ANTONIO MONTALBÁN Theorem

I am interested in studying the complexity of mathematical practice. In mathematics, as we all know, some structures are more complicated than others, some constructions more complicated than others, and some proofs more complicated than others. I am interested in understanding how to measure this complexity and in measuring it. The motivations for this come from different areas. Form a foundational viewpoint, we want to know what assumptions we really need to do mathematics (ZFC is way much more than we usually use), and we are also interested in knowing what assumptions are used in the different areas of mathematics. Form a computational viewpoint, it is important to know what part of mathematics can be done by mechanical algorithms, and, even for the part that can’t be done mechanically, we want to know how constructive are the objects we deal with. Furthermore, it is sometimes the case that this computational analysis allows us to find connections between constructions in different areas of mathematics, and in many cases to obtain a deeper understanding of mathematical objects being analyzed. My work is quite diverse in terms of the techniques I have used, the approaches I have taken, and the areas of mathematics that I have analyzed. However, my background area is Computability Theory, and most of my work can be considered as part of this 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 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 studies the partial ordering induced by the relation “computable from” in an abstract way. I have also written papers in other areas like effective randomness, automata theory, the lattice of Π1-classes, etc.. If I had to choose favorites, tools that are recurrent over my work are the iterations of the Turing jump and hyperarithmetic theory. Structures that are recurrent over my work are linear orderings, well-quasi-orderings and Boolean algebras. In the next three sections I describe my work in each of the areas of Computable Mathematics, Reverse Mathematics, and Turing Degree Theory. The fourth section is dedicated to the part of my work that does not fit in this classification. Each of these sections starts by describing the general ideas of the subject and becomes more technical inside each subsection; for the most part, the reader can skip sections and subsections without loosing in understanding. All my papers are available on my web page at www.math.uchicago.edu/∼antonio.