I've taken a graduate course in model theory and I like it so much that I can imagine doing research in this area. Are there survey articles or review papers on the current research topics in model theory? Where can I find them?

Also, I wish there is literature about the common proof techniques and tricks one uses in the current research. (Sometimes I have the feeling that everybody just writes down their proofs and nobody writes down the essential ideas or an overview of techniques used.) Do you know of any such text?

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    $\begingroup$ Have you looked at David Marker's book? $\endgroup$ – Todd Trimble Aug 18 '19 at 5:15
  • $\begingroup$ @ToddTrimble: I have heard of it, but it seems to be a regular graduate-level textbook. Does it have a special emphasize on current research and points to current research papers and questions? $\endgroup$ – user144513 Aug 18 '19 at 10:39
  • $\begingroup$ This was in response to your second paragraph: "Also, I wish there is literature about the common proof techniques and tricks one uses in the current research" -- it seemed to me Marker's book fits that description. $\endgroup$ – Todd Trimble Aug 18 '19 at 12:48
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    $\begingroup$ What you should really do is find a research advisor who works in model theory - e.g. the person who taught your graduate course? $\endgroup$ – Alex Kruckman Aug 18 '19 at 23:10
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    $\begingroup$ I'm not a model theorist, but from the Zentralblatt review: "The author’s intended audience for this high level introduction to model theory is graduate students contemplating research in model theory, graduate students in logic, and mathematicians who are not logicians but who are in areas where model theory has interesting applications." The first edition is from 2002. You might just have a look. In general, you might get better response to your question if you say what books you've already looked at, what topics you studied, etc. Of course model theory is quite a big field... $\endgroup$ – Todd Trimble Aug 19 '19 at 10:44

This answer is partly an answer to your questions, and also partly a response to the conversation unfolding in the comments. So I am just going to list a somewhat disconnected list of resources.

One thing I'll say first though is in response to the query "or are they just the polished versions of the results of the last century" made in a comment. As a "stand alone" field, model theory is not even a century old (barely more than half a century by some accounts). So, yes, textbooks such as Marker's book on model theory are relevant for learning about current research. Other good texts are:

  • A Course in Model Theory, by K. Tent and M. Ziegler

  • A Course in Model Theory, by B. Poizat

  • Stable Groups, by B. Poizat

  • An Introduction to Stability Theory, by A. Pillay

  • Model Theory, by W. Hodges

Another remark is that, in my opinion, it could be difficult to compile a list of common techniques and "tricks" used in model theory research. Perhaps it's because of how new the field is, and so there hasn't been enough time for such things to congeal. It could also be because a major portion of current research in model theory focuses on applications and interactions with other areas of math, and so the "techniques and tricks" are widely varied.

At any rate, here are more papers, books, and notes. These aren't in any particular order, and certainly this list is incomplete. But I think the list covers a lot, and provides starting points into several active areas. The Forking and Dividing website also has many definitions and examples from model theory with citations.

  1. Notes and survey articles. Survey articles on recent research in model theory can be scarce, but you should look through recent volumes of the Bulletin of Symbolic Logic. That being said, here are some survey-type articles that I like.

    a. Stability theory and its variants, by B. Hart

    b. Seminar and talk notes by E. Casanovas (you can find many topics of current research).

    c. Diophantine geometry from model theory, by T. Scanlon

    d. Approximate groups (after Hrushovski, and Breuillard, Green, Tao), by L. van den Dries (more "applied" model theory)

    e. Notes on the model theory of finite and pseudo-finite fields, by Z. Chatzidakis

    f. Unpublished notes of H. Adler

  2. Papers and books on specific research topics.

    a. A Guide to NIP Theories, by P. Simon.

    b. Simple Theories, by B. Kim

    c. Tame Topology and O-minimal Structures, by L. van den Dries

    d. Model Theory of Fields, by D. Marker, M. Messmer, and A. Pillay.

    e. Vapnik-Chervonenkis density in some theories without the independence property, I and II, by M. Aschenbrenner, A. Dolich, D. Haskell, D. Macpherson, and S. Starchenko

    f. Theories without the tree property of the second kind, by A. Chernikov

    g. A geometric introduction to forking and thorn-forking, by H. Adler

    h. Model theory for metric structures, by I. Ben Yaacov, A. Bernstein, C. W. Henson, and A. Usvyatsov

    i. Topological dynamics of definable group actions, by L. Newelski

Lastly, one can read Classification theory and the number of non-isomorphic models by S. Shelah.

  • $\begingroup$ What marks the beginning of model theory as a stand-alone field? Henkin’s 1949 paper on completeness is a good candidate, coming just after Tarski’s first version of the decision method for elementary algebra and geometry, in 1948. So my estimate of model theory’s age as a stand-alone field is roughly 70 years. $\endgroup$ – Matt F. Aug 20 '19 at 3:25
  • $\begingroup$ @MattF. I don’t know the answer to that. Your suggestion makes sense. I’m partial to work on building elementary extensions. This could start with Maltsev, although that’s probably too early to use the phrase “stand alone”. The early 60’s work of Vaught and Morley on saturated extensions definitely counts in my mind. That makes a range of 55-85 years. It might also be quite another question to ask when model theory became widely recognized as a stand alone field by mathematicians in other fields. $\endgroup$ – Gabe Conant Aug 20 '19 at 4:14
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    $\begingroup$ Simple Theories and Hyperimaginaries by Casanovas and Simple Theories by Wagner are alternatives to Kim's book on simple theories. I always enjoy Casanovas's style, so I'm partial to the first one. $\endgroup$ – Alex Kruckman Aug 21 '19 at 3:33
  • $\begingroup$ @GabeConant: Thank you so much! I guess you answered my question as best as possible. :-) $\endgroup$ – user144513 Aug 21 '19 at 10:35

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