The aim of the study group is to learn the basics of o-minimality up to Wilkie's theorem that $\mathbb{R}_{\mathrm{an,exp}}$ is an o-minimal structure. The guiding thread of the choice of references is to adopt a geometric (rather than model-theoretic) point of view. Thus, no background in logic is expected.
We will first examine the axioms of structures and o-minimal structures and infer basic facts about definable sets and definable mappings, based on [3]. We shall show that semi-linear sets form an o-minimal structure (the corresponding sets and mappings are those studied in PL-topology). Next, we will review facts about polynomials and infer the Tarski-Seidenberg theorem that semi-algebraic sets form an o-minimal structure, also based on [3]. We may cover related topics, such as Newton-Puiseux series.
Next, we will develop the general theory as it follows from the axioms, based on several references, including [3] and [5]. The topics are: first order formulae; interior, closure, and boundary; oscillation of continuous definable functions; limits along fibers and partial derivatives; definable continuous extensions; definable functions of one variable; curve selection; Kurdyka-Łojasiewicz inequalities.
Based on [3], we may then study the cell decomposition theorem. Some of its consequences to be reviewed are: definable inverse and implicit function theorem; fiber cutting; existence of basic definable stratification of definable sets.
We then ought to have brief look at the notion of Hardy field and draw some consequences, such as the link with the modulus of continuity of continuous definable functions, polynomially bounded o-minimal structures, and the exact estimates in Łojasiewicz inequalities for semi-algebraic sets.
We may or may not make an excursion in the world of consequences of Gödel coding, namely that if a structure defines $\mathbb{N}$ (the natural numbers) then it defines all Borel-measurable sets. This could lead to a discussion about the lack of stability of o-minimal structures under anti-differentiation.
Then we must study basic facts about real-analytic functions and it is probably most efficient to do it via holomorphic functions as in [4]. The main topic along these lines is the Weierstrass preparation theorem and some of its consequences that are useful for what comes next.
Finally, our goal is to prove the Gabrielov theorem of the complement (that globally subanalytic sets form an o-minimal structure; polynomially bounded, hence, proving the classical Łojasiewicz inequalities) and the theorem of Wilkie (that adding the graph of $\exp$ to the previous o-minimal structure generates an o-minimal structure). The proof we will study is through proving a preparation theorem for functions in certain classes, following Lion-Rolin [6]. This is probably the only proof of Wilkie's theorem that avoids hard model-theoretic arguments.
References
[1] E. Bierstone and P. D. Milman. "Semianalytic and subanalytic sets". In: Inst. Hautes Études Sci. Publ. Math. 67 (1988), pp. 5-42.
[2] D. Brink. "Hölder continuity of roots of complex and $p$-adic polynomials". In: Comm. Algebra 38.5 (2010), pp. 1658-1662.
[3] L. van den Dries. Tame topology and o-minimal structures. Vol. 248. London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 1998, pp. x+180.
[4] R. C. Gunning and H. Rossi. Analytic functions of several complex variables. Prentice-Hall, Inc., Englewood Cliffs, NJ, 1965, pp. xiv+317.
[5] K. Kurdyka. "On gradients of functions definable in o-minimal structures". In: Ann. Inst. Fourier (Grenoble) 48.3 (1998), pp. 769-783.
[6] J.-M. Lion and J.-P. Rolin. "Théorème de préparation pour les fonctions logarithmico-exponentielles". In: Ann. Inst. Fourier (Grenoble) 47.3 (1997), pp. 859-884.
[7] C. Miller. "Exponentiation is hard to avoid". In: Proc. Amer. Math. Soc. 122.1 (1994), pp. 257-259.
[8] C. P. Rourke and B. J. Sanderson. Introduction to piecewise-linear topology. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 69. New York: Springer-Verlag, 1972, pp. viii+123.
