Geometry of Banach spaces and metric spaces

首席研究员和他的研究生将调查问题的几何Banach空间和几何度量空间。要考虑的问题分为几个子类别，即，算子的Banach空间，非线性因子分解的线性算子，非线性分类的Banach空间，离散度量几何，逼近性质，和杂项问题。样本问题是：1.如果一个Banach空间有Pelczynski分解，并且该空间上的有界线性算子空间有唯一的极大理想，那么每个不是交换子的有界线性算子都必须是一个非零标量加上一个极大理想中的算子之和吗？2.可分Banach空间的Lipschitz可补子空间一定是线性可补的吗？3.估计最小维数k，使得可积函数空间的每个n点子集嵌入到由坐标绝对值之和赋范的k维空间中，失真最多为D。 (Here k将取决于子集中的点数n和允许的失真D）。在这个项目中处理的主题与分析的许多领域和分析以外的领域联系，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，高维数据集映射到广泛用于科学和工程的赋范向量空间。这项工作和其他方面的离散度量几何部分的项目可以用于算法的设计。此外，降维的积极成果将在压缩感知中得到应用。非线性算子的因式分解问题与几何分析中经常出现的问题有关，而线性有限维问题的研究属于凸几何和分析。 非线性分类问题的各个方面对于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.