General info:
Every Wednesday, from 1 April 2026, 16:30-18:30 Beijing time.
Westlake University, Yungu Campus, Building E-14, Room 301
The seminar is streamed on Zoom, if you are interested please contact me for the link.
Abstract:
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-Lojasiewicz 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 Lojasiewicz 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 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 Lojasiewicz 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.
[1] E. Bierstone and P. D. Milman. “Semianalytic and subanalytic sets”. In: Inst. Hautes Etudes Sci. Publ. Math. 67 (1988), pp. 5–42.
[2] D. Brink. “H¨older 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. “Theoreme de preparation 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.
Diary of the seminar room:
1, 1/4/2026, Thierry De Pauw - After a very brief and subjective introduction to the subject, I will talk about Boolean algebras of concrete sets; E-sets; axioms of structures and o-minimal structures; the structure of semi-linear sets; and elementary facts about definability. Emphasis will be on cylindrical decomposition, a tool that will be used repeatedly.
2, 8/4/2026, Thierry De Pauw - After defining what are a definable set and a definable function in a structure, I will list and prove some elementary facts about those, sometimes assuming the structure is o-minimal. Next, I will review rings of polynomials and prove the (Hölder) continuity of roots of a polynomial of one variable with complex coefficients (this continuity property will be instrumental in our proof of the Tarski-Seidenberg theorem).
3, 15/4/2026, Francesco Tropeano - In preparation for the proof of o-minimality of the structure of semi-algebraic sets, this lecture develops tools to control the roots of polynomials. We begin with the theory of discriminants to study the number of distinct complex roots. We then refine this analysis by considering families of polynomials, with particular attention to the behaviour of real roots. Next, we introduce some elements of the theory of symmetric polynomials and show how they can be used to construct semi-algebraic sets encoding the distribution of roots within such families. These results provide a foundation for cylindrical decomposition in the structure of semi-algebraic sets, and play a key role in the proof of the Tarski–Seidenberg theorem.
4, 22/4/2026, Francesco Tropeano - We prove that the semi-algebraic sets form an o-minimal structure. Using the results on the distribution of real roots from the previous lecture, we first study the decomposition of real space associated with a single polynomial, and then extend it to finite families of polynomials. This leads to the construction of a cylindrical cell decomposition for semi-algebraic sets. As a consequence, we establish the Tarski–Seidenberg theorem, thereby concluding the proof.
5, 29/4/2026, Yihan Chen - In this lecture, we study Puiseux series over the real numbers. We first use Hensel’s lemma to prove a decomposition theorem for polynomials whose coefficients are real analytic functions. From this, we deduce that any continuous solution to a polynomial equation with coefficients in the field of Puiseux series must be a Puiseux series (on a sufficiently small interval). Finally, we present an application of Puiseux’s theorem to the smoothness of continuous solutions.
6, 6/5/2026, Thierry De Pauw - We first prove a general result on definability of the infimum of a definable function along definable fibers. Then we study some of its corollaries, namely: definability of the distance function and consequences; definable continuous extensions; and definability of the modulus of continuity of a definable uniformly continuous function. We briefly discuss hints that uniformly continuous semi-algebraic functions are Hölder continuous.
7, 13/5/2026, Thierry De Pauw - We will study limits of definable functions along definable fibers and apply this to showing that definable functions are continuous on a definable subset of their domain and that their partial derivative, defined on a definable subset of their domain as well, is definable. Next, we study definable functions of one variable and prove a theorem due to van den Dries, namely that for any n, any sub-interval of their domain can be partitioned in finitely many sub-intervals such that the restriction of f to the interior of any of these is of class $C^n$ and all derivatives of f order up to n have a sign there.
8, 20/5/2026, Nuno Hultberg - Structures on the real numbers that do not satisfy o-minimality can be very wild. In fact, as soon as the natural numbers are definable all Borel sets are definable. We prove that the minimal class of definable sets are the so-called projective sets by relying on ideas going back to Gödel.
9, 27/5/2026, Nuno Hultberg - Continuation of the previous lecture.
10, 3/6/2026, Paolo De Donato - The idea behind the stratification of a topological object is to partition the object into components, called stratas, which are themselves smooth manifolds with a lower dimension such that the boundary to each strata is entirely composed by other stratas with a strictly lower dimension. Usually, the stratified object is the singular set of a more regular object, for example an algebraic variety or a minimal surface, in order to retrieve more information from each strata of the singular set. In this talk I will present the definition of stratification, the Verdier's and Whitney's conditions (a) and (b), and some elementary results for stratifications.