Overview

Date: Sunday, March 30, 2014
Time: 10:15 am–4:15 pm
Place: Academic Center, Room 113, Olin College (Directions, Campus map)

Schedule

10:15–11:00	Coffee
10:30–11:00	Informal discussion with Olin undergraduates
11:00–12:00	Damir Dzhafarov (University of Connecticut)	Notions of robust information-coding
12:00–2:00	Lunch and discussion
2:00–3:00	Jesse Johnson (Westfield State University)	Quasiminimality and computability
3:00–3:15	Break
3:15–4:15	Russell Miller (Queens College CUNY)	Spectra of Differentially Closed Fields

Abstracts

Damir Dzhafarov

We study several notions of computability-theoretic reducibility between subsets of natural numbers that are “robust” in the following sense: given a notion of largeness, a set B is reducible to a set A if every set that agrees with A on a large domain uniformly computes a set that agrees with B on a large domain. A familiar example is 1-reducibility, for which “large” can be taken to mean “cofinite”. This framework encompasses a wide range of reducibilities, including generic reducibility, for which “large” means “of density 1”, and which was first studied by Jockusch and Schupp. We obtain some new results concerning this reducibility, and introduce several related reducibilities that behave quite counterintuitively. For instance, infinite-information reducibility, for which “large” simply means “infinite”, enjoys the unusual property of admitting maximal pairs. I will discuss this and other results about these reducibilities, and sketch some of the techniques needed to prove them. This is joint work with Greg Igusa.

Jesse Johnson

We briefly define computability on uncountable domains via $\Sigma_1$-recursion, with an emphasis on $\omega_1$-computability. We will then apply this notion of uncountable computability to quasiminimal excellent classes. We will concentrate on the pseudo-exponential fields and the topological group covers of $C$. We show that the group covers are relatively $\omega_1$-computably categorical while the pseudo-exponential fields are not even $\omega_1$-computably categorical.

Russell Miller

The spectrum Spec$(M)$ of a countable structure $M$ is the set of all Turing degrees of structures isomorphic to $M$. This topic has been the focus of much research. Here we describe the spectra of countable differentially closed fields of characteristic $0$: they are precisely the preimages $\{ \boldsymbol{d}~:~\boldsymbol{d’} \in \text{Spec}(G) \}$ of spectra of arbitrary countable graphs $G$, under the jump operation. To establish this, we describe the proofs of two theorems: one showing how to build the appropriate differential field $K$ from a given graph $G$, and the other showing that every low model of the theory \textbf{DCF}$_0$ is isomorphic to a computable one. The latter theorem (which relativizes, to give the main result above) resembles the famous result of Downey and Jockusch on Boolean algebras, but the proof is different, yielding a $\Delta_2$ isomorphism between the low model and its computable copy; moreover, our first theorem shows that the extension of the result to the low$_4$ case for Boolean algebras does not hold for $\textbf{DCF}_0$.