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. Around 2002 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 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

– Chunaev P, Mateu J & Tolsa X 2020, “A family of singular integral operators which control the Cauchy transform“. Mathematische Zeitschrift 294, p. 1283-1340.

– Azzam J, Mourgoglou M & Tolsa X 2020, “A two-phase free boundary problem for harmonic measure and uniform rectifiability“. Trans. Amer. Math. Soc., vol. 373, no. 6, 2020, 4359–4388

– Prats, M., Tolsa, X. ‘The two-phase problem for harmonic measure in VMO‘. Calculus of Variations and Partial Differential Equations 59, 102:3 (2020).

– Jonas Azzam, Xavier Tolsa, and Tatiana Toro. “Characterization of rectifiable measures in terms of α-numbers“. Transactions of the American Mathematical Society, no. 11, 7991-8037

Xavier Tolsa. “Jump formulas for singular integrals and layer potentials on rectifiable sets”. Proceedings of the American Mathematical Society 148(11) (2020), 4755-4767.

– J. Azzam, S. Hofmann, J.M. Martell, M. Mourgoglou, and X. Tolsa. Harmonic measure and quantitative connectivity: geometric characterization of the Lp-solvability of the Dirichlet problem. Inventiones Mathematicae 222 (2020), no. 3, 881-993

– M. Mourgoglou and X. Tolsa.Harmonic measure and Riesz transform in uniform and general domains“. Journal für die reine und angewandte Mathematik 758 (2020), 183–221.

– B. Jaye, F. Nazarov, M.C. Reguera, and X. Tolsa. ‘The Riesz transform of codimension smaller than one and the Wolff energy.’ Mem. Amer. Math. Soc. 266 (2020), no. 1293.

– Alan Chang and Xavier Tolsa.Analytic capacity and projections“. Journal of the European Mathematical Society (JEMS) 22 (2020), no. 12, 4121-4159.

Selected research activities

Plenary talks in conferences, seminars and courses

– Dynamics, Analysis, Geometry, and Probability. Simons Center for Geometry and Physics. USA.

– Monroe Lecture 2020, Johns Hopkins Univ. (USA).

– One-world fractals and related topics (on-line conference), France.

– Square functions and rectifiability. On-line minicourse. Shanghai.