Title: Function spaces and applications to partial differential equations. Coordinator: Prof. V. Burenkov (Cardiff). The project is dedicated to the rapidly developing contemporary direction in analysis, the theory of spaces of differentiable functions and its applications to partial differential equations. Altogether there are 11 teams from 11 countries: Cardiff, Jena, Luleå, Padova, Paris, Prague, Astana, Baku, Moscow, Tbilisi, Yerevan, and 65 participants. Among them there are 28 professors (5 of them are members or corresponding members of national academies of sciences) and 17 participants are of age less that 35 (26%). All teams involved in the project are highly qualified and have long ago established research links, including mutual visits and many joint publications. The programme consists of two tasks: "Function spaces" and "Applications to partial differential equations" and 14 subtasks dedicated to interpolation, embedding and extension theorems, operators of classical analysis in Morrey-type spaces, asymptotic behaviour of the approximation and entropy numbers for operators of classical analysis, rearrangement invariant envelopes for spaces with generalised smoothness, composition operators, weighted inequalities for general integral and differential operators, asymptotic behaviour of solutions to linear and nonlinear elliptic equations near isolated singularities of the boundary and coefficients, spectral stability of differential operators, Weyl asymptotic formula for domains with rough boundaries, boundary values problems in domains with non-smooth boundary, regularity properties of solutions to hypoelliptic and more general equations, application of Morrey-type spaces to the problem of regularity of solutions to elliptic and parabolic and Navier-Stokes equations. In the majority of subtasks methodology would be based on the methods described in 29 books written by the participants. They are quite diverse: approximation theory, harmonic analysis, Fourier multipliers, singulars integrals, integral representations, various series expansions, mollifiers etc. However the necessity will arise in working out new approaches. An important characteristic feature of the programme is tight connection of the topics related to the theory of function spaces and to the theory of partial differential equations. Some of the subtasks of the second task are designed as direct applications of subtasks of the first task. In all other subtasks of the second task the results and technique of the theory of function spaces will be essentially used. The new knowledge generated by carrying out the project will be disseminated by publications in leading mathematical journals (it is expected that more than 100 papers will be published or submitted for publication) and by presentations at international conferences (it is expected that more than 100 talks on the research results of the project will be given).