Series: Penn State Logic Seminar Date: Tuesday, November 13, 2001 Time: 2:30 - 3:45 PM Place: 306 Boucke Building Speaker: William Calhoun, Bloomsburg University, Mathematics Title: The Computably Enumerable Turing Degrees: An Algebraic Approach Abstract: A set of natural numbers is computably enumerable (c.e.) if it can be generated by a Turing machine that is allow to run forever. We say A is Turing reducible to B if B can be computed by a Turing Machine with an oracle for A. The c.e. sets are partially ordered by Turing reducibility, and the equivalence classes are called c.e. Turing degrees. It has been almost 50 years since the independent proofs by Friedberg [1957] and Muchnik [1956] that there are incomparable c.e. sets. In that time, the structure of the c.e. degrees has been studied intensively. A few nice global properties have been discovered, but many results indicate the complexity of the structure. Over the last decade a new direction for research has been to study the ideals and homomorphic images of the structure. Perhaps this algebraic approach will provide a clearer picture of the global structure. This talk will be a nontechnical survey of "algebraic" computability theory, including recent work by Englert, Lerman and Wald and by Lerman and myself.