Ene 05 2009

## Viana

### UMALCA AWARD IN MATHEMATICS 2000

### Marcelo Viana

Marcelo Viana is one of the most brilliant mathematicians of his generation in worlwide terms. It is to be noted that he has been invited to give talks at two consecutive International Congresses of Mathematicians, Zurich-94 and Berlín-98. The first one as Section Speaker and the second one as Plenary Lecturer. Besides, he has also been invited to give a Plenary Lecture at the International Congress on Mathematical Physics, París-94.

He was born in Río de Janeiro on March 4, 1962 and got a Ph.D. degree at the Institute of Pure and Applied Mathematics (IMPA) after graduating at the University of Porto, Portugal. At the time he was concluding his doctor thesis in such as the Abundance of strange attractors, extending to great generality a very difficult and very basic work of Benedicks and Carleson, for quadratics maps of the plane. Such excellent work, showing that strange attractors occur whenever a homoclinic tangency is unfolded was published in Acta Mathematica, a first-rate journal. The same goes for his extension to full generality in arbitrary dimensions of Newhouse’s famous result on the simultaneous existence of in ntely many periodic attractors. This paper has been published in the Annals of Mathematics.

He then went on to explore the existence of the above attractors in a persistent way in another kind of basic bifurcation of dynamics, saddle-node cycles. The corresponding paper was published in Inventiones Matehmaticae. Again appearing in the Annals of Mathematics, is the very original work on spiraling chaotic attractors, for vector elds going through spiraling homoclinic bifurcations. The problem was reduced to strange attractors for maps with in nitely many critical points, a case which had not been dealt with before.

Recently ge proved that the above mentioned chaotic attractors, called Henonlike attractors (Henon was the first to suggest their existence), are stochastically stable and have no wholes in their basin of attraction, answering an old question of Eckman-Ruelle and Sinai. A first article on these topics has just been accepted in Inventiones Mathematicae. On the other hand, he is exhibiting new kinds of the famous Lorenz attractor in dimensions greater than three, namely attractors with two-dimension unstable directions. This also gives a positive answer to an old question. Before he had provided new and very original kind of Henon-like attractors, again with multi-dimensional unstable directions. This work has appeared in a very first-rate journal, Publications Mathematiques, Institut des Hautes Etudes Scientifiques. More recently, he has also been studying the statistical properties of non hyperbolic systems, specially the construction of Sinai-Ruelle-Bowers measures, and niteness results for Sinai-Ruelle-Bowen measures for non -uniformly hyperbolic maps, in another article published in Inventiones Mathematicae. He was awarded a Guggenheim fellowship to work at UCLA and Princeton, in 1993 and since the he has been a visitor to main centers around the world. He has received the Third World Academy Price in Mathematics in 1998 and at his early age he is a member of that academy and of the Brazilian Academy of Science. He is considered a superb lecturer and he has already supervised six Ph.D. thesis.