## Research lines

The group has a long tradition dating back to the start of the millennium when Juan Antonio Barceló and Alberto Ruiz became interested in inverse problems; that is to say, given the answer, can you guess the question? Initially this sounds like a somewhat esoteric pursuit, however this kind of process is what underpins our understanding of the natural world. Not only do particle physicists attempt to guess the properties of a subatomic particle (the question) from measurements of the scattered beam (the answer), but even our brains guess the objects that have distorted the light that arrives to our eyes. Other physical inverse processes that we are currently studying include Calderón's conductivity problem and elasticity.

The mathematics we employ includes deep results from Fourier analysis, microlocal analysis, quasiconformal mappings, differential geometry, as well as numerical analysis. Indeed, although inverse problems brought us together, we have since realised that we have many more common interests. The cooperation has been fruitful to settle a framework where young researchers are exposed to various mathematical theories, broadening their future horizons.

Beyond their potential application to inverse problems, below is a nonexclusive list of areas that we are currently interested in, mainly for their own sake:

- The Kakeya, restriction and Bochner-Riesz problems.
- Geometric measure theory including Falconer's distance problem.
- Pointwise convergence problems for the Schrödinger and wave equations.
- Semiclassical analysis: global geometry and high-frequency solutions of dispersive PDE.
- Controllability of PDE.
- Spectral theory and spectral geometry.
- Metric and Riemannian geometry.
- Conformal geometry, conformally invariant tensors and quasiconformal mappings.
- Vectorial calculus of variations: Rank-one convexity, quasiconvexity and the Morrey problem.
- Mechanics of solids: Peridynamics and variational theories of cracks and fracture.
- Fluid mechanics: Instability and turbulence modelled via convex integration.
- Scattering theory, wave propagation and mathematical quantum physics.
- Particle methods in hydrodynamics.
- Tokamaks and sun flares.

## Oportunidades

### Oportunidades pasadas

- Beca FPI para realizar una tesis doctoral en nuestro grupo de investigación. Convocatoria