Model theory of analytic functions: some historical comments

Model theorists have been studying analytic functions since the late 1970s. Highlights include the seminal work of Denef and van den Dries on the theory of the p-adics with restricted analytic functions, Wilkie’s proof of o-minimality of the theory of the reals with the exponential function, and the formulation of Zilber’s conjecture for the complex exponential. My goal in this talk is to survey these main developments and to reflect on today’s open problems, in particular for theories of valued fields. When I was invited to give this talk for the ASL annual meeting 2011, I decided it would be a good opportunity to review the history and the development of ideas that has led to today’s rich area of research into the study of analytic functions from a model theoretic point of view (and viceversa, as interesting questions in model theory arise from the geometric understanding of analytic functions). As I started reading, and wondering what I had let myself in for, I very quickly had to make decisions about what I would not be talking about. One very major area I am not talking about is model theory and analysis, in the sense of the general study of topological spaces equipped with a metric. There is much that is going on in this area, from descriptive set theory to continuous logic, and it would make a very interesting talk to hear about the historical development of these ideas. However, it is not this talk. What I do want to talk about is the model-theoretic study of analytic functions which begins with Tarski. In the early 1930s, Tarski proved the decidability of the real numbers as a field. This work was finally published in 1948 as a Rand volume, and reprinted with annotations in 1951 by the University of California press (this is the version which appears in the collected works) [36]. In a discussion of related decision problems, Tarski says . . . the decision problem is open . . . for the system obtained by introducing the operation of exponentiation. and comments on its potential interest. What we will see, in the course of this talk, is the continuing role that the exponential function has played. Of Received March 20, 2012. c © 2012, Association for Symbolic Logic 1079-8986/12/1803-0002/$2.40