Definability, automorphisms, and dynamic properties of computably enumerable sets

We announce and explain recent results on the computably enumerable (c.e.) sets, especially their definability properties (as sets in the spirit of Cantor), their automorphisms (in the spirit of FelixKlein’sErlanger Programm), their dynamic properties, expressed in terms of how quickly elements enter them relative to elements entering other sets, and theMartin Invariance Conjecture on their Turing degrees, i.e., their information content with respect to relative computability (Turing reducibility). §