Mathematical Sciences: Geometric Properties Determined by Random Walks

Ball将研究度量空间的几何性质，特别是lp空间，它描述了这些空间中随机漫步的速率。他已经证明了这些性质与分解和扩展Lipschitz映射的问题密切相关。目的是发展一个非线性理论，它与线性空间中强大的类型和共类型理论平行。这样的理论可以预期有许多应用，以分析各种类型的度量空间出现在数学的各个领域。巴拿赫空间理论是数学的一部分，它试图将三维欧几里得空间（即普通空间）的结构推广到无限多维。巴拿赫空间中距离函数的公理比欧几里得距离的公理更宽松（例如，“平行四边形定律”不需要成立），因此，巴拿赫空间的“几何”可能相当奇特。该领域的研究主要集中在Banach空间的结构理论研究上。

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.