Mathematical Sciences: Geometric Properties Determined by Random Walks

球将研究几何性质的度量空间，在 特别是Lp空间，它描述了随机 在这些空间里散步可以漫步。 他指出，这些 属性与以下问题密切相关： 分解和扩展Lipschitz映射。 目的是发展一个 非线性理论，它与强大的类型理论平行， 线性空间中的同构 这样的理论可以预期有 许多应用程序的分析，各种类型的 度量空间出现在数学的所有领域。 Banach空间理论是数学的一部分， 试图将其推广到无限多个维度， 三维欧氏空间的结构。 的 Banach空间中距离函数的公理更多 比欧几里得距离的那些更宽松（例如， “法律”不需要持有。），结果， 巴拿赫空间的“几何学”可以是相当奇异的。 大部分 这方面的研究涉及到结构理论的研究 Banach空间。

Ball will study geometric properties of metric spaces, in particular the Lp-spaces, which describe the rate at which random walks in these spaces can wander. He has shown that these properties are intimately connected with questions about factoring and extending Lipschitz maps. The aim is to develop a non-linear theory which parallels the powerful theory of type and cotype in linear spaces. Such a theory can be expected to have numerous applications to the analysis of the various types of metric spaces which arise in all areas of mathematics. Banach space theory is that part of mathematics that attempts to generalize to infinitely many dimensions the structure of 3-dimensional Euclidean (i.e.ordinary) space. The axioms for the distance function in a Banach space are more relaxed than those for Euclidean distance (For example, the "parallelogram law" is not required to hold.), and as a result, the "geometry" of a Banach space can be quite exotic. Much of the research in this area concerns studying the structure theory of Banach spaces.