Computable embeddings for pairs of linear orders

We study computable embeddings for pairs of structures, i.e. for classes containing precisely two non-isomorphic structures. Surprisingly, even for some pairs of simple linear orders, computable embeddings induce a non-trivial degree structure. Our main result shows that $\{\omega \cdot k,\omega^\star \cdot k\}$ is computably embeddable in $\{\omega \cdot t, \omega^\star \cdot t\}$ iff $k$ divides $t$.Comment: 20 page

Computable Procedures for Fields

This tutorial will introduce listeners to many questions that can be asked about computable processes on fields, and will present the answers that are known, sometimes with proofs. This is not original work. The questions in greatest focus here include decision procedures for the existence of roots of polynomials in specific fields, for the irreducibility of polynomials over those fields, and for transcendence of specific elements over the prime subfield. Several of these questions are related to the construction of algebraic closures, making Rabin\u27s Theorem prominent

Computability Theory

Computability and computable enumerability are two of the fundamental notions of mathematics. Interest in effectiveness is already apparent in the famous Hilbert problems, in particular the second and tenth, and in early 20th century work of Dehn, initiating the study of word problems in group theory. The last decade has seen both completely new subareas develop as well as remarkable growth in two-way interactions between classical computability theory and areas of applications. There is also a great deal of work on algorithmic randomness, reverse mathematics, computable analysis, and in computable structure theory/computable model theory. The goal of this workshop is to bring together researchers representing different aspects of computability theory to discuss recent advances, and to stimulate future work

A topological approach to undefinability in algebraic extensions of Q\mathbb{Q}

For any subset $Z \subseteq \mathbb{Q}$, consider the set $S_Z$ of subfields $L\subseteq \overline{\mathbb{Q}}$ which contain a co-infinite subset $C \subseteq L$ that is universally definable in $L$ such that $C \cap \mathbb{Q}=Z$. Placing a natural topology on the set $\text{Sub}(\overline{\mathbb{Q}})$ of subfields of $\overline{\mathbb{Q}}$, we show that if $Z$ is not thin in $\mathbb{Q}$, then $S_Z$ is meager in $\text{Sub}(\overline{\mathbb{Q}})$. Here, thin and meager both mean "small", in terms of arithmetic geometry and topology, respectively. For example, this implies that only a meager set of fields $L$ have the property that the ring of algebraic integers $\mathcal{O}_L$ is universally definable in $L$. The main tools are Hilbert's Irreducibility Theorem and a new normal form theorem for existential definitions. The normal form theorem, which may be of independent interest, says roughly that every $\exists$-definable subset of an algebraic extension of $\mathbb Q$ is a finite union of single points and projections of hypersurfaces defined by absolutely irreducible polynomials.Comment: 24 pages. Introduction has been rewritte

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