Xavier Tolsa

Universitat Autònoma de Barcelona

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 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. 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, which is another of my main interests.

Selected publications

- Tolsa X 2017, 'Rectifiable measures, square functions involving densities, and the Cauchy transform', Memoirs of the American Mathematical Society,  vol. 245, no. 1158.

- Azzam J, Mourgoglou M & Tolsa X 2017, 'Singular sets for harmonic measure on locally flat domains with locally finite surface measure'. Int Math Res Notices, (12): 3751-3773.

- Mas A  & Tolsa X 2017, 'Lp-estimates for the variation for singular integrals on uniformly rectifiable sets', Trans. Amer. Math. Soc. 369, no. 11, 8239-8275.

- Azzam J, Mourgoglou M & Tolsa X 2017, 'Mutual absolute continuity of interior and exterior harmonic measure implies rectifiability'. Communications on Pure and Applied Mathematics, Vol. LXX, 2121–2163.

- Azzam J, Mourgoglou M & Tolsa X 2017, 'The one-phase problem for harmonic measure in two-sided NTA domains', Analysis & PDE 10-3, 559-588. 

Selected research activities

- Congreso Bienal de la RSME. Zaragoza, January 2017. Plenary speaker: "Rectifiability, Riesz transforms, and harmonic measure".

- Spring School of Analysis. Bedlewo (Poland), March 2017. Minicourse.: "The Riesz transform, rectifiabilty, and harmonic measure".

- Geometry, Analysis and Probability. Conference in honor of Peter Jones. Seoul (South Korea), May 2017. Plenary speaker: "Harmonic measure, Riesz transforms, and uniform rectifiability".

- Recent Developments in Harmonic Analysis. MSRI, Berkeley, May 2017. Plenary speaker: "Uniform rectifiability, bounded harmonic functions, and elliptic PDE's".

- Real Analysis, Harmonic Analysis, and Applications. Oberwolfach (Germany), July 2017.  "Riesz transforms, square functions, and rectifiability".

- CIMPA2017 Research School - IX Escuela Santaló: Harmonic Analysis, Geometric Measure Theory and Applications, Buenos Aires, August 2017. Minicourse.: "The Riesz transform, rectifiabilty, and harmonic measure''.

- Harmonic Analysis and Geometric Measure Theory. Marseille, October 2017. Plenary speaker: "Harmonic and elliptic measures, and uniform rectifiability".