### Xavier Tolsa (Universidade Autonoma de Barcelona, Espanha)

I work in mathematical analysis. My research deals with harmonic analysis, potential theory, 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 in the plane. 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 studied the connection between analytic capacity and the curvature of measures introduced by Melnikov, and the semiadditivity of analytic capacity. The relationship between analytic capacity and Favard length is another question in this area which is not fully understood and that still calls my attention.

I have also studied related problems in higher dimensions, such as the so-called David-Semmes problem about the Riesz transforms and rectifiability. More recently, I have applied some of the techniques developed for the David-Semmes problem to study different questions on harmonic measure, which is another of my main interests at present, together with some neighboring topics on free boundary problems,

## ENSPM 2022 Presentation

**Title: Rectifiability, square functions, and a conjecture of Carleson**

**Abstract: ** In this talk we will review some classical characterizations of the notion of rectifiability and other more recent related results connected to the travelling salesman theorem of Peter Jones in terms of the so-called -numbers.

We will also explain the application of these techniques to other areas of analysis and PDE’s. One such application is the solution of the conjecture of Carleson by Jaye, Villa, and myself about the characterization of tangent points of a Jordan curve in terms of the so-called -square function.