Xavier Tolsa Domènech

Xavier Tolsa Domènech

Universitat Autònoma de Barcelona

Experimental Sciences & Mathematics

Xavier Tolsa first studied engineering and later he turned to mathematics. After obtaining his PhD in mathematics in 1998 at the Universitat Autonoma de Barcelona (UAB), he spent about one year in Goteborg (University of Goteborg – Chalmers) and another year in Paris (Université de Paris-Sud). Afterwards, he returned to the UAB as a Ramon y Cajal felow in 2001. Since 2003 he is ICREA Research Professor at the UAB. He has received several awards for his achievements, such as the Salem Prize (2002), the Prize of the European Mathematical Society (2004), or the Prize Rei Jaume I (2019, first time awarded in the field of mathematics). He was invited lecturer at the European Congress of Mathematics in Stockholm (2004) and at the International Conference of Mathematicians in Madrid (2006). He has also been PI of two ERC Advanced Grants (2013-2018 and 2021-2026).

Research interests

Xavier Tolsa works in mathematical analysis. His research deals with harmonic analysis, geometric measure theory, and potential theory. More recently he has also become interested in elliptic PDE's and free boundary problems.
Particularly, he is interested in the relationship between analytic notions such as analytic capacity or harmonic measure, and geometric concepts like rectifiability. Around 2002 he proved that analytic capacity is semiadditive. This was an open problem since the early 1960s. Later on he studied related problems in higher dimensions. In particular, in a collaboration with F. Nazarov and A. Volberg he 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 Dirichlet problem for the Laplace equation, which are other main interests in his research.

Selected publications

- Molero A, Mourgoglou M, Puliatti C & Tolsa X 2023, 'L-2-Boundedness of Gradients of Single Layer Potentials for Elliptic Operators with Coefficients of Dini Mean Oscillation-Type (vol 247, 38, 2023)', Archive For Rational Mechanics And Analysis, 247, 38.

- Tolsa X 2023, 'Unique continuation at the boundary for harmonic functions in C1 domains and Lipschitz domains with small constant', Comm. Pure Appl. Math. 76 (2), pp. 305-336.

- Azzam J, Garnett J, Mourgoglou M & Tolsa X 2022, 'Uniform Rectifiability, Elliptic Measure, Square Functions, and epsilon-Approximability Via an ACF Monotonicity Formula', International Mathematics Research Notices, 2023, 13, 10837–10941.

Selected research activities

  • "Harmonic measure and free boundary problems", BGSMath (Barcelona). November-December 2023.
  • "Introduction to harmonic measure", University of Jyväskylä (Finland). August 2023.
Plenary talks in the conferences:
  • Harmonic Analysis, PDEs, and GMT in Bilbao 2023. June 2023.
  • Harmonic and Complex Analysis: modern and classical, Bar-Ilan University, Ramat Gan, Israel. June 2023.
  • Conference in honor of S. Treil and A. Volberg, University of Würzsburg (Germany). June 2023.
  • Workshop of the LMS Harmonic Analysis & PDE network, Warwick (UK). February 2023.