Mathematical Sciences: The Lie Theory of Semigroups

该项目的目的是在新兴的半群李论学科中开展基础数学研究，并探索和发展该理论与其他学科（如几何、控制论和物理学中的因果关系概念）的联系点。在控制领域，人们考虑李群上的右不变向量场，并试图理解关于李饱和的各种可达性问题，如可控性和极大性。在随意性中，半群李论中的切线对象，即李楔，以一种自然的方式产生齐次随意性流形。当得到的流形具有强偶然性或全局双曲性（作为在流形上求解偏微分方程的有用性质）等理想的物理性质时，人们寻求更好的理解。另一条建议的研究路线是通过埃利斯半群发展拓扑动力学的半群方法。群的李理论是一个高度发达的理论。它将现代分析和现代代数应用于数学物理和微分方程理论中出现的几何对象。这个理论在回答困难和重要的问题方面非常成功，同样，在提出有希望的研究方向方面也非常成功。半群是与群相关但没有那么多结构的代数系统。它们在数学和一些应用中发挥着作用，但它们不具有群体所享有的核心重要性。他们的谎言理论还没有得到如此彻底的研究，部分原因是这个，部分原因是很难看到如何进行。劳森教授已经克服了其中的一些困难，并取得了令人印象深刻的进展，使半群的李论有了坚实的基础。他将继续在几何、控制理论和数学物理等相关领域进行研究。

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.