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The references listed at http://en.wikipedia.org/wiki/Computable_analysis have all been published 30-15 years ago. Are the approaches which these references expose still up-to-date and relevant to the current paths of research in computable analysis? (In case they are not, where can I find a good introduction to the current trends?)

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    $\begingroup$ I suggest that you copy the titles of the references in the Wikipedia article into, e.g., Google Scholar and look for recent works that cite these works. Seems like there are plenty... $\endgroup$
    – Dirk
    Commented Jan 25, 2015 at 14:19
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    $\begingroup$ Yes, it is still an active research area. It however is spread out throughout a number of camps (traditions): The Weihrauch camp, the reverse math camp, the computability theory camp, the randomness camp, the proof theory camp, and a few different constructive math camps. (Also, see many of the quantitative results in classical analysis.) Most researchers span two or more camps, and I don't mean to imply there is a feud or anything. However, there isn't necessarily an organized central list of open problems or a central agenda. You may want to check out cca-net.de as a starting point. $\endgroup$
    – Jason Rute
    Commented Jan 25, 2015 at 15:08
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    $\begingroup$ @JasonRute This is a pretty good answer. Maybe there are a few more details to add, but it's a really good summary of the current state of affairs. Why not post it as an answer? $\endgroup$
    – François G. Dorais
    Commented Jan 25, 2015 at 21:45

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(At François's request, my comment in now an answer.)

Yes, it is still an active research area. It however is spread out throughout a number of camps (traditions): The Weihrauch camp, the reverse math camp, the computability theory camp, the randomness camp, the proof theory camp, and a few different constructive math camps. (Also, see many of the quantitative results in classical analysis.) Most researchers span two or more camps, and I don't mean to imply there is a feud or anything. However, there isn't necessarily an organized central list of open problems or a central agenda. You may want to check out cca-net.de as a starting point.

Edit: You may also want to check out this survey by Brattka and Avigad. It does a good job of explaining many of the different traditions.

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The references you mention are all monographs (Abert, Pour-El and Richards, Simpson, Weihrauch). Here are some more recent (at most 5 years old) monographs which border on computable analysis:

  • Kohlenbach, Applied Proof Theory: Proof Interpretations and their Use in Mathematics. (A lot of the applications are in analysis.) 2008.
  • Nies, Computability and randomness. 2009.
  • Downey and Hirschfeldt: Algorithmic randomness and complexity. 2010.
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