## Contact Info

Pure MathematicsUniversity of Waterloo

200 University Avenue West

Waterloo, Ontario, Canada

N2L 3G1

Departmental office: MC 5304

Phone: 519 888 4567 x43484

Fax: 519 725 0160

Email: puremath@uwaterloo.ca

Thursday, May 2, 2019 — 10:00 AM EDT

**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

University of Waterloo

200 University Avenue West

Waterloo, Ontario, Canada

N2L 3G1

Departmental office: MC 5304

Phone: 519 888 4567 x43484

Fax: 519 725 0160

Email: puremath@uwaterloo.ca

University of Waterloo

University of Waterloo

43.471468

-80.544205

200 University Avenue West

Waterloo,
ON,
Canada
N2L 3G1

The University of Waterloo acknowledges that much of our work takes place on the traditional territory of the Neutral, Anishinaabeg and Haudenosaunee peoples. Our main campus is situated on the Haldimand Tract, the land granted to the Six Nations that includes six miles on each side of the Grand River. Our active work toward reconciliation takes place across our campuses through research, learning, teaching, and community building, and is centralized within our Office of Indigenous Relations.