Real Sociedad Matemática Española All that Math Portraits of mathematicians as young readers Editors Antonio Córdoba José L. Fernández Pablo Fernández Portraits of mathematicians as young readers Contributors Antonio Ambrosetti, Kari Astala, Luis Barreira, Pilar Bayer, Christopher J. Bishop, Aline Bonami, Joaquim Bruna, Luis A. Caffarelli, Fernando Chamizo, Sun-Yung Alice Chang, Antonio Córdoba, David Drasin, Charles Fefferman, José L. Fernández, Pedro Luis García, Evarist Giné, Tadeusz Iwaniec, Joseph J. Kohn, Pierre-Gilles Lemarié-Rieusset, Rafael de la Llave, Gaven Martin, Pertti Mattila, Yves Meyer, David Nualart, Antonio Ros, Stephen Semmes, Luis Á. Seco, Elias M. Stein, Gunther Ulhmann, Alexander Volberg, Stephen Wainger, All that Math Jang-Mei Wu, Paul Yang, Marc Yor. Celebrating the Centennial of Real Sociedad Matemática Española ISBN 978-84- 615-2900-1 9 7 8 8 4 6 1 5 2 9 0 0 1 Revista Matemática Iberoamericana All that Math Portraits of mathematicians as young readers Revista Matem´atica Iberoamericana All that Math Portraits of mathematicians as young readers Celebrating the Centennial of Real Sociedad Matem´atica Espa˜nola Antonio C´ordoba, Jos´eL.Fern´andez, Pablo Fern´andez (Editors) Reservados todos los derechos. Prohibida la reproducci´on total o parcial del contenido de este libro, incluso para uso privado, sin permiso escrito de los editores. c 2011 Revista Matem´atica Iberoamericana ISBN: 978-84-615-2900-1 Dise˜no de la edici´on:PFG. Dise˜no de cubierta: Pepe Herrera Taller Gr´afico. http://www.tallergraficopepeherrera.com. Ilustraci´on de la cubierta. “La quinta luz”, Santiago de la Quintana, 2011. Impresi´on:AventuraGr´afica. C/ Lepanto 12-13, 19004-Guadalajara. http://www.aventuragrafica.com. Libro impreso en papel estucado semimate de 115 gramos fabricado con pastas libres de cloro elemental. La quinta luz Santiago de la Quintana, 2011. Preface As mathematicians we work on our research program, write up and publish our results, deliver lectures here and there disseminating our theorems and, once in a while, we attend the lectures and read the papers of our fellow mathematicians. Some of us read a lot and some just rarely. But all of us have had that special experience of reading a paper (attending a lecture or having a mathematical discussion) which has been a personal turning point, perhaps because its techniques and elegance have illuminated ongoing work or be- cause it has led us to explore a whole new area of mathematics or because it has given us a new set of mathematical objectives. We all treasure those occasions and some of us, in the right context, even like to share them. On the occasion of its centennial, the Real Sociedad Matem´atica Espa- nola˜ asked the editors of the Revista Matem´atica Iberoamericana to con- tribute to the occasion by editing and publishing a commemorative issue of the journal. We decided to take the opportunity to ask distinguished math- ematicians to write down their personal experiences as readers of research papers. We are convinced that such a collection of essays, written by a set of world-class researchers, will be interesting reading for mathematicians in general and that it will contribute to understanding the role of research journals in the development of mathematics. We asked longtime friends of the Revista Matem´atica Iberoamericana, both authors and members of past and present editorial boards who have fostered and cared for Revista, to contribute with an essay about a paper –not necessarily the most important publication in the field– which, in one way or another, had a deep impact on their own mathematical careers, espe- cially at its early stages, and giving the special reasons why “that particular publication” got their attention and affected their research. We have been very pleased, even overwhelmed, with the response of the friends of the Revista to our call. The reader will find among the articles in this book a surprisingly wide and rich variety of points of view, styles and approaches to this topic, all of which we appreciate greatly and have fully respected. Some are personal recollections with interesting anecdotes and charming stories about some well-known mathematician, while others present the state of the art of some specific field. Some papers are exquisitely technical while some others are perfect bedtime stories. We offer our deepest thanks to all of the authors for sharing these con- tributions with all of us. The Editors Authors Antonio Ambrosetti ............................................... 1 Kari Astala ......................................................... 9 Luis Barreira ...................................................... 19 Pilar Bayer ........................................................ 