The institute has several researchers working in this line in a wide spectrum of subjects, some of which, like for example the study of the problems of fluid mechanics and their numerical simulation, have far reaching physical applications. A large part of the work of this group lies at the interface between the more theoretical aspects of mathematical analysis and the more applied theory of differential equations. Indeed, in part, it is this interaction between differing mathematical cultures which has made the group so internationally recognised.

**Mathematical Analysis**

Harmonic analysis is a corner stone in the solution of many problems coming from the natural and social sciences. The analysis group at the ICMAT works actively on directional maximal operators, convergence of Fourier series and integrals, noncommutative Calderón-Zygmund theory, harmonic analysis on discrete groups, restriction of the Fourier transform to surfaces, inverse problems, geometric measure theory and weighted norm inequalities. These areas have applications in the recovery of potentials from scattered plane waves, elliptic and dispersive partial differential equations, number theory, as well as quantum probability and quantum information theory.

**Differential Equations and Applications **

This research line has grown to be one of the richest areas of mathematics. Ordinary and partial differential equations are very powerful tools for modelling many diverse phenomena, in physics, mechanics, chemistry, biology, economics, etc. Moreover, the emergence of computers has made these models even more efficient and useful by predictive numerical simulations. The researchers of the ICMat involved in this area work in the well-posedness and formation of singularities of PDEs, mathematical theory and numerical computations of Euler, Navier-Stokes and related equations of fluid mechanics, as well as the study of the mathematical theory of kinetic models related to statistical physics and convective instabilities in geophysical problems.