Definability and Automorphisms in Computability Theory

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AbstractAward: DMS-0245167Principal Investigator: Peter A. CholakThe principal investigator plans to study the relationshipbetween definability and automorphisms in various structuresarising in computability theory. Primarily, but not totally, hewill focus on the collection of all computably enumerable sets.The principal investigator will also consider the collection ofall Pi01 classes and the computably enumerable degrees. A longrange goal is to provide a complete understanding of thesestructures and their automorphisms and definable orbits. Somerelated projects in computable structure theory and models ofsecond order arithmetic are also planned.The main focus of these projects is on definability andcomputability. These notions are both important in measuring thecomplexity of an answer to a mathematical problem. Problems suchas "Is there a computer program which can solve all questions ofthis type?" One develops an intertwined hierarchy ofdefinability and computability. Only answers which lay on thelowest level are computable and even then they are not alwaysfeasibly computable given today's computers. Answers of highercomplexity provide useful mathematical information, allow one totest the limits of mathematical techniques, reveal whether thewrong techniques are being used, and, in some very rare cases,can be useful for encoding/decoding information.