Contemporary Mathematics 2000; 320 pp; softcover Volume: 257 ISBN10: 0821819224 ISBN13: 9780821819227 List Price: US$96 Member Price: US$76.80 Order Code: CONM/257
 This collection of articles presents a snapshot of the status of computability theory at the end of the millennium and a list of fruitful directions for future research. The papers represent the works of experts in the field who were invited speakers at the AMSIMSSIAM Joint Summer Conference on Computability Theory and Applications held at the University of Colorado (Boulder). The conference focused on open problems in computability theory and on some related areas in which the ideas, methods, and/or results of computability theory play a role. Some presentations are narrowly focused; others cover a wider area. Topics included from "pure" computability theory are the computably enumerable degrees (M. Lerman), the computably enumerable sets (P. Cholak, R. Soare), definability issues in the c.e. and Turing degrees (A. Nies, R. Shore) and other degree structures (M. Arslanov, S. Badaev and S. Goncharov, P. Odifreddi, A. Sorbi). The topics involving relations between computability and other areas of logic and mathematics are reverse mathematics and proof theory (D. Cenzer and C. Jockusch, C. Chong and Y. Yang, H. Friedman and S. Simpson), set theory (R. Dougherty and A. Kechris, M. Groszek, T. Slaman) and computable mathematics and model theory (K. AmbosSpies and A. Kučera, R. Downey and J. Remmel, S. Goncharov and B. Khoussainov, J. Knight, M. Peretyat'kin, A. Shlapentokh). Readership Graduate students and mathematicians working in or interested in computability theory and its applications. Table of Contents  K. AmbosSpies and A. Kučera  Randomness in computability theory
 M. Arslanov  Open questions about the \(n\)c.e. degrees
 S. Badaev and S. Goncharov  The theory of numberings: Open problems
 D. Cenzer and C. G. Jockusch, Jr.  \(\mathrm{\Pi}^0_1\) classes  Structure and applications
 P. A. Cholak  The global structure of computably enumerable sets
 C. T. Chong and Y. Yang  Computability theory in arithmetic: Provability, structure and techniques
 R. Dougherty and A. S. Kechris  How many Turing degrees are there?
 R. Downey and J. B. Remmel  Questions in computable algebra and combinatorics
 H. Friedman and S. G. Simpson  Issues and problems in reverse mathematics
 S. Goncharov and B. Khoussainov  Open problems in the theory of constructive algebraic systems
 M. Groszek  Independence results from ZFC in computability theory: Some open problems
 J. F. Knight  Problems related to arithmetic
 M. Lerman  Embeddings into the computably enumerable degrees
 A. Nies  Definability in the c.e. degrees: Questions and results
 P. Odifreddi  Strong reducibilities, again
 M. Peretyat'kin  Finitely axiomatizable theories and Lindenbaum algebras of semantic classes
 A. Shlapentokh  Towards an analog of Hilbert's tenth problem for a number field
 R. A. Shore  Natural definability in degree structures
 T. A. Slaman  Recursion theory in set theory
 R. I. Soare  Extensions, automorphisms, and definability
 A. Sorbi  Open problems in the enumeration degrees
