Date: Saturday, October 18, 2014
Time: 10:00 am–4:20 pm
Place: Assumption College (Visitor information)
|10:30–11:30||Marcia Groszek (Dartmouth University)||The Implicitly Constructible Universe|
|11:40–12:10||Caleb Martin (University of Connecticut)||The Degree Spectra of Orders on a Computable Abelian Group|
|2:00–2:30||Leah Marshall (George Washington University)||Computability Theoretic Properties of Isomorphisms between Partial Computable Injection Structures|
|2:40–3:10||Jacob Suggs (University of Connecticut)||Low for omega and Equivalence Class Isomorphism Properties|
|3:20–4:30||Reed Solomon (University of Connecticut)||Ramsey’s Theorem applied to infinite traceable graphs|
The implicitly constructible universe, IMP, defined by Hamkins and Leahy, is produced by iterating implicit definability through the ordinals. IMP is an inner model intermediate between L and HOD. We look at some consistency questions about the nature of IMP.
For any computable abelian group, the collection of possible linear orderings on the group can be represented as a Pi^0_1 class. However, not all Pi^0_1 classes can occur in this way, and we investigate the possible Turing degrees of their members. First, we describe the connection between orders for a group and bases for the group. Additionally, we extend a previously known result regarding the Turing degrees not represented in the space of orders of some group.
A partial computable injection structure consists of a computable set of natural numbers and a partial computable, injective (1-1) function. Clearly, the isomorphism type of such a structure is determined by the kinds and number of orbits of the elements under the function application. First, we investigate partial computable injection structures always having computable isomorphisms to other isomorphic structures, and we analyze what goes wrong in structures without such computable isomorphisms. Additionally, we do the same for partial computable injection structures under Delta_2-isomorphisms and Delta_3-isomorphisms.
We look at the property of being low for isomorphism, restricted to certain classes of structures – if C is a class of structures, a set D is low for C isomorphism iff for any two structures in C, if D computes an isomorphism between them then there is a computable isomorphism between them. In particular we will show that exactly those sets which cannot compute non-zero delta 2 degrees are low for omega isomorphism (when omega is viewed purely as an order), and we will show that no set which computes a non-zero delta 2 set or which computes a separating set for any two computably inseparable c.e. degrees has is low for equivalence class isomorphism.
We consider three applications of Ramsey’s Theorem to infinite traceable graphs and finitely generated lattices from the point of view of reverse mathematics. For two of the applications, we will show that Ramsey’s Theorem is necessary while for the third application, it is not necessary. We will conclude with some related open questions.