Title: Enumerating subgroups, and the computational complexity of recognising torsion-freeness, in finitely presented groups.
Abstract: We give a construction for a finitely presented group G for which the set of finitely presented groups that embed into G is not recursively enumerable. We do this by showing that from a description of a recursively enumerable set A, we can uniformly construct a finitely presented group G(A) such that the set of orders of torsion elements in G(A) is one-one equivalent to A. As a corollary we show an earlier result by Lempp that the set of torsion-free finitely presented groups is Pi-0-2 complete in the Arithmetic Hierarchy.
Title: A w-REA Set Forming A Minimal Pair With 0'
Abstract: It is easy to see that no n-REA set can form a (non-trivial) minimal pair with 0' and only slightly more difficult to observe that no w-REA set can form a (non-trivial) minimal pair with 0''. Shore has asked whether this can be improved to show that no w-REA set forms a (non-trivial) minimal pair with 0'. We show that no such improvement is possible by constructing a w-REA set C forming a minimal pair with 0'.
Title: Reverse mathematics and equivalents of the axiom of choice.
Abstract: I will discuss recent joint work with Carl Mummert studying the reverse mathematics of various maximality principles classically equivalent to the axiom of choice. We show that these principles have a wide range of strengths in the context of second-order arithmetic, from being equivalent to Z_2, to lying below ACA_0 and being incomparable with WKL_0. Principles of the latter kind form a rich and complicated structure. I will discuss some recent development in its study, and how choice principles fit into it. For example, our results show that modulo \Sigma^0_2 induction, the principle FIP, which asserts that every family of nonempty sets has a maximal subfamily with the property that any finite intersection of its members is nonempty, lies strictly below the principle AMT studied by Hirschfeldt, Shore, and Slaman , and implies the principle OPT. This gives a surprising connection between the reverse mathematical content of model theoretic principles on the one hand, and of set-theoretic principles on the other.
Title: The search for natural definability in the Turing degrees
Abstract: While the definability of all jump classes other than low has been established through coding techniques, there remains a conspicuous lack of
any natural definability results in the Turing degrees -- I shall detail the state of affairs in a program which looks to address this issue by systematically analyzing the order theoretic properties satisfied by the degrees in the various jump classes.