Welcome to the homepage of the team funded by the ERC Advanced Grant “Geometric Analysis in the Euclidean Space” (Principal Investigator: Xavier Tolsa). Our team studies different questions in the area of the so called geometric analysis. Many of the topics we are interested in deal with the connection between the behaviour of singular integrals and the geometry of sets and measures. The study of this connection has shown to be extremely useful for the solution of certain long standing problems in the last years, such as the Painlevé problem about the removability for bounded analytic functions.
In particular, we are interested in the relationship between the L2 boundedness of singular integrals associated with Riesz and other related kernels, and rectifiability and other geometric notions. The so called David-Semmes problem is probably the main open problem in this area. Some of the techniques used to study with this problem come from multi-scale analysis and involve ideas from Littlewood-Paley theory and quantitative techniques of rectifiability.
The research of our team also takes place in other related areas in mathematical analysis with a “geometric flavor”, like quasiconformal mappings, mass transport, or potential theory.