Mathematical Sciences: The Lie Theory of Semigroups

这个项目的目的是在半群的李理论这一新兴学科中进行基本的数学研究，并探索和发展该理论与其他学科的接触点，如几何、控制论和物理学中的因果关系概念。在控制领域，人们考虑李群上的右不变向量场，并试图将各种可达性问题理解为关于李饱和的可控性和极大性。在随机性中，半群的李理论中的切线对象，即李楔形，以一种自然的方式产生齐次偶然流形。人们试图更好地理解所得到的流形何时具有诸如强烈偶然或全局双曲的理想物理性质(作为求解流形上的偏微分方程时的有用性质)。另一项拟议的研究集中在通过Ellis半群开发半群方法来研究拓扑动力学。群的李理论是一个高度发展的理论。它将现代分析和现代代数应用于数学物理和微分方程式理论中出现的几何对象。这一理论在回答困难而重要的问题方面非常成功，同样地，在提出有希望的调查路线方面也非常成功。半群是与群相关的代数系统，但没有如此多的结构。它们在数学和一些应用程序中可以发挥作用，但它们并不具有群体所享有的核心重要性。他们的谎言理论没有得到如此彻底的研究，部分是因为这个原因，部分是因为看不清如何继续下去。劳森教授克服了其中的一些困难，并在将半群的李理论建立在坚实的基础上取得了令人印象深刻的进展。他将继续在几何学、控制论和数学物理的相关领域探索这一问题及其后果。

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.