Xavier Tolsa

Xavier Tolsa

Universitat Autònoma de Barcelona

Experimental Sciences & Mathematics

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 Goteborg (University of Goteborg - 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 Dirichlet problem for the Laplace equation, which are other main interests. 

Selected publications

- Mateu J, Prat L & Tolsa X 2022, 'Removable singularities for Lipschitz caloric functions in time varying domains', Rev. Mat. Iberoam. 38, 2, pp. 547-588.

Selected research activities

- Organization of the Trimester Program "Interactions between Geometric measure theory, Singular integrals, and PDE" and stay of two months in the Hausdorff Institute of Mathematics (Bonn).

Plenary and invited conferences:

- Encontro Nacional SPM2022 (Tomar, Portugal).

- PRIMA Congress 2022. Harmonic Functions And Laplace Eigenfunctions (Vancouver, Canada).

- Fractal and Related Fields IV (Porquerolles, France).

- Kick-Off Meeting for the Maria de Maeztu grant. Dissemination talk (CRM, Barcelona). 

Invited seminars:

- "Harmonic measure and Hausdorff measures". Analysis and PDE Seminar, Stanford University.
- "The measures with L^2-bounded Riesz transform and the Painlevé problem for Lipschitz harmonic functions". Analysis Seminar of Princeton University (online).