Xavier Tolsa (born 1966) is a Catalan mathematician, specializing in analysis. Tolsa is a professor at the

Autonomous University of Barcelona The Autonomous University of Barcelona (; Spanish: ; ; UAB) is a public university mostly located in Cerdanyola del Vallès, near the city of Barcelona in Catalonia, Spain. , the university consists of 57 departments in the experimental, lif ...

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 investigating the connections between a function and its representation in frequency. The frequency representation is found by using the Fourier transform for functions on unbounded do ...

(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 functions of complex numbers. It is helpful in many branches of mathematics, including algebraic ...

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 surfac ...

, 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 the two fundamental forces of nature known at the time, namely g ...

. 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 France, Prime Minister of the French Third Republic, Third Republic: 12 September – 13 November 1917 and 17 A ...

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 Mark may refer to: In the Bible * Mark the Evangelist (5–68), traditionally ascribed author of the Gospel of Mark * Gospel of Mark, one of the four canonical gospels and one of the three synoptic gospels Currencies * Mark (currency), a currenc ...

in 1995. Tolsa's proof involves estimates of Cauchy transforms. He has also done research on the so-called

David David (; , "beloved one") was a king of ancient Israel and Judah and the third king of the United Monarchy, according to the Hebrew Bible and Old Testament. The Tel Dan stele, an Aramaic-inscribed stone erected by a king of Aram-Dam ...

- 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 ar ...

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 fou ...

. In 2006 in Madrid he was an Invited Speaker at the

ICM ICM may refer to: Organizations * Irish Church Missions, an Anglican mission * Institut du Cerveau, the Paris Brain Institute, a research center * Interdisciplinary Centre for Mathematical and Computational Modelling, University of Warsaw * Interna ...

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 curren ...

and was an Invited Lecturer at the 2004

ECM ECM may refer to the following: Economics and commerce * Engineering change management * Equity capital markets * Error correction model, an econometric model * European Common Market Mathematics * Lenstra's Elliptic curve method for factor ...

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 Spanish mathematician Ferran Sunyer i Balaguer (1912–1967), a self-taught Catalan mathematician who, despite significant physical disability, was ...

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

* * *

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 Ferran Sunyer i Balaguer Prize recipients