IPI group is mainly devoted to three independent research lines: inverse problems in PDEs, concentration inequalities and exceptional orthogonal polynomials. A major common characteristic of the three topics is that they are transversal to many other mathematical fields, as well as to other sciences.

Inverse problems in PDEs

The study of inverse problems is one of the most active fields of modern applied mathematics. Inside the context of partial differential equations (PDE), many inverse problems deal with the reconstruction of the (properties of the) interior of a medium in a non-invasive and non-destructive way. Their study makes use and contributes to the development of other areas of mathematics such as harmonic analysis, PDEs themselves, or differential geometry.

A prototypical example of inverse problem is the Calderón problem, which studies the determination of the electrical conductivity of a medium from voltage and current measurements on its boundary. Besides its applications in medical imaging and geophysics, the study of the Calderón problem is leading in the area of inverse problems for PDE for the last 40 years. 

Concentration inequalities

Functional inequalities play a crucial role in several mathematical fields, including PDEs, calculus of variations and mathematical physics. In particular, concentration inequalities may be considered from different perspectives including uncertainty principles, mathematical physics or signal processing. When the concentration is given in terms of the Wehrl entropy, the question on its optimality goes back to the late 1970s and involves the notion of coherent states.

Exceptional orthogonal polynomials

Orthogonal polynomials play a relevant role in many interesting problems related to differential equations, approximation theory, numerical analysis or quantum mechanics. For a long time, it was thought that the classical polynomials families of Hermite, Laguerre and Jacobi were the only complete sets of orthogonal polynomials that are eigenfunctions of a Sturm-Liouville problem. However, there are other families that satisfy these properties but, unlike the previous ones, their degree sequences have some gaps. The so-called exceptional orthogonal polynomials immediately received the attention of the community in orthogonal polynomials and special functions, and of mathematical physicists, who noted their natural presence in quantum mechanics problems.