in 1979: Does a point dissipative a-contraction posses a maximal compact invariant set? Nowadays this question is considered a well known fact as it is shown to us by Hale in his book “Asymptotic Behavior of Dissipative Systems", Mathematical Surveys and Monographs, n o 25 (AMS), Providence 1988. Dan was like that. In 1994 he published another fundamental paper “Exponential Dichotomies, the shadowing lemma and homoclinic orbits in Banach Spaces" ( Resenhas nº4, vol 1, 381-401, 1984) again because we kept discussing properties of a C 1 -map (not a diffeomorphism) that enable us to analyze certain ows of retarded and neutral functional differential equations.
Jack Hale emphasized, in the present e-book, the importance of the mathematical contribution of Dan Henry. So, I decided to write something about Dan's personality and his unusual way of life. Everyday Dan Henry used to appear in the Department and stay in the Library most of his time reading mathematics. He never payed to much attention to the material aspects of his life. When he arrived in Brazil, the country had an accelerated ination, so everybody had to take care with money. But he stayed during almost all his first year receiving checks of his salary and never go to the bank to cash them. So, José Carlos Oliveira advised Dan to buy an apartment with garage close to the Campus. Henry succeeded but he had always forgotten to pay taxes and joint ownership expenses. Finally José Carlos rented to him the garage. The rent was paid by check, every month, but Dan put the checks in the pockets of pants and sent the pants to the laundry; of course he lost the checks. After some months the occupant of his garage, a good man, complained just saying that Dan never cashed the checks. Again José Carlos helped him deciding to ask the good man to pay the rent directly to the responsible of the joint ownership expenses, and so one more big problem was solved. I used to visit Dan, often, in his place to discuss mathematical questions with him. One day he told me that we should go to talk in the lobby of the building because down stairs we had more light. I did not understand very well but I did agree with him. The question was that he forgot to pay the bills of electricity. The doorman told me that he could not work at home after 5 pm because he was without light but early in the morning he was working hard again. So, we decided to make another agreement and the doorman started to cash and pay all the bills for him. On Christmas I used to invite Dan Henry and Carlos Harle to stay with us in my house. I never forget a nice Christmas when Angela and I had together with us Dan, Harle, Jack and Hazel. Dan was really happy that night. Another time I drove to the farm with Harle, Dan and Ivan Kupka. Dan and Kupka were very excited with a ight of termites trying to develop a mathematical model for that nice biological fact. They started to make some computations during the night and should finish in the next morning after evaluating the number of termites in some blocks. Unfortunately they forgot an important kind of predators, the chickens, that ate all the termites, early in the morning. The biological experience then failed. Dan Henry had a very long agony. His life with the terminal cancer had the assistance of a series of many dedicated friends. We will never forget the efforts of Antonio Luiz Pereira, Luiz Augusto de Oliveira and Sergio Oliva, between others, and mainly of Helena Castro, a true mother, who devoted most of her time to Dan specially during the last two years of Dan's sickness. I think that the MAP-department of USP will always be grateful to these friends who helped someone so important for the mathematical research. São Carlos, February 2006 |
