Mathematical Logic Quarterly, 53, 2007, pp. 483–492. We examine the concept of almost everywhere domination from the viewpoint of mass problems. Let AED and MLR be the set of reals which are almost everywhere dominating and Martin-Löf random, respectively. Let b1, b2, b3 be the degrees of unsolvability of the mass problems associated with the sets AED, MLR×AED, MLR∩AED respectively. Let Pw be the lattice of degrees of unsolvability of mass problems associated with nonempty Π 0 1 subsets of 2 ω.Let1 and 0 be the top and bottom elements of Pw. We show that inf(b1, 1) andinf(b2, 1) andinf(b3, 1) belongtoPw and that 0 < inf(b1, 1) < inf(b2, 1) < inf(b3, 1) < 1. Under the natural embedding of the recursively enumerable Turing degrees into Pw, weshow that inf(b1, 1) andinf(b3, 1) but not inf(b2, 1) are comparable with some recursively enumerable Turing degrees other than 0 and 0 ′. In order to make this paper more self-contained, we exposit the proofs of some recent

Recall that RT is the upper semilattice of recursively enumerable Turing degrees. We consider two fundamental, classical, unresolved issues concerning RT. The first issue is to find a specific, natural, recursively enumerable Turing degree a ∈ RT which is> 0 and < 0 ′. The second issue is to find a “smallness property ” of an infinite, co-recursively enumerable set A ⊆ ω which ensures that the Turing degree deg T (A) = a ∈ RT is> 0 and < 0 ′. In order to address these issues, we embed RT into a slightly larger degree structure, Pw, which is much better behaved. Namely, Pw is the lattice of weak degrees of mass problems associated with nonempty Π 0 1 subsets of 2 ω. We define a specific, natural embedding of RT into Pw, and we present some recent and new research results.

Abstract. We show that there exists an almost everywhere (a.e.) dominating computably enumerable (c.e.) degree which is half of a minimal pair. 1.

Abstract. We study the reverse mathematics and computability-theoretic strength of (stable) Ramsey’s Theorem for pairs and the related principles COH and DNR. We show that SRT 2 2 implies DNR over RCA0 but COH does not, and answer a question of Mileti by showing that every computable stable 2-coloring of pairs has an incomplete ∆ 0 2 infinite homogeneous set. We also give some extensions of the latter result, and relate it to potential approaches to showing that SRT 2 2 does not imply RT 2 2. 1.

Research supported by NSF grants DMS-0600823 and DMS-0652637.

My research interests are driven by much the same considerations that inspired me to work in computability theory in the first place: the challenge of building a solution to an interesting problem or pinpointing the reason it can’t be done. Thus, just like computability theory itself, my interest is not in any particular class of objects, but in a certain kind of construction oriented approach to mathematical problems. Aesthetic sensibilities, mathematical or otherwise, are nearly impossible to capture in writing so the best I can do is observe that I’m often attracted to the interplay between computational properties and some non-computational notion (set containment, function domination, paths through trees, etc..). So when it came time to start my serious graduate research, I was attracted to a problem posed by Groszek and Slaman [7] about the relation between computational properties of perfect trees and those of their paths. In particular the problem asked whether there was a non-computable perfect tree T ⊂ 2 <ω so that every path f ∈ 2 ω through T computable by T was actually computable. Of course the downside to attacking problems with large intricate constructions is they provide few partial results. A lesson that I learned when, after getting repeatedly stuck