Descripción del proyecto
Un nuevo método para resolver el problema lorentziano de Calderón
La teoría matemática de los problemas inversos es un campo de investigación interdisciplinar que se sitúa entre las matemáticas puras y las matemáticas aplicadas. Uno de los problemas inversos más importantes es el problema lorentziano de Calderón. En comparación con problemas inversos similares para ecuaciones de onda no lineales, el problema lorentziano de Calderón no se conoce bien. El equipo del proyecto LoCal, financiado con fondos europeos, desarrollará un nuevo método para resolver el problema lorentziano de Calderón. Explorará las condiciones geométricas que lo permiten, es decir, los medios que son visibles a las ondas de sondeo, así como los contraejemplos que violan tales condiciones y conducen a la invisibilidad. Para lograrlo, el equipo desarrollará técnicas basadas en la intersección de las ecuaciones diferenciales parciales y la geometría, con afinidad a la teoría de control y la relatividad general.
Objetivo
This project addresses questions in the mathematical theory of inverse problems, a research field at the interface between pure and applied mathematics. The techniques that will be developed lie at the intersection of partial differential equations and geometry, with affinity to control theory and general relativity.
The Lorentzian Calderon problem is central to the proposal. A physical interpretation of the problem asks us to recover a moving medium given data generated by acoustic waves probing the medium, and seen from the mathematical point of view, it is the simplest formulation of an inverse boundary value problem for a linear wave equation that is expressed in a generally covariant fashion. The project explores the geometric conditions under which the problem can be solved, that is, media that are visible to probing waves, as well as counterexamples violating such conditions, leading to invisibility.
The Lorentzian Calderon problem is poorly understood in comparison to similar inverse problems for nonlinear wave equations. One of the guiding ideas in the project is to adapt techniques from the theory of these problems, developed recently by PI and others, to the Lorentzian Calderon problem. Another source of inspiration is the recent solution of the Lorentzian Calderon problem under curvature bounds by PI and his coauthors.
The project develops a new approach to solve the Lorentzian Calderon problem, and this may also lead to a breakthrough in the resolution of the Riemannian version of the problem, often called the anisotropic Calderon problem. The latter problem has remained open for more than 30 years. In addition to being the hyperbolic analogue of this well-known problem, the Lorentzian Calderon problem can be viewed as a generalization of the even more classical inverse problem studied by Gelfand and Levitan in the 1950s.
Palabras clave
Programa(s)
- HORIZON.1.1 - European Research Council (ERC) Main Programme
Régimen de financiación
HORIZON-ERC - HORIZON ERC GrantsInstitución de acogida
00014 HELSINGIN YLIOPISTO
Finlandia