Interactions between the geometry of Banach spaces and other areas

Banach 空间的几何形状与其他区域之间的相互作用

• 批准号: 0968813
• 负责人: Alan Reid
• 金额: $ 16.74万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-09-15 至 2013-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0968813&HistoricalAwards=false
• 关键词: Interactions; between; geometry; Banach; spaces

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.

巴拿赫空间的几何结构位于中心位置，可以成功地与许多数学领域进行交互，包括分析和应用数学。这个建议涉及的问题，特别是与集合论，组合学，近似理论和算子理论相互作用。与集合论的互动来自于Banach空间中的嵌入定理，如果没有Borel博弈是确定的Martin定理，这些定理就无法成立。反过来，它们的存在又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，而这个提议包含了这样的问题。特别是，这些结果对可能的算子结构以及某些“异常值”如何比以前假设的更广泛地存在有影响。巴拿赫空间的几何在逼近理论家对贪心逼近的各种概念的发展中起着重要的作用。该建议还包括扩展该地区已知边界的问题。与组合学的互动是通过拉姆齐理论及其与几何概念“部分无条件”的关系。近30年来，这一直是一个棘手的前沿问题，但随着这两个领域的研究人员之间的互动增加，这一问题被成功攻克的可能性也在增加。这个提议最终可能会对数学物理产生影响，因为尚未被证明（但现在更有可能）存在一个非常接近希尔伯特空间的空间，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，可能能够使用巴拿赫空间几何学者所研究的空间的弱形式，如果他们能够识别空间上的算子。这样的空间可能存在很少的算子，这些算子很容易被理论家处理。这个建议的一部分是为了证明这种极少算子空间的广泛存在。这是实现上述目标的第一步。本文的另一部分讨论了贪心逼近中的问题。这又与信号处理中的问题有关。中心问题是如何准确有效地传递信息。这个提议的研究的数学联系是实质性的和广泛的，特别是在集合论，分析，近似理论，算子理论和组合。将促进这些领域的研究人员之间增加交流，并可能导致进一步的联系。