Knowledge may be applicable or may increase understanding, sometimes both. But where is it? I entered college as an engineering student, found some courses boring and shifted to mathematics, which required less credits. During my third and last year as an undergraduate, I was an intern at IBGE, the federal institution encharged of the census, and was very moved by the available data. I became a vegetarian and considered a M.Sc. in agricultural planning, but I soon realized that was a highly theoretical career in Brazil. Luck interfered: P.Deift, from NYU, came to visit our Department and in 1979 I started my PhD studies under his wonderful supervision. Deift remained a constant inspiration for the rest of my career. In 1982, through a Gibbs instructorship at Yale, I had the additional luck of meeting two other sources of inspiration, R. Beals and R. Coifman: there was no way I could do something different from math in life. At the Department of Mathematics here at PUC, my intellectual freedom allowed me to place strong bets – some successful – and to appreciate a diversity of mathematical forms and styles, a wealth of problems and colleagues.
My current research stems from years of concentration in certain subjects. My interests in the intertwining of spectral theory with integrable systems led me to the study of concrete eigenvalue algorithms through a conceptual approach to numerical analysis. A concrete example is the interplay between the so called Toda Lattice and the QR method, dating back to the late seventies, which used symplectic geometry as the foundational background. Further developments in the eighties and nineties brought a deeper understanding of familiar algorithms. And recently, a Lie algebraic approach yielded simpler theoretical tools: the dynamics at relevant points of phase space has been reduced to standard local theory. Fundamental numerical methods which have been used for decades are now thoroughly understood, as are basic transversality properties of the underlying vector fields. In a similar vein, my interest on nonlinear functions in the plane, which started in the nineties, developed into the consideration of a collection of cases, ranging from finite dimensional maps associated to spectral properties, to ordinary and partial differential operators. A constant attention is given to both theoretical and numerical implications. The list of collaborators increased accordingly.