Mathematical Sciences: Logic and Analysis

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The project will concern mathematical logic and its connections with other areas of mathematics, especially algebra and analysis. C. Ward Henson will study the model theory of the exponental function and will apply nonstandard analysis and techniques from model theory to the study of Banach spaces and other topics from analysis. In addition, he will attempt to determine the computational complexity of various specific first-order theories and to characterize those theories which are computable in polynomial space. He will also investigate highly homogeneous structures and their groups of automorphisms. Carl Jockusch will work in classical recursion theory and in particular will study the degrees of unsolvability of diagonally nonrecursive functions, relative recursive enumerability, and recursively enumerable degrees relative to which certain permitting arguments may be performed. Lou Van den Dries will study algebraic decision problems such as the word problem for commutative rings and the problem of finding generators for the group of units of a finitely generated commutative ring. He also will continue his work on definability in algebraic-analytic structures. Some of the investigations will refine the techniques of mathematical logic, and some will apply them to questions often viewed as theoretical computer science or as algebra. Jockusch's work will be of the former kind, while a portion of Henson's and of van den Dries' work will be of the latter. For example, Henson will continue to assess the relative complexity of different mathematical theories and will seek ways to recognize easily ones with lower measures of complexity.