Mathematical Sciences: Isometries and Small Bound Isomorphisms of Banach Spaces and Algebras

Jarosz教授将研究一些关于各种函数空间和代数之间的映射的开放问题，这些映射保留或仅轻微扭曲度量结构。当一个空间到另一个空间存在等距或小界同构时，共同的主线是发现两个这样的空间共享的代数、几何、晶格或其他性质。这些问题与Banach代数的小摄动、Banach- stone定理的推广、Banach空间的几何以及拟共形映射密切相关。巴拿赫空间的理论是关于无限多维的分析。对于每一个有限维n，只有一个n维空间，在大多数情况下，当n为1、2或3时，一般情况下都很容易理解。它们用于跟踪只有有限多个自由度的情况。当自由度的数量变得无限时，例如描述一根弦的结构是必要的，对于执行分析的环境空间有许多可能的合理选择。区分这些空间的一个关键结构元素是测量距离的方式。Jarosz教授的项目涉及距离测量与这些空间中的代数和几何性质的关系，特别是这些性质在距离的微小扭曲下的持久性。

Professor Jarosz will investigate a number of open problems concerning maps between various function spaces and algebras that preserve or only slightly distort metric structure. The common thread is to discover algebraic, geometric, lattice, or other properties shared by two such spaces when there is an isometry or small bound isomorphism from one onto the other. These problems are closely related to small perturbations of Banach algebras, generalizations of the Banach-Stone theorem, geometry of Banach spaces, and quasiconformal mappings. The theory of Banach spaces is about analysis in infinitely many dimensions. For each finite dimension n, there is only one n-dimensional space, in most respects as well understood in general as in the familiar cases when n is 1, 2, or 3. These are used to keep track of situations in which there are only finitely many degrees of freedom. When the number of degrees of freedom becomes infinite, as would for instance be necessary to describe the configuration of a piece of string, there are many possible reasonable choices for the ambient space in which to perform analysis. A key element of structure that distinguishes these spaces one from another is the way in which distance is measured. Professor Jarosz's project is concerned with the relationship of distance measurement to algebraic and geometric properties in such spaces, in particular the persistence of these properties under small distortions of distance.