Universitat Autònoma de Barcelona (UAB)
Experimental Sciences & Mathematics
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Research interests
I work in mathematical analysis. My research deals with harmonic analysis, geometric measure theory, and elliptic PDE's. 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. Later on I 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 and the Dirichlet problem for the Laplace equation, which are other main interests.