Final Activity Report Summary - LOGIC AND ANALYSIS (Set Theoretic Methods in Analysis) A number of results have been obtained in the framework of functional analysis, general topology and set theory, and especially in applications of set theoretic methods in functional analysis. Two joint works of the fellow Aviles and supervisor Todorcevic have been produced. In the first one we solved an open question regarding zero subspaces of polynomials in complex Banach spaces by using fine partition techniques developed by Todorcevic. In the second one, still in preparation, motivated by a problem about extension of operators in Banach spaces, we have developed a systematic study of what we called multiple gaps, which is a multidimensional version of the classical notion of gap in set theory. The fellow has also obtained several results on his own or in collaboration with other researchers. This includes his joint work with Kalenda in which he developed a technique to associate ordered sets to compact spaces, with applications to the homeomorphic classification of some natural compacta. The fellow wrote three further papers related to this subject with applications of this techniques to problems relating the norm and weak topology of a Banach space, or to the possibility of mapping a finite product of compacta onto another. In a joint work with Arvanitakis, possible counterexamples to the problem of RN compacta are proposed. He also wrote a paper about descriptive complexity of Banach spaces, showing that spaces of arbitrarily high complexity exist. His works in collaboration with several co-authors related to extensions of operators and operator equations respectively complete the picture of the research of the fellow during this period.