Turing degree spectra of differentially closed fields

Model theory seminarFriday, March 7, 201412:30 pmGC 6417

Russell Miller

Turing degree spectra of differentially closed fields

City University of New York

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 { d : d’ ∈ 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 DCF0 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 Δ2 isomorphism between the low model and its computable copy; moreover, our first theorem shows that the extension of the result to the low4 case for Boolean algebras does not hold for DCF0.

This is joint work by Dave Marker and the speaker. The slides for this talk are available here.

Russell Miller is professor of mathematics at Queens College of CUNY and also at the CUNY Graduate Center.  He conducts research in mathematical logic, especially computability theory and its interaction with other areas of mathematics, as in computable model theory. He received his doctorate from the University of Chicago in 2000, as a student of Robert Soare, and subsequently held a postdoctoral position at Cornell University until 2003, when he came to CUNY.

Posted by on February 16th, 2014