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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 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.

Selected publications

– Prat L, Puliatti C & Tolsa X 2021, “L-2-boundedness of gradients of single layer potentials and uniform rectifiability“.  Analysis & PDE 14 (2021), no. 3, 717-791.

Tolsa X 2021, “The mutual singularity of harmonic measure and Hausdorff measure of codimension smaller than one“. Internat. Math. Res. Not., vol.  (18), 13783-13811.

– Jaye B, Tolsa XV& Villa M 2021, ‘A proof of Carleson’s ε2-conjecture‘. Annals of Mathematics (2) 194, no. 1, 97-161.

Selected research activities

Invited and plenary talks in conferences

– Analysis days in Sirius, Sochi (Russia), on-line talk.

– Rajchman Zygmund Marcinkiewicz Conference, Warsaw, Poland.

– Workshop “Bounded mean oscillation” (on-line), Focus Program on Analytic Function Spaces and their Applications. Fields Institute, Toronto.

– Geometric Measure Theory and Applications. Cortona, Italy.

Direction of a PhD thesis

– Damian Dabrowski, “Rectifiability of Radon measures”, UAB.

ERC Advanced Grant “Geometric analysis and Potential Theory” (2021-2026)

ICREA Memoir 2021