Xavier Tolsa

Universitat Autònoma de Barcelona (UAB)

Experimental Sciences & Mathematics

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 interests

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.

Selected publications

– Chousionis V, Garnett J, Le T & Tolsa X 2016, ‘Square functions and uniform rectifiability’, Trans. Amer. Math. Soc., 368, 6063-6102.

– Chousionis V, Prat L & Tolsa X 2016, ‘Square Functions of Fractional Homogeneity and Wolff Potentials’, Int Math Res Notices, Vol. 2016 2239-2294.

– Azzam J, Hofmann S, Martell JM, Mayboroda S, Mourgoglou M, Tolsa X & Volberg A 2016,  ‘Rectifiability of harmonic measure’, Geometric and Functional Analysis, 26(3), 703-728.

– Reguera MC & Tolsa X 2016, ‘Riesz tranforms of non-integer homogeneity on uniformly disocnnected sets’, Transactions Of The American Mathematical Society, 368, 10, 7045 – 7095.

Selected research activities

Conferences and minicourses

– International Conference on Harmonic Analysis and PDE’s. El Escorial (Madrid), June 2016. Plenary speaker: “The Riesz transform, quantitative rectifiabilty, and a two-phase problem for harmonic measure”.

– Harmonic analysis, complex analysis, spectral theory and all that. Bedlewo (Poland), August 2016. Plenary speaker: “The Riesz transform, quantitative rectifiabilty, and a two-phase problem for harmonic measure”

– Harmonic Analysis and its Applications. Matsumoto (Japan), August 2016. Minicourse: “The Riesz transform, rectifiabilty, and harmonic measure”.

– Journées du GdR Analyse Fonctionnelle, Harmonique et Probabilités. Toulouse, October 2016. Plenary speaker: “Riesz transforms, square functions, and rectifiability”.

– Spaces of analytic functions and singular integrals. Chebyshev Laboratory, St. Petersburg, October 2016. Minicourse: “The Riesz transform, rectifiabilty, and harmonic measure”.

– Albert Einstein Institute (Potsdam), January 2016. Seminar: “Riesz transform and quantitative rectifiabilty for general Radon measures”.


Direction of the PhD thesis by Daniel Girela Sarrión “Singular integrals and rectifiability”, Universitat Autònoma de Barcelona (2016). Excellent cum Laude.

President of the Scientific Committee of the Barcelona Analysis Conference 2016.