Xavier Tolsa (born 1966) is a

Catalan Catalan may refer to: Catalonia From, or related to Catalonia: * Catalan language, a Romance language * Catalans, an ethnic group formed by the people from, or with origins in, Northern or southern Catalonia Places * 13178 Catalan, asteroid #1 ...

mathematician, specializing in analysis. Tolsa is a professor at the

Autonomous University of Barcelona The Autonomous University of Barcelona ( ca, Universitat Autònoma de Barcelona; , es, Universidad Autónoma de Barcelona; UAB), is a public university mostly located in Cerdanyola del Vallès, near the city of Barcelona in Catalonia, Spain. ...

and at the ''Institució Catalana de Recerca i Estudis Avançats'' (ICREA), the Catalan Institute for Advanced Scientific Studies. Tolsa does research on

harmonic analysis Harmonic analysis is a branch of mathematics concerned with the representation of Function (mathematics), functions or signals as the Superposition principle, superposition of basic waves, and the study of and generalization of the notions of Fo ...

(Calderón-Zygmund theory),

complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...

geometric measure theory In mathematics, geometric measure theory (GMT) is the study of geometric properties of sets (typically in Euclidean space) through measure theory. It allows mathematicians to extend tools from differential geometry to a much larger class of surfa ...

, and

potential theory In mathematics and mathematical physics, potential theory is the study of harmonic functions. The term "potential theory" was coined in 19th-century physics when it was realized that two fundamental forces of nature known at the time, namely gravi ...

. Specifically, he is known for his research on

analytic capacity In the mathematical discipline of complex analysis, the analytic capacity of a compact subset ''K'' of the complex plane is a number that denotes "how big" a bounded analytic function on C \ ''K'' can become. Roughly speaking, ''γ''(''K ...

and removable sets. He solved the problem of A. G. Vitushkin about the semi-additivity of analytic capacity. This enabled him to solve an even older problem of

Paul Painlevé Paul Painlevé (; 5 December 1863 – 29 October 1933) was a French mathematician and statesman. He served twice as Prime Minister of the Third Republic: 12 September – 13 November 1917 and 17 April – 22 November 1925. His entry into politic ...

on the geometric characterization of removable sets. Tolsa succeeded in solving the Painlevé problem by using the concept of so-called curvatures of measures introduced by Mark Melnikov in 1995. Tolsa's proof involves estimates of Cauchy transforms. He has also done research on the so-called

David David (; , "beloved one") (traditional spelling), , ''Dāwūd''; grc-koi, Δαυΐδ, Dauíd; la, Davidus, David; gez , ዳዊት, ''Dawit''; xcl, Դաւիթ, ''Dawitʿ''; cu, Давíдъ, ''Davidŭ''; possibly meaning "beloved one". w ...

- Semmes problem involving

Riesz transform In the mathematical theory of harmonic analysis, the Riesz transforms are a family of generalizations of the Hilbert transform to Euclidean spaces of dimension ''d'' > 1. They are a type of singular integral operator, meaning that they a ...

s and rectifiability. In 2002 he was awarded the

Salem Prize The Salem Prize, in memory of Raphael Salem, is awarded each year to young researchers for outstanding contributions to the field of analysis. It is awarded by the School of Mathematics at the Institute for Advanced Study in Princeton and was fo ...

. In 2006 in Madrid he was an Invited Speaker at the ICM with talk ''Analytic capacity, rectifiability, and the Cauchy integral''. He received in 2004 the

EMS Prize The European Mathematical Society (EMS) is a European organization dedicated to the development of mathematics in Europe. Its members are different mathematical societies in Europe, academic institutions and individual mathematicians. The current ...

and was an Invited Lecturer at the 2004 ECM with talk ''Painlevé's problem, analytic capacity and curvature of measures''. In 2013 he received the

Ferran Sunyer i Balaguer Prize The Ferran Sunyer i Balaguer Prize is a prize in mathematics, first awarded in 1993. It honors the memory of Ferran Sunyer i Balaguer (1912–1967), a self-taught Catalan mathematician who, despite a serious physical disability, was very active ...

for his monograph ''Analytic capacity, the Cauchy transform, and non-homogeneous Calderón-Zygmund theory'' (Birkhäuser Verlag, 2013}. In 2019 he received the Rei Jaume I prize for his contributions to Mathematics.

Selected publications

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References

{{DEFAULTSORT:Tolsa, Xavier 20th-century Spanish mathematicians 21st-century Spanish mathematicians Mathematical analysts Academic staff of the University of Barcelona 1966 births Living people