PIF2025 - Inverse problems and funcional inequalities - (PID2024-156055NA-I00)

The study of inverse problems is one of the most active fields of modern applied mathematics. A prototypical example 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. We plan to contribute to several open questions about this problem by exploring new methods that put aside the restrictions of previous approaches. Our objectives are closely related with unique continuation properties, nonlocal equations and differential geometry. Functional inequalities play a crucial role in several mathematical fields, including partial differential equations, calculus of variations and mathematical physics. In particular, concentration inequalities may be considered from different perspectives including uncertainty principles, mathematical physics or signal processing. We aim to study the extremal properties of coherent states in different spaces, answering positively to a Wehrl-type entropy conjecture. The thesis project may focus on one or both topics.