LOGIC AND ANALYSIS · Set theoretic methods in analysis
FP6 — Marie Curie Actions (Human Resources and Mobility)
- Duration
- 2006-11-02 → 2008-11-01
- EU contribution
- €143,433
- Participants
- 1
- Scheme
- EIF
Lines connect the coordinator with its partners. CORDIS does not always give exact coordinates for projects before 2014. These points are placed at city or country level.
Results in brief
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.
Data: CORDIS, © European Union
Project objective
In this project we propose to use set-theoretic methods in order to study two sets of problems coming from functional analysis. The first concerns with the problem of the continuous images of Radon-Nikodym compact spaces posed by Namioka some thirty years ago. This class of compacta has been isolated in the early 1970 and apos;s in the course of studying a Radon-Nikodym property of vector measures. Particularly useful was the characterization in terms of Frechet differentiability leading to a purely topological formulation of the problem. The other is the problem of the uniform regularity of Borel measures, posed by Pol for Rosenthal compacta and by Fremlin for general first countable compacta (it is one prominent item of his famous list of problems).We plan to supplement the standard techniques in dealing with this kind of problems with advanced methods of set theory and Ramsey theory, and particularly the method of forcing. It is generally believed that one of the reasons why these problems have been open for so long time is the fact that so far too elementary set-theoretic methods have been applied. During the last thirty or forty years set theory has advanced considerably and specially its method of forcing invented by Paul J. Cohen some forty years ago in order to prove the independence of the Continuum Hypothesis. By now this method has been developed to the point of being able to obtain results that are not necessarily consistency results.Moreover in recent times this method has been successfully merged with another powerful method, the Ramsey-theoretic method. We hope that the combination of these two methods is likely to give us a substantial progress on these two long- standing open problems of functional analysis.
Original text from CORDIS.
Participants
- UNIVERSITE PARIS 7 DENIS DIDEROT · PARISCoordinatorCity levelFrance
Links
Data: CORDIS, © European Union