2 Antonio Montalbán In

My area of research centers around Computability Theory, a branch of Mathematical Logic. Inside computability theory, I have worked in various different areas. I have been particularly interested in the programs of Computable Mathematics, Reverse Mathematics and Turing Degree Theory. The former one studies the computability aspects of mathematical theorems and structures. The second one analyzes the complexity of mathematical theorems in terms of the complexity of the constructions needed for their proofs. The latter, which is considered Pure Computability Theory, studies the partial ordering induced by the relation “computable from”. I have also written a papers in other areas like Effective Randomness, automata theory, the structure of linear orderings, the lattice of Π1-classes, and Borel structures. However, most of my work is on the three programs described before. In Computable and Reverse Mathematics, my research has concentrated on linear orderings, wellquasi-orderings and Boolean algebras. But I have also worked with other type of structures like torsionfree abelian groups, vector spaces, and on computable model theory where we consider general types of structures. I now have a student working on Artinian Rings. Results in Computable and Reverse Mathematics usually require a deeper understanding of the objects from classical mathematics. For instance I have obtained interesting results purely on the structure of the embeddability relation on linear orderings as explained below. I have recently written a survey paper [Mon07] about my results on linear orderings. Various approaches have been taken to understanding the shape of the Turing Degree Structure. One is to study the algebraic properties of the structure. Once people realized the structure is a quite complicated one, methods from logic started being used to study the complexity of the structure. Another approach has been studying how algebraic properties of certain Turing degrees in this structure relate to properties about the computational power of the degree. There is a lot of interaction between these approaches and I have been interested in this program in general. I have written a survey paper [Mona] on the history of the study of the Turing Degree Structure via embeddability results where I mention my contributions to the area until 2006. Hyperarithmetic theory appears as a tool all over my work. In the next subsection I quickly introduce the basic concepts of Computability Theory. In the next three sections I describe my work in each of the areas of Turing Degree Theory, Computable Mathematics and Reverse Mathematics. Each of these sections starts by describing the general ideas of the subject and becomes more technical at the end. Some of my plans for future work are described along the way.