The geometry of Banach spaces is centrally located to successfully interact with many areas of mathematics including analysis and applied mathematics. This proposal involves problems that, in particular, interact with set theory, combinatorics, approximation theory and operator theory. The interaction with set theory comes about from embedding theorems in Banach spaces that could not hold without Martin's theorem that Borel games are determined. In turn their existence has had implications for set theory. Recent surprising embedding theorems of the investigator and co-authors are ripe for further development and application and this proposal contains such problems. In particular these results have implications on possible operator structures and how certain "outliers" exist much more widely than previously supposed. The geometry of Banach spaces has played a big role in the development by approximation theorists of various notions of greedy approximations. Problems extending the boundary of what is known in this region are also included in the proposal. The interaction with combinatorics is through Ramsey theory and its relation to the geometric notion of "partial unconditionality." This has remained a tough frontier for nearly 30 years, but as the interactions between researchers in both areas increase so is the likelihood that this can be successfully attacked.
This proposal could ultimately have impact in mathematical physics through the yet unproven (but now much more likely) existence of a space very close to Hilbert space where all the operators are simply written down. Physicists, rather than using Hilbert (i. e. Euclidean) space as a model for natural phenomena might be able to use a weak form of the space studied by Banach space geometers, if they can identify the operators on the space. Such a space may exist with very few operators, ones easily handled by theorists. Part of this proposal is to prove the widespread existence of such very few operator spaces. This is a first step towards the goal above. Another part of this proposal deals with problems in greedy approximation. This is, in turn, connected with problems in signal processing. The central problem there is to transmit information accurately and efficiently. The mathematical connections of this proposed research are substantial and widespread, touching, in particular upon set theory, analysis, approximation theory, operator theory and combinatorics. Increased communication between researchers in these areas will be fostered and should likely lead to further connections.
The original PI for this award passed away in January 2013. The original PI worked in the area of Banach spaces. This involves the study of spaces with a distance function that allows the computation of length and distance. These are basic objects in the study of spaces of functions. Regarding Intellectual Merit: The original PI continued to make important breakthroughs right up until the time of his death. Without question he has left an outstanding mathematical legacy through his work on Banach spaces and this will surely continue to have impact on postdocs and new generations of students interested in the subject. Beyond that, and in connection with Broader Impact he original PI brought his love of research to the classroom where he impacted many students lives particularly through his involvement in Inquiry Based Learning methods. Finally the broader impact can also be clearly seen through the original PI's longstanding reputation of being a great mentor to many young mathematicians in the field. Given the untmely death of the PI, the remaining funds in the grant were used to help support two young mathematicans. The two young mathematicians gave invited talks on their work, both here in the US and overseas in Europe and Israel. Thus their work was disseminated to the mathematical community.
