Mathematical Sciences: The Lie Theory of Semigroups

该项目的目的是进行基础数学 在李理论的新兴学科的研究， 半群和探索和发展的接触点， 理论与其他学科，如几何，控制理论， 和物理学中的因果关系概念。 在控制领域， 考虑李群上的右不变向量场， 将各种可达性问题理解为可控性， 和Lie饱和度的极大值。 在政治上， 半群的李理论中的切对象，李楔， 以一种自然的方式产生齐次偶然流形。 一 寻求更好地理解当产生的流形有 这些理想的物理性质，如强烈的偶然性或 全局双曲（作为求解部分 流形上的微分方程）。 另一行 建议的研究中心的发展，半群 通过Ellis半群的拓扑动力学方法。 李群理论是一个高度发展的理论。 它 将现代分析和现代代数引入几何学 数学物理学中出现的物体， 微分方程 这一理论非常成功， 回答困难和重要的问题，同样， 暗示着有希望的调查线索 半群是代数的 与群体相关但没有太多结构的系统。 他们 在数学和一些应用中发挥作用，但 它们没有群体所具有的核心重要性， 享受. 他们的谎言理论还没有被彻底研究过， 部分是因为这个原因，部分是因为很难看到 如何进行。 劳森教授已经克服了其中的一些问题 困难，并取得了令人印象深刻的进展， 半群理论的坚实基础。 他将继续 继续这一调查路线及其在相关领域的影响， 几何学、控制理论和数学物理学领域。

The aim of this project is to carry out basic mathematical research in the emerging discipline of the Lie theory of semigroups and to explore and develop points of contact of the theory with other disciplines such as geometry, control theory, and notions of causality in physics. In the area of control one considers right invariant vector fields on a Lie group and seeks to understand various attainability questions as controllability and maximality with respect to Lie saturation. In casuality, the tangent objects in the Lie theory of semigroups, the Lie wedges, give rise in a natural way to homogeneous casual manifolds. One seeks a better understanding of when the resulting manifolds have such desirable physical properties as being strongly casual or globally hyperbolic (as useful property for solving partial differential equations on the manifold). Another line of proposed research centers on the development of a semigroup approach to topological dynamics via the Ellis semigroup. The Lie theory of groups is a highly developed theory. It brings modern analysis and modern algebra to bear upon geometric objects which arise in mathematical physics and in the theory of differential equations. This theory has been very successful in answering difficult and important questions, and equally, in suggesting promising lines of inquiry. Semigroups are algebraic systems related to groups but without so much structure. They have a role to play in mathematics and some applications, but they do not have the central importance that groups have come to enjoy. Their Lie theory has not been so thoroughly investigated, partly for this reason, partly because of difficulties in seeing how to proceed. Professor Lawson has overcome some of these difficulties and has made impressive progress in putting the Lie theory of semigroups on a firm footing. He will continue to pursue this line of inquiry and its ramifications in related areas of geometry, control theory, and mathematical physics.