Mathematical Sciences: Logic and Analysis

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9503398 Jockusch This project concerns mathematical logic and its connections with other areas of mathematics, especially algebra and analysis. Carl Jockusch will work in computability theory (recursion theory) and its connections with other areas. In particular, he will study degrees of unsolvability, both in their own right and in connection with effective combinatorics, group theory, and algebra. L. van den Dries will investigate expansions of the real field by quasi-analytic classes of functions that are familiar from analysis. The goal is to prove o-minimality and polynomial boundedness of such expansions. This is part of an effort to create a framework for solving Hilbert's 16th problem on limit cycles and similar problems. Mathematical logic provides a precise framework for mathematical assertions and deductive systems. It also provides a precise concept of computability by abstract computers, i.e. ones without limitations of running time or memory space. Jockusch will study the theory of computability and its connections with other branches of mathematics. Theoretical limitations on what can be computed and the possibility of computing information given access to other information are of central interest for this research. In particular, he will analyze questions related to computability in algebra and number theory. Lou van den Dries will work in analytic geometry, using tools from mathematical logic, especially model theory. These investigations are expected to shed new light on questions about analytic vector fields. Information gained in current research on real numbers is presently being applied to many areas, including robotics and neural networks. ***