Maurice Chiodo
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.
Peter Gerdes
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'.
Damir Dzhafarov
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 [2009], 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.
Andy Lewis
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.