Conference: MAMLS (Mid-Atlantic Mathematical Logic Seminar) Funded by the National Science Foundation Hofstra University, Hempstead, Long Island, NY March 6, 2004 http://mamls.org/Hofstra2004 Invited Speaker: Stephen G. Simpson Pennsylvania State University t20@psu.edu http://www.personal.psu.edu/t20/ Title: Two topics in the theory of computability and unsolvability Abstract: (1) Almost everywhere domination. This is joint work with Natasha Dobrinen. A Turing degree is said to be almost everywhere dominating if it contains functions which dominate all functions that are computable from a random Turing oracle. We consider the problem of characterizing the Turing degrees which are almost everywhere dominating. We relate this to some questions in the reverse mathematics of measure theory. (2) An extension of the recursively enumerable Turing degrees. The semilattice of r. e. Turing degrees has been studied intensively for almost 50 years, yet no natural examples of r. e. Turing degrees are known beyond the two originally noted by Turing: 0 = the degree of solvable problems, and 0' = the degree of the Halting Problem. In order to overcome this obstacle, we present a natural embedding of the r. e. Turing degrees into the Muchnik degrees of mass problems given by nonempty Pi^0_1 subsets of 2^omega. We exhibit some natural examples of such degrees, and we present a result due to Stephen Binns showing that the lattice of all such degrees is well behaved.