Mathematical Sciences: Topics in Logic and Set Theory

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9505118 Blass The research project of Andreas Blass is in four areas: (1) Cardinal characteristics of the continuum, including both the general theory of such characteristics (involving Galois-Tukey connections) and the detailed study of particular characteristics, especially the recently introduced evasion and prediction numbers. (2) Applications of set theory, especially in the theory of abelian groups. (3) Semantics of linear logic, especially game semantics and its variants. (4) Logic of topoi, particularly the logical properties of geometric morphisms. The research project of Glen Whitney concerns interpreting general recursive definitions, especially in non-determinisitic and concurrent contexts. Specific questions include: (1) Finding a universal class of structures with a recursion operation, so that every structure will be contained in one from this class. (2) Comparing proposals for the semantics of concurrent programming languages. (3) Seeking connections with fixpoint logics used in finite model theory. (4) Extending the current theory to allow more detailed modelings of parameter and message passing between subprograms. Both the formal language of recursion FLR, which Whitney will study, and Girard's linear logic LL, one of Blass' research topics, can serve as mathematical models of certain aspects of computing. FLR models recursion, i.e., programs using subroutines identical to the programs themselves. Whitney's project is directed toward improved understanding of recursion in the context of parallel computing, i.e. computations done collaboratively by many machines operating simultaneously. LL models data types; its game semantics models the situation where accessing data involves more of an interaction (access protocol) between the provider and the user of the data than mere transmission of data. Both theories are currently at the stage of mathematical exploration, but it is hoped that the information obtained w ill be of use in the design of future programming languages and in the formal verification of correctness of programs. Cardinal characteristics of the continuum are a way to exhibit and clarify the combinatorial content of difficult problems from various areas of mathematics -- originally topology and analysis, more recently algebra as well. They provide connections between these different areas and thus allow progress on a problem in one area to be applied in another. In addition, they make these problems amenable to treatment by the methods of modern set theory. The final topic in Blass' proposed research, topos theory, also serves to connect apparently disparate areas, mostly within mathematical logic. The particular topic of geometric morphisms of topoi combines ideas from set theory, from information systems (introduced as a model of computation), from sheaf theory (a part of topology), and from constructive logic. ***