The principal investigator plans to work on three problems. The first is to find relationships between the topology of a real algebraic variety and its complex geometry, more precisely its Kodaira dimension. The ultimate is to prove that a topologicaly complicated (for intance hyperbolic) threefold can be represented only by a variety of maximal Kodaira dimension. The second project is to do a systematic investigation of the connections between hyperbolic differential equations and real algebraic geometry. Many instances of this have been noted in the past, but no systematic theory was ever developed. The third project is the study of arithmetic properties of rationally connected varieties, especially the existence of rational points and curves over fields which are not algebraically closed. It is quite likely that geometry governs these questions over local fields, while the problems over global fields are more arithmetic in nature.
Hyperbolic differential equations describe processes that change with time. For instance the heating up of a furnace, the flow of water through a turbine and the spreading of bacteria in a medium can all be described, more or less accurately, by hyperbolic differential equations. These differential equations and their solutions are rather complicated, both theoretically and computationally. It has been noticed that many questions about these differential equations can be approached through simple algebraic manipulations. It is the principal investigator's plan to put these diverse observations into a general conceptual framework. The principal investigator expects that this will lead to several new applications in the theory of hyperbolic differential equations. Conversely, relating the abstract algebraic machinery to physical phenomena in a new way should also provide insights to the behaviour of algebraic systems.
