Tolsa DomÈnech, Xavier

Biography

I was born in Barcelona in 1966. First I studied engineering, but later I turned to mathematics. After obtaining my PhD in mathematics in 1998 (UAB), I spent about one year in Gotteborg (University of Gotteborg - Chalmers) and another year in Paris (Université de Paris-Sud), until I came back to Barcelona (UAB) by means of a "Ramón y Cajal" position. In 2002 I was awarded the Salem Prize by the Institute of Advanced Study and Princeton University for the proof of the semiadditivity of analytic capacity and my works in the so called Painlevé problem. Since 2003 I am an ICREA Research Professor. In 2004 I received the prize of the European Mathematical Society for young researchers. In 2012 I was awarded an ERC Advanced Grant to develop the project ''Geometric analysis in the Euclidean space''. My current research in mathematics focuses in Fourier analysis, geometric measure theory, and potential theory.

Research Interest

I work in mathematical analysis. My research deals with complex analysis, Fourier analysis, geometric measure theory, and quasiconformal mappings. Particularly, I am interested in the relationship between analytic notions such as analytic capacity and geometric concepts like rectifiability. In a sense, analytic capacity measures how much a set in the plane is visible or invisible for analytic functions. On the other hand, rectifiability tells you if a set is contained in a (countable) collection of curves with finite length. Some years ago, I proved that analytic capacity is semiadditive. This means that the analytic capacity of the union of two sets A and B is smaller or equal than some constant times the addition of the analytic capacites of A and B. This was an open problem since the early 1960s. More recently I have studied related problems in higher dimensions. In particular, in a recent collaboration with F. Nazarov and A. Volberg I have proved the so called David-Semmes conjecture in the codimension 1 case. This result has important applications to the study of harmonic measure.