29 Christopher J. Bishop ............................................ 35 Aline Bonami....................................................... 55 Joaquim Bruna ..................................................... 65 Luis A. Caffarelli................................................. 79 Fernando Chamizo ................................................ 91 Sung-Yun Alice Chang .......................................... 101 Antonio Cordoba´ ................................................ 107 David Drasin ...................................................... 119 Charles Fefferman .............................................. 131 JoseL.Fern´ andez´ ................................................ 137 Pedro Luis Garc´ıa ............................................... 153 Evarist Gine´ ...................................................... 163 Tadeusz Iwaniec .................................................. 171 Joseph J. Kohn ................................................... 187 Pierre Gilles Lemarie-Rieusset´ ................................ 199 Rafael de la Llave .............................................. 205 Gaven Martin..................................................... 223 Pertti Mattila ................................................... 235 Yves Meyer ....................................................... 247 David Nualart .................................................... 259 Antonio Ros ...................................................... 267 Luis A.´ Seco....................................................... 275 Stephen Semmes .................................................. 281 Elias M. Stein .................................................... 289 Gunther Uhlmann................................................ 295 Alexander Volberg .............................................. 303 Stephen Wainger ................................................. 319 Yang-Mei Wu ..................................................... 323 Paul Yang ......................................................... 333 Marc Yor ......................................................... 339 Antonio Ambrosetti was born in Bari (Italy), in 1945. He is currently Full Professor at SISSA, Trieste. His research field is nonlinear analysis. On a paper by Giovanni Prodi on bifurcation Antonio Ambrosetti∗ 1. Introduction Many papers have influenced my work, but the very first one is a survey paper by G. Prodi [3] deal- ing with bifurcation theory. I was fascinated by the interplay be- tween abstract methods which are motivated by, and hence applied to, concrete problems like fluid- dynamics and elasticity. Among other things, I learned that par- tial differential equations aris- ing in mathematical physics are strongly linked to nonlinear func- tional analysis. The correct ap- proach to this field must have its starting point in concrete prob- lems modeled by differential equa- tions. These applications moti- vate the right choice of the ab- stract tools and the advances to be carried out in order to obtain the expected results. Many physical problems can be formulated as an operator equation F (λ, u)=0whereu belongs to an appropriate Banach function space X and λ is a real parameter with a specific physical meaning. Often there is a threshold λ∗ such that new solutions arise when λ crosses λ∗, which is called a bifurcation value for the problem. Prodi outlines three different approaches that can be used to find bifur- cation points: (i) analytical methods, (ii) degree theoretical arguments and, in the case that the problem is variational, (iii) Morse theory. The abstract results are used to discuss two main applications: (i) the motions of a viscous fluid between two cylinders and (ii) the equilibria of an elastic clamped plate. ∗[email protected]. 3 4 Antonio Ambrosetti 2. Analytical methods Let X, Y be Banach spaces and let F ∈ C2(R × X, Y ) (weaker regularity assumption could be given). We assume the equation F (λ, u)=0hasthe trivial solution u =0forallλ ∈ R. Let us denote by S the set of nontrivial solutions of F (λ, u) = 0. A value λ∗ is a bifurcation point for F =0if (λ∗, 0) ∈ S, the closure of S in R × X. It is easy to see that if λ∗ is ∗ a bifurcation point then DuF (λ , 0) cannot be invertible as a linear map from X to Y . One main purpose of bifurcation theory is to find sufficient conditions for the existence of bifurcation points. Suppose that ∗ ∗ ∗ (F1) there exists φ ∈ X such that ker[DuF (λ , 0)] = Rφ ; ∗ (F2) range[DuF (λ , 0)] is closed and has codimension 1. Using the Lyapunov-Schmidt reduction one proves Theorem 1 Suppose that (F1) and (F2) hold. Moreover, letting M := ∗ ∗ ∗ ∗ Du,λF (λ , 0), assume that M(φ ) ∈ Range[DuF (λ , 0)].Thenλ is a bifur- cation point for F =0. Remark 1 If X = Y and F (λ, u)=λu − G(u), the previous assumptions are verified provided