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

雅罗什教授将研究一些悬而未决的问题 关于各种函数空间和代数之间的映射， 保持或仅略微扭曲度量结构。共同 线程是发现代数，几何，晶格，或其他 两个这样的空间共享的属性，当存在等距或 从一个到另一个的小界同构。这些问题 与Banach代数的小扰动密切相关， 巴拿赫-斯通定理的推广，巴拿赫几何 空间和拟共形映射。 Banach空间的理论是关于无限空间中的分析 多维度。对于每个有限维n，只有一个 n维空间，在大多数方面， 一般如在n为1、2或3的熟悉情况中。这些是 用来记录只有一个人的情况， 多个自由度。当自由度的数量 变成了无限的，例如， 一根绳子的配置，有许多可能 合理选择表演的环境空间 分析.一个关键的结构要素，区分这些 一个与另一个的距离是测量距离的方式。 雅罗什教授的项目关注的是 距离测量的代数和几何性质， 这种空间，特别是这些属性的持久性 在距离的微小扭曲下。

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.