Computable Structure Theory

University Of Notre Dame, Notre Dame IN

Investigators

Abstract

Knight and her students will work on various aspects of computable structure theory. Knight plans to continue work with D'Aquino on weak fragments of arithmetic. She expects also to continue work with Goncharov and Harizanov, with Young, and with Shore, on a collection of inter-related problems involving complicated relations on computable structures, structures of high rank, and paths through Kleene's O. She is currently working with Csima, Hirschfeldt, and Soare on prime models and lowness properties. She plans to work with Lempp, McCoy, and Solomon on Boolean algebras. The funds are mainly for student travel and books. The remainder is for partial support of the Notre Dame Logic Seminar. Currently, Knight has two students. Andrew Arana, who may finish this summer, has a number of results on complexity in arithmetic, using some new independent sentences (variants of the Godel-Rosser sentence). Arana is now concentrating on problems of a more foundational nature. Wesley Calvert, a second year student, is working on complexity of the isomorphism problem for various familiar classes of computable structures. He already has results for several classes of fields. Knight may acquire new students during the period of the grant. The goal of Knight's research, and that of students working with her, is to determine which aspects of mathematical structures are computable, and for those which are not computable, whether there is some nice, computable approximation. The grant will facilitate new work of this kind. There is a larger goal---enabling students to become first-rate research scientists. The grant will be used mainly for student travel and books. Without money for books, Knight's students would hardly own any. She has found that, given money to buy the most important references, her students read much more than they would otherwise. They see how the problems they are working on arose, and what their work means for the field as a whole. Knight's past students, and other logic students at Notre Dame, have benefited tremendously from opportunities to travel to meetings. They meet researchers from other universities and hear about new developments. As soon as they have results of their own, they give talks, and they get valuable suggestions. In short, they have the opportunity, as students, to join the community of research mathematicians.