Mohammad Mahmoud, Department of Pure Mathematics, University of Waterloo
"Degrees of Categoricity, the Isomorphism Problem, and the Turing Ordinal"
We are going to talk about some notions of complexity in computable structure theory. We will talk about degrees of categoricity, the isomorphism problem and the Turing Ordinal. For degrees of categoricity, first we will focus on computable tree structures, then we will talk about degrees that are c.e. in and above $\mathbf{0}^{(\alpha)}$, for $\alpha$ a limit ordinal. From our work on degrees of categoricity of computable trees we will be able to conclude some results about the isomorphism problem for classes of computable trees. Finally, we will talk about the Turing ordinal. We observed that the definition of the Turing ordinal has two parts each of which alone can define a specific ordinal which we now call the upper and lower Turing ordinals. The Turing ordinal exists if and only if these two ordinals exist and are equal. We will discuss the possibilities of having the two ordinals existent but different.
MC 5479