Computable Structure Theory

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Knight and her students will work on various aspects ofcomputable structure theory. Knight plans to continue work withD'Aquino on weak fragments of arithmetic. She expects also tocontinue work with Goncharov and Harizanov, with Young, and withShore, on a collection of inter-related problems involvingcomplicated relations on computable structures, structures ofhigh rank, and paths through Kleene's O. She is currentlyworking with Csima, Hirschfeldt, and Soare on prime models andlowness properties. She plans to work with Lempp, McCoy, andSolomon on Boolean algebras. The funds are mainly for studenttravel and books. The remainder is for partial support of theNotre Dame Logic Seminar. Currently, Knight has two students.Andrew Arana, who may finish this summer, has a number of resultson complexity in arithmetic, using some new independent sentences(variants of the Godel-Rosser sentence). Arana is nowconcentrating on problems of a more foundational nature. WesleyCalvert, a second year student, is working on complexity of theisomorphism problem for various familiar classes of computablestructures. He already has results for several classes offields. Knight may acquire new students during the period of thegrant.The goal of Knight's research, and that of students working withher, is to determine which aspects of mathematical structures arecomputable, and for those which are not computable, whether thereis some nice, computable approximation. The grant willfacilitate new work of this kind. There is a largergoal---enabling students to become first-rate researchscientists. The grant will be used mainly for student travel andbooks. Without money for books, Knight's students would hardlyown any. She has found that, given money to buy the mostimportant references, her students read much more than they wouldotherwise. They see how the problems they are working on arose,and what their work means for the field as a whole. Knight'spast students, and other logic students at Notre Dame, havebenefited tremendously from opportunities to travel to meetings.They meet researchers from other universities and hear about newdevelopments. As soon as they have results of their own, theygive talks, and they get valuable suggestions. In short, theyhave the opportunity, as students, to join the community ofresearch mathematicians.