Computability Theory

Summary

Computability theory grew from work to understand effectiveness in mathematics. Sophisticated tools have been developed towards this task. For a while, the area tended to be concerned with internal considerations such as the structure of the various hierarchies developed for the tasks of calibrations. More recently, this machinery has seen modern applications into areas such as model theory, algorithmic randomness, analysis, ergodic theory, number theory and the like; and the tools have been used to answer several classical questions. Seminar 17081 was an opportunity for researchers in several areas of modern computability theory to get together and interact.

The format was for 2-3 lectures in the morning with at least one being an overview, and a similar number of 3-4 in the afternoon, with Wednesday afternoon and Friday afternoon free. The weather being miserable, participants opted to stay at the Schloss Wednesday afternoon, and quite a bit of work was done in pairs in the time left free, on the Wednesday afternoon in particular. At least one problem seen as significant in the area was solved (one concerning the strength of Ramsey's Theorem for Pairs in reverse mathematics), and the organizers know of several other papers in preparation resulting from the seminar.

The lectures were from various areas, but the greatest concentration was around

• classification tools in computable analysis (the Weihrauch Lattice) and Reverse Mathematics (on what proof-theoretic strength is needed to establish results in mathematics), and these areas' relationship with generating algorithms, such as in proof mining;
• computable model theory (looking at structures such as groups, rings, or abstract algebraic structures, endowing them with effectivity and seeing what else is algorithmic). Notable was the new work on effective uncountable structures such as uncountable free groups, and on topological groups;
• algorithmic randomness: Here one seeks to give meaning to randomness for individual strings and infinite sequences. Talks given explored the relationship of calibrations of randomness to computational power, and online notions of randomness.

Of course, these are not separate areas but are inter-related, and the talks reflected these inter-relationships.

Currently, computability theory is quite vibrant with many new applications being found, and a number of young and talented researchers entering the field. This was reflected in the age of the presenters of many of the lectures, as well as the significant number of people we could have invited in addition.

All in all, the meeting was a great success and should have significant impact on the development of the area.