Modern computer systems are too complex to be analyzed directly, and often mathematical models are employed to analyze such computer systems to understand how well they function and whether they operate correctly. The area of computational models is devoted to constructing such models, and to their application to solve problems in computing, such as specifying computational devices designed to solve specific problems. Quite distinct from this, the area of Diophantine approximation from number theory studies how well real numbers can be approximated by rational numbers - ones that can be expressed a quotient of two integers. Despite dating back to the time of Diophantus of Alexandria, there are many deep questions here that remain unsolved. Finally, the study of classical communication channels forms the basis for the area of classical information theory that plays a fundamental role in many aspects of everyday life. This project will explore relationships between emerging techniques for building computational models, and research in classical information and in Diophantine approximation. These represent very diverse areas of mathematics, computer science and information theory, and a goal of the project is to develop links between them that reveal common underlying principles that can be applied to solve problems in each area. Such principles will lead to methods that can be used from one area to solve problems in the other areas.
The proposed work spans a number of research areas: domain theory and computational models; topological games; classical information and families of classical channels; analysis of complex systems; and applications to Schmidt games and Diophantine approximation. Recent results on channel capacity that underpin the work are based on a topological view of channels and their capacity that offers a unique way to analyze related families of channels. The proposed analysis of existing approaches to constructing domain models of spaces - the co-algebraic approach as opposed to one based on Choquet games - will explore the relationship between these approaches, and will provide insights into their applicability to new and interesting problems in the range of areas listed above. Using domain-theoretic approaches to analyze Schmidt games is a novel idea that should prove useful in understanding these games and how they differ from ones such as Choquet games.
