2010 Balzan Prize for Mathematics (Pure and Applied)
Acceptance Speech – Rome, 19.11.2010
Mr. President of the Italian Republic,
Chairmen of the Balzan Foundation,
Ladies and Gentlemen,
I was recently lecturing at the University of Warwick in the United Kingdom, presenting a broad view of my scientific work over the last twenty years or so, when I received the news that I had been awarded the 2010 Balzan Prize in Mathematics. It was one of the most precious moments in my life. The fact that the present ceremony takes place in this beautiful country that embodies the true attributes of science, culture and arts, makes the present moment even more precious. Thank you very much, Mr. President, for this inspiring gesture. Above all, I wish to express my deepest gratitude to the Balzan Foundation and the General Prize Committee Members for the great honour that they have bestowed on me. On this special occasion, I would like to praise the Italian Government for generously and wisely supporting two important global institutions, the Academy of Sciences for the Developing World and the International Center for Theoretical Physics, which I have been much involved with. I should also like to thank the Accademia Nazionale dei Lincei for welcoming me as a foreign member just a week ago. How can I pay tribute to my own institution, namely the National Institute for Pure and Applied Mathematics, and my colleagues there, as well as the more than forty Ph.D. students that I have advised and their many Ph.D. students spread over a number of institutions and countries, several of them being outstanding mathematicians at the highest level? It is a world jewel, with a wonderfully stimulating scientific atmosphere. To the National Institute for Pure and Applied Mathematics I have dedicated my lifetime as a mathematician and to it I owe so much. More broadly, I want to recognize the remarkable advancement of science in my country and I am happy to share this moment with our scientific community. In the development of my scientific work, I have travelled a long way, from the global stability of dynamical systems, to the bifurcations of Poincaré’s cycles and fractal dimensions and finally, to a meaningful proposal of a global scenario for dynamics, comprising chaotic systems. I have begun to prove that gradient-like dynamical systems in lower dimensions are stable, meaning that the structure of their orbit remains qualitatively the same under small perturbations of the law of evolution. This has considerably extended a previous remarkable work by Peixoto. The methods of proof used, have embodied a new geometric approach that will fundamentally influence subsequent developments in this area of research: the creation of the notion of stable foliations being partially subfoliated to include those of critical points or isolated period motions of higher indices where they accumulate. Immediately after that, this result was extended to all dimensions in a joint work with Smale, who had been my thesis adviser. Together, we formulated the well-known stability conjecture that became a major topic of research in the area, partially solved, about two decades later, in a remarkable exposition by Mañé, one of my students, and subsequently by Hyashi in a similar context. Liao also contributed to the solution of this question. Since the early seventies, I have become very interested in the theory of bifurcations of Poincaré’s cycles, together with Newhouse and, sometime later, Takens. It became clear that fractal dimensions would play a key role in understanding such bifurcations. My early collaboration with Newhouse and Takens was later extended to Yoccoz and Viana in the eighties and then Moreira in the nineties. Together, we have proved that fractal dimensions indeed determine the frequency of stability in homoclinic bifurcations in all dimensions. The case of heterodimensional cycles in higher dimensions have been successfully considered by Bonatti, Diaz and Rocha, among others. Based on previous work, I was able to formulate by 1995 a bold series of ideas and conjectures that encompassed a global view of dynamics with much more of a probabilistic character than attempted before. The Russian school of Kolmogorov and Sinai, among others, was inspiring. Such a programme has spured significant activity and some successful results derived from this can be seen in works by Lyubich, de Melo, Avila, Moreira, Martens, Viana, and also Abdenur, Bonatti, Diaz, Rocha, Crovisier, Pujals, Sambarino and Wen among others, as well as Yoccoz and myself. Again, I am most grateful to the Balzan Foundation and I am happy to accept the 2010 Balzan Prize, which will give me the pportunity to further develop this line of research and related topics with the second half of the Prize which will support a research project that will involve very talented young colleagues, as well as more experienced mathematicians of renown. Finally, I wish to thank my family, especially my wife Suely, for a lifetime of joy and support.