ABSTRACT

	AMS BOULDER MEETING ON COMPUTABILITY THEORY 
	            JUNE 12-18, 1999

SPEAKER: Robert I. Soare, University of Chicago
TITLE: Automorphisms and Extension Theorems for C.E. Sets
ABSTRACT:

We study automorphisms of the structure of the computably enumerable
(c.e.) sets under set inclusion. The principal tools for constructing
such automorphisms are the extension theorems. We begin with a sketch
of the very recent New Extension Theorem (N.E.T.), which is much
simpler to state and much more powerful than the preceding ones,
including the original Soare Extension Theorem of 1974 and the Cholak
Translation Theorem.

From N.E.T., we very easily derive some earlier difficult results, such
as the fact that maximal sets, hemi-maximal sets, and others form
orbits. We show show how N.E.T. is used in recent results, including
those of type 1, type 2, and type 3.  N.E.T. suggests a number of open
questions and approaches, which we explore in detail.

We also discuss non-automorphism results given by definability
properties which demonstrate that certain automorphisms cannot exist.
These have been much neglected during the past decade in comparison
with automorphism results.