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

- Martikainen H, Mourgoglou M & Tolsa X 2018, 'Improved Cotlar's inequality in the context of local Tb theorems'. J. Funct. Anal. 274, no.5, 1255-1275

- Jaye B, Nazarov F & Tolsa X 2018, 'The measures with an associated square function operator bounded in L2'. Advances in Mathematics 339, 60–112

- Tolsa X & Volberg A 2018, 'On Tsirelson's theorem about triple points for harmonic measure'. International Mathematics Research Notices, Vol. 2018, No. 12, pp. 3671–3683

- Garnett J, Mourgoglou M & Tolsa X 2018, "Uniform rectifiability in terms of Carleson measure estimates and e-approximability of bounded harmonic functions". Duke Mathematical Journal, Vol. 167, No. 8, 1473 -1524.

- Girela-Sarrión D & Tolsa X 2018, 'The Riesz transform and quantitative rectifiability for general Radon measures'. Calc. Var. PDE, 57:16.

Selected research activities

Plenary talks in conferences and workshops, and minicourses

- Harmonic Analysis of Elliptic and Parabolic Partial Differential Equations. CIRM, Marseille, April 2018.

- Minicourse "Harmonic measure via blow up methods and monotonicity formulas". ICMAT (Madrid), May 2018.
- Workshop on Real Harmonic Analysis and its Applications to PDE's and Geometric Measure Theory: on the occasion of the 60th birthday of Steve Hofmann.  ICMAT, Madrid, May 2018.

- Geo­met­ric Meas­ure The­ory and its Con­nec­tions Conference. Helsinki, June 2018.

- Geometric Aspects of Harmonic Analysis (in honor of Fulvio Ricci). Cortona, Italy, June 2018.

- Research Program in Harmonic Analysis. Park City Mathematics Institute (IAS Princeton), Utah, July 2018 (stay of three weeks and an invited talk).

- CMI at 20. Analysis and Probability Workshop. Clay Mathematics Institute. Oxford, September 2018.

- Conference "PDEs and Geometric Measure Theory". ETH Zürich, October 2018.


Direction of the PhD Thesis "Singular Integral Operators and Rectifiability", by Petr Chunaev. June 2018, UAB.