http://www.dagstuhl.de/17081

February 19 – 24 , 2017, Dagstuhl Seminar 17081

Computability Theory

Organizers

Klaus Ambos-Spies (Universität Heidelberg, DE)
Vasco Brattka (Universität der Bundeswehr – München, DE)
Rodney Downey (Victoria University – Wellington, NZ)
Steffen Lempp (University of Wisconsin – Madison, US)

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Dagstuhl Service Team

Documents

Dagstuhl Report, Volume 7, Issue 2 Dagstuhl Report
Aims & Scope
List of Participants
Shared Documents
Dagstuhl Seminar Schedule [pdf]

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.

License
  Creative Commons BY 3.0 Unported license
  Klaus Ambos-Spies, Vasco Brattka, Rodney Downey, and Steffen Lempp

Classification

  • Data Structures / Algorithms / Complexity

Keywords

  • Computability theory
  • Generic case complexity
  • Computable analysis
  • Computable algebra
  • Proof mining
  • Algorithmic randomness

Book exhibition

Books from the participants of the current Seminar 

Book exhibition in the library, ground floor, during the seminar week.

Documentation

In the series Dagstuhl Reports each Dagstuhl Seminar and Dagstuhl Perspectives Workshop is documented. The seminar organizers, in cooperation with the collector, prepare a report that includes contributions from the participants' talks together with a summary of the seminar.

 

Download overview leaflet (PDF).

Publications

Furthermore, a comprehensive peer-reviewed collection of research papers can be published in the series Dagstuhl Follow-Ups.

Dagstuhl's Impact

Please inform us when a publication was published as a result from your seminar. These publications are listed in the category Dagstuhl's Impact and are presented on a special shelf on the ground floor of the library.

NSF young researcher support