Geometry of Banach spaces and metric spaces

首席研究员和他的研究生将研究巴拿赫空间几何和度量空间几何中的问题。要考虑的问题可分为几个小类别；即，Banach空间上算子的对易子，线性算子的非线性分解，Banach空间的非线性分类，离散度量几何，近似性质，以及其他问题。示例问题有：1。如果一个巴拿赫空间有一个Pelczynski分解并且空间上有界线性算子的空间有一个唯一的极大理想，那么每一个非对易子的有界线性算子必须是一个非零标量加一个极大理想中的算子的和吗？2. 可分离巴拿赫空间的Lipschitz补子空间必须是线性补的吗？3. 估计最小的维度k，使得可积函数空间的每个n点子集嵌入到由坐标的绝对值和赋范的k维空间中，其失真不超过D。（这里k将取决于n，子集中的点的数量，和D，允许的失真）。在这个项目中处理的主题涉及分析的许多领域和分析之外的领域，包括算子理论、群论、几何分析和理论计算机科学。其中一名研究生参与开发概念框架，用于理解大规模，高维数据集到赋范向量空间的映射，这在科学和工程中广泛使用。本工作和其它方面的离散度量几何部分的课题可以用于算法的设计中。此外，在降维方面的积极成果将在压缩感知中得到应用。非线性算子的分解问题是几何分析中反复出现的问题，而线性有限维研究既属于凸几何，也属于分析。非线性分类问题的某些方面对几何群论和几何泛函分析同样重要。

The principal investigator and his graduate students will investigate problems in the geometry of Banach spaces and the geometry of metric spaces. The problems to be considered fall into several subcategories; namely, commutators of operators on a Banach space, non linear factorization of linear operators, non linear classification of Banach spaces, discrete metric geometry, approximation properties, and miscellaneous problems. Sample problems are: 1. If a Banach space has a Pelczynski decomposition and the space of bounded linear operators on the space has a unique maximal ideal, must every bounded linear operator that is not a commutator be the sum of a non zero scalar plus an operator in the maximal ideal? 2. Must a Lipschitz complemented subspace of a separable Banach space be linearly complemented? 3. Estimate the smallest dimension k such that every n point subset of the space of integrable functions embeds, with distortion at most D, into the k dimensional space normed by the sum of the absolute values of the coordinates. (Here k will depend on both n, the number of points in the subset, and D, the allowable distortion).The topics treated in this project make contact with many areas of analysis and areas outside of analysis, including operator theory, group theory, geometric analysis, and theoretical computer science. One of the graduate students is involved in developing the conceptual framework for understandings mappings of large scale, high dimensional data sets into the normed vector spaces that are widely used in science and engineering. This work and other aspects of the discrete metric geometry part of the project can be used in the design of algorithms. Moreover, positive results on dimension reduction will have applications in compressed sensing. The questions on the factorization of non linear operators are connected to recurring problems in geometric analysis, while the linear finite dimensional investigation belong as much to convex geometry as to analysis. Aspects of the non linear classification problems may be as important for geometric group theory as for geometric functional analysis.