Institute of Information Science, Academia Sinica

The general area of effective algebra is devoted to studying the algorithmic content of mathematical structures. The standard model for such investigations is a (Turing) computable presentation of an algebraic structure. A computable structure S is computably categorical if for any computable presentation A of S, there is a Turing computable isomorphism from A onto S. The notion of categoricity spectrum, which is built upon computable categoricity, captures the complexity of isomorphisms between different computable presentations of a given structure. For a class of structures K, effective classification gives a measure of complexity for computably presentable members of K. In the talk, we give a review of recent results on categoricity spectra and related effective classification problems.