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.
