Infinite-Dimensional Banach and Operator Spaces

奥德尔 本课题涉及几何泛函分析中的几个公开问题。 1)是否存在有界畸变的畸变Banach空间？ 2)若X的任一给定基的块基的所有扩散模型都1-等价于某个l_p空间，那么X是否包含l_p？ 3)每一个Banach空间都必须有一个“好的”扩散模型吗？ 4)Banach空间序l_1（+）指标的研究 在通常的空间几何学中，人们可以用木匠的卷尺测量两点之间的距离（距离的测量是“直线距离”）。 然而，数学家遇到的其他自然距离。 例如，这个表面上两点之间的“出租车距离”是由只能水平或垂直行驶的出租车测量的两点之间的距离（想象所有道路都是南北走向或东西走向）。 数学家必须研究这些不同距离概念的几何学，以便解决这些几何学产生的问题。 此外，许多数学问题需要在三维以上的空间中工作，甚至是无限维空间（例如量子力学微分方程）。 本项目涉及这些问题。 例如，作者和Th。Schlumprecht证明，普通（卷尺）无限维空间可以有其几何改变（“扭曲”），以便是一个无法夺回原来的几何形状，以任何给定的多个在任何无限维“切片”。 这对于出租车几何学来说是错误的。但是，是否存在一种可以改变但又不太坏的几何，这仍然是一个著名的悬而未决的问题。 这个项目的其他问题涉及到确定有限的“渐近”（小于无限维，但超过有限维）信息的几何无限维子结构。

ABSTRACTOdell This project involves several open problems in geometric functional analysis. 1) Does there exist a distortable Banach space of bounded distortion? 2) If all spreading models of any block basis of a given basis for X are 1-equivalent to some l_p space does X contain l_p? 3) Must every Banach space admit a "nice" spreading model? 4) The study of the ordinal l_1(+) index of a Banach space. In the usual geometry of space one can measure the distance between two points by means of a carpenter's tape measure (distance is measured "as the crow flies"). There are however other natural distances which a mathematician encounters. For example the "taxicab distance" between two points on this surface would be the distance between the points as measured by a taxi that could only travel horizontally or vertically (imagine all roads run N-S or E-W). Mathematicians must study the geometry of these different notions of distance in order to solve those problems from whence the geometry arose. Furthermore many mathematical problems require one to work in spaces of higher than three dimensions, even infinite dimensional space (for example in quantum mechanics differential equations,...). This project concerns such problems. For example the author and Th. Schlumprecht proved that ordinary (tape measure) infinite dimensional space could have its geometry altered ("distorted") so as to be that one could not recapture the original geometry up to any given multiple in any infinite dimensional "slice". This is false for the taxicab geometry. But it remains a famous open problem as to whether or not there exists a geometry which can be altered but not too badly. Other problems of this project involve determining infinite dimensional substructures of a geometry given limited "asymptotical" (less than infinite dimensional but more than finite dimensional) information.