Interactions between the geometry of Banach spaces and other areas

Banach空间的几何位于中心位置，以成功地与许多数学领域相互作用，包括分析和应用数学。这一建议涉及的问题，特别是与集合论、组合学、逼近理论和算子理论相互作用的问题。与集合论的相互作用来自于Banach空间中的嵌入定理，如果没有确定Borel对策的Martin定理，这些定理是不成立的。反过来，它们的存在也对集合论产生了影响。最近研究者和合著者令人惊讶的嵌入定理已经成熟，可以进一步发展和应用，而这个建议就包含了这样的问题。特别是，这些结果对可能的运算符结构以及某些“离群值”如何比先前假设的更广泛地存在具有一定的影响。Banach空间的几何在逼近理论家提出的各种贪婪逼近概念的发展中发挥了重要作用。该提案还包括扩大这一地区已知范围的问题。与组合学的相互作用是通过拉姆齐理论及其与“部分无条件”几何概念的关系来实现的。近30年来，这一直是一个艰难的前沿，但随着两个领域研究人员之间互动的增加，这一点被成功攻破的可能性也在增加。这一提议最终可能会对数学物理产生影响，因为它存在一个非常接近希尔伯特空间的空间(但现在更有可能)，在那里，所有的算子都被简单地写下来。物理学家，而不是使用希尔伯特(即欧几里得)空间作为自然现象的模型，也许能够使用Banach空间几何学家研究的弱空间形式，如果他们能识别空间上的算子的话。这样的空间可能只有很少的运算符，很容易被理论家处理。这一建议的一部分是为了证明这种极少数算子空间的广泛存在。这是朝着上述目标迈出的第一步。这项建议的另一部分涉及贪婪近似问题。这反过来又与信号处理中的问题有关。那里的中心问题是准确而高效地传递信息。这项研究的数学联系是丰富和广泛的，涉及到集合论、分析、逼近理论、算子理论和组合学。将促进这些领域研究人员之间的更多交流，并可能导致进一步的联系。

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.