Harmonic analysis meets inverse problems

This research proposal lies in the areas of harmonic analysis and inverse problems. The proposal mostly suggests and investigates problems in these two main thematic areas, using a wide variety of tools from the theory of partial differential equations, functional analysis, geometry and combinatorics, among others.

Many problems in theoretical and applied mathematics are initially stated in a geometric or combinatorial language. For example, we want to understand a set if we know that it contains many simple patterns, such as line segments or planes. Harmonic analysis deals with properties of functions and operators; in this proposal we encode the geometric or combinatorial information about the set into a quantitative question about the size of a such an operator, for which we aim at proving optimal bounds.

Another classical theme in harmonic analysis is the dual description of functions (signals), in space (time), and frequency. We want to be able to go from one realization of a function to the other. One of the very important harmonic analysis achievements of the previous century, the Carleson theorem on the convergence of Fourier series, tells us that in many cases we can pass from physical space to frequency in a natural (pointwise) way, and proves that we will not be wrong in doing so. In this proposal we explore operators which encode geometric and combinatorial information as described above: for example the operators are acting on certain patterns such as lines or planes. We explore surprising connections between  quantitative properties of such operators and higher dimensional versions of Carleson-type theorems.

Inverse problems arising in the analysis of partial differential equations aim at determining the coefficients of a given equations from knowledge of their solutions. Solving inverse problems has two important consequences. The first one has physical implications, since equations usually model certain phenomena ---as dynamics of electromagnetic waves or quantum particles--- in specific media, and their coefficients encode the physical properties of such media. Thus, when solving an inverse problem we are detecting properties of a given medium in a non-invasive manner. For this reason inverse problems have important applications in remote sensing strategies, nondestructive testing, or medical imaging techniques. The second consequence of inverse problems has a theoretical character and states how differential operators are determined by some subspaces of functions in their domains of definition, such as their kernels.

A very famous inverse problem that has driven the field in the last sixty years is known as the Calderón problem. It consists in reconstructing the electric conductivity of a given body by measuring the voltage of specific currents on its surface. In this proposal, we analyse some capital questions about this problem that remain open, and consider related problems for quantum systems.