___________________JOINT COLLOQUIUM______________________ Speaker: Dr Sergey Ajiev, (UNSW) Title: Homogeneous right inverses and metric projections. Date: Friday, August 21, 2009 Time: 2:00 pm Venue: Room 275 Carslaw Building, University of Sydney Preceded by lunch, meeting at 12:30 p.m., outside foyer, second-floor Carslaw building. Abstract: Some problems, for example, in PDE can be reduced to the invertibility of a closed operator with a non-trivial and non-complemented kernel defined, for example, on a Sobolev space. We construct homogeneous (non-linear) right-inverse operators and establish explicit estimates for the exponents of their Hölder regularity and the corresponding Hölder seminorms in both abstract and particular settings. In the setting of Banach spaces, our right inverses provide optimal solution for the equation Af=g. One deals with various types of Besov and Lizorkin-Triebel spaces of functions on an arbitrary open subset of an Euclidean space defined in terms of local approximations, differences, wavelet expansions, or a functional calculus and also with their duals, subspaces, quotients and a wide class of "independently generated spaces". We also discuss the relations with the classical problem of the regularity of the metric projection map and establish the global regularity of this map in a quantitative manner. Attention is paid to the sharpness of some results, natural generalisations and the limitations of the linear and Lipschitz settings.