Mathematical Sciences: The Lie Theory of Semigroups

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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. ***