A COMPUTABLE FUNCTOR FROM GRAPHS TO FIELDS

Fried and Kollár constructed a fully faithful functor from the category of graphs to the category of fields. We give a new construction of such a functor and use it to resolve a longstanding open problem in computable model theory, by showing that for every nontrivial countable structure S, there exists a countable field F of arbitrary characteristic with the same essential computable-model-theoretic properties as. Along the way, we develop a new computable category theory, and prove that our functor and its partially defined inverse (restricted to the categories of countable graphs and countable fields) are computable functors.National Science Foundation (U.S.) (Grant DMS-1069236

Computability Theory (hybrid meeting)

Over the last decade computability theory has seen many new and fascinating developments that have linked the subject much closer to other mathematical disciplines inside and outside of logic. This includes, for instance, work on enumeration degrees that has revealed deep and surprising relations to general topology, the work on algorithmic randomness that is closely tied to symbolic dynamics and geometric measure theory. Inside logic there are connections to model theory, set theory, effective descriptive set theory, computable analysis and reverse mathematics. In some of these cases the bridges to seemingly distant mathematical fields have yielded completely new proofs or even solutions of open problems in the respective fields. Thus, over the last decade, computability theory has formed vibrant and beneficial interactions with other mathematical fields. The goal of this workshop was to bring together researchers representing different aspects of computability theory to discuss recent advances, and to stimulate future work

Computable structure theory on Banach spaces

In this dissertation we investigate computability notions on several different Banach spaces, namely the separable $L^p$-spaces and $C[0,1]$. It was demonstrated by McNicholl \cite{TM} that the halting problem is a necessary and sufficient condition for the existence of computable isometric isomorphisms between any two computable representations of the purely atomic $L^p$-spaces (e.g. $\ell^p$) where the underlying measure space is generated by finitely many atoms. In the case where the underlying measure space is generated by finitely many atoms (such as in $\ell^p_n$), McNicholl also proved that it is always possible to find an algorithm that computes isometric isomorphisms between any two computable representations. Clanin, McNicholl, and Stull \cite{CMS} proved a similar result. Namely they proved that for any two computable representations of a non-atomic $L^p$-space (e.g. $L^p[0,1]$) there is always a computable isometric isomorphism between them. We both continue and complete the classification of the separable $L^p$-spaces up to degree of categoricity by investigating the hybrid $L^p$-spaces, whose underlying measure spaces consist of both atomic and non-atomic parts, and determine how much computational power is necessary and sufficient to compute isometric isomorphisms between any two copies of these spaces. Secondly, we continue a line of inquiry initialized by Melnikov and Ng in 2014, who proved that for $C[0,1]$ (i.e. the Banach space of all continuous functions on the closed unit interval) there is a pair of computable representations between which there is no computable isometric isomorphism. They achieved this by constructing one of the representations in such a manner that the constant unit function \textbf{1} is not computable, contrasting with the other representation in which \textbf{1} is computable. We show in Chapter 5 that given any computable representation of $C[0,1]$ as a Banach space the halting set always computes \textbf{1}. We also determine how much extra computational power beyond that of the halting set is sufficient to compute the modulus operator $|\cdot|$ within any computable representation. Lastly, we use these two results to determine how much power is sufficient to compute an isometric isomorphism between any two computable representations of a restricted class of representations of $C[0,1]$

Sharp Vaught's Conjecture for Some Classes of Partial Orders

Matatyahu Rubin has shown that a sharp version of Vaught's conjecture, $I({\mathcal T},\omega )\in \{ 0,1,{\mathfrak{c}}\}$, holds for each complete theory of linear order ${\mathcal T}$. We show that the same is true for each complete theory of partial order having a model in the the minimal class of partial orders containing the class of linear orders and which is closed under finite products and finite disjoint unions. The same holds for the extension of the class of rooted trees admitting a finite monomorphic decomposition, obtained in the same way. The sharp version of Vaught's conjecture also holds for the theories of trees which are infinite disjoint unions of linear orders.Comment: 27 page