Xavier Tolsa

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 and geometric measure theory. 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 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

– Azzam J, Mourgoglou M, Tolsa X & Volberg A 2019. “On a two-phase problem for harmonic measure in general domains“. Amer. J. Math., vol.141(5),1259-1279.

– Conde-Alonso JM, Mourgoglou M & Tolsa X 2019 ‘Failure of L2 boundedness of gradients of single layer potentials for measures with zero low density‘, Matematische Annalen, 373, 1-2, 253 – 285.

Tolsa X 2019. “Rectifiability of measures and the βp coefficients”. Publicacions Matemàtiques 63, 491-519.

– Paramonov  PV & Tolsa X 2019, ‘On C1-approximability of functions by solutions of second order elliptic equations on plane compact sets and C-analytic capacity‘. Analysis and Mathematical Physics, Volume 9, Issue 3, pp 1133–1161

– Chunaev P, Mateu J & Tolsa X 2019, “Singular integrals unsuitable for the curvature method whose L^2-boundedness still implies rectifiability”. Journal d’Analyse Mathématique 138, no.2, 741-764.


Selected research activities

Plenary talks in conferences

– Spaces of Analytic Functions: Approximation, Interpolation, Sampling. Centre de Recerca Matemàtica, Barcelona. November 2019.

– Complex and Fourier Analysis, and Operator Theory. INdAM – Istituto Nazionale di Alta Matematica Francesco Severi. Città Universitaria “La Sapienza”. September 2019.

– Modern Aspects of Complex Analysis and Its Applications (in honor of Don Marshall and John Garnett). University of Washington, Seattle. August 2019.

– Harmonic Analysis in non-homogeneous settings and applications. Birmingham. June 2019.

– Harmonic Analysis and PDEs. Helsinki. June 2019.

– Complex analysis and operator theory. Saint Petersburg. May 2019.

Dissemination talks

– Colloquium of the Facultat de Matemàtiques de la Universitat de València, November 2019.

– Inaugural lesson of the course of the Societat Catalana de Matemàtiques, November 2019.

Codirection of the PhD thesis of Carmelo Puliatti “Singular integrals, rectifiability, and elliptic measure”, UAB, December 2019.

Prize Rei Jaume I of Basic Science, 2019, awarded by Generalitat Valenciana and FVEA.