Mathematical Sciences: The Lie Theory of Semigroups

9623724劳森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，并探索和扩大该理论与其他学科的联系，如几何、控制论和物理学中的因果关系概念。一般理论通过引入秩序或“实证性”的概念，为经典的谎言理论提供了新的方向和动力，从而为其他学科开辟了新的接触点和应用。在群水平上，合适的研究对象是由其单参数子半群生成的李群的子半群。在李代数水平上，它是一类特殊的锥体，称为李楔。在齐性空间的层次上，是齐性因果序和齐性因果结构的相互联系的概念。这些对应结构的理论是本研究的主要焦点。经典李氏理论已被证明是一种极其通用和强大的数学工具，在几何学、数学物理、微分方程和其他许多学科中都有广泛的应用。在这个项目所涉及的新方向上，Lie理论已经有了实质性的发展和应用，系统的最优控制理论是一个非常成功的应用领域，预计该研究计划将显著提高理论和它的适用性。***

9623724 Lawson 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 extend contact of the theory with other disciplines such as geometry, control theory, and notions of causality in physics. The general theory has provided a new direction and impetus to classical Lie theory by introducing a notion of order or "positivity" to the theory and thereby opened up new contact points and applications to other disciplines. At the group level, the appropriate objects to be investigated are those subsemigroups of Lie groups that are generated by their one-parameter subsemigroups. At the Lie algebra level, it is a special class of cones called Lie wedges. And at the level of homogeneous spaces, it is the interrelated notions of homogeneous causal orders and homogeneous causal structures. The theory of these corresponding structures is the principal focus of this investigation. Classical Lie theory has proved itself an extremely versatile and powerful mathematical tool with a host of applications in geometry, mathematical physics, differential equations, and a wide variety of other disciplines. There has already been a substantial development and application of Lie theory in the new directions with which this project is concerned, the theory of optimal control of systems being an area of notably successful application, and it is anticipated that the program of research will significantly enhance both the theory and its applicability. ***