# Tolsa Domènech, Xavier

ICREA Research Professor at Universitat Autònoma de Barcelona (UAB).

Experimental Sciences & Mathematics

#### Short biography

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'' and in 2020 another for the project "Geometric Analysis and Potential Theory". My current research in mathematics focuses in Fourier analysis, geometric measure theory, potential theory, and elliptic PDE's.

#### Research interests

I work in mathematical analysis. My research deals with harmonic analysis, geometric measure theory, and elliptic PDE's. Particularly, I am interested in the relationship between analytic notions such as analytic capacity or harmonic measure, 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. Around 2002 I proved that analytic capacity is semiadditive. This was an open problem since the early 1960s. Later on I studied related problems in higher dimensions. In particular, in a 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 and the Dirchlet problem for the Laplace equation, which are other main interests.