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Über dieses Buch

​This volume was produced in conjunction with the Thematic Program in o-Minimal Structures and Real Analytic Geometry, held from January to June of 2009 at the Fields Institute. Five of the six contributions consist of notes from graduate courses associated with the program: Felipe Cano on a new proof of resolution of singularities for planar analytic vector fields; Chris Miller on o-minimality and Hardy fields; Jean-Philippe Rolin on the construction of o-minimal structures from quasianalytic classes; Fernando Sanz on non-oscillatory trajectories of vector fields; and Patrick Speissegger on pfaffian sets. The sixth contribution, by Antongiulio Fornasiero and Tamara Servi, is an adaptation to the nonstandard setting of A.J. Wilkie's construction of o-minimal structures from infinitely differentiable functions. Most of this material is either unavailable elsewhere or spread across many different sources such as research papers, conference proceedings and PhD theses.

This book will be a useful tool for graduate students or researchers from related fields who want to learn about expansions of o-minimal structures by solutions, or images thereof, of definable systems of differential equations. ​



Blowings-Up of Vector Fields

A new proof of the reduction of singularities for planar vector fields is presented. The idea is to adapt Zariski’s local uniformisation method to the vector field setting.
Felipe Cano

Basics of O-minimality and Hardy Fields

This paper consists of lecture notes on some fundamental results about the asymptotic analysis of unary functions definable in o-minimal expansions of the field of real numbers.
Chris Miller

Construction of O-minimal Structures from Quasianalytic Classes

I present the method of constructing o-minimal structures based on local reduction of singularities for quasianalytic classes.
Jean-Philippe Rolin

Course on Non-oscillatory Trajectories

Non-oscillatory trajectories of vector fields are discussed, and some sufficient conditions are established that make the expansion of the real field by such a trajectory o-minimal.
Fernando Sanz Sánchez

Pfaffian Sets and O-minimality

Recent developments in the theory of pfaffian sets are presented from a model-theoretic point of view. In particular, the current state of affairs for Van den Dries’s model-completeness conjecture is discussed in some detail. I prove the o-minimality of the pfaffian closure of an o-minimal structure, and I extend a weak model completeness result, originally proved as Theorem 5.1 in (J.-M. Lion and P. Speissegger, Duke Math J 103:215–231, 2000), to certain reducts of the pfaffian closure, such as the reduct generated by a single pfaffian chain.
Patrick Speissegger

Theorems of the Complement

This is an expository paper on a Theorem of the Complement, due to Wilkie, and its generalisations. Wilkie (Sel Math (NS) 5:397–421, 1999) gave necessary and sufficient conditions for an expansion of the real field by C-infinity functions to be o-minimal. Karpinski and Macintyre (Sel Math (NS) 5:507–516, 1999) weakened the original smoothness hypotheses of Wilkie’s theorem. Here we exhibit the proof of a generalised Wilkie’s result, where we further weaken the smoothness assumptions and show that the proof can be carried out not only over the real numbers but more generally in a non-Archimedean context, i.e. for definably complete Baire structures, which we introduced in 2008 and which form an axiomatizable class. Furthermore we give necessary and sufficient conditions for a definably complete Baire expansion of an o-minimal structure by C-infinity functions to be o-minimal.
Mathematics Subject Classification (2010): Primary 58A17, Secondary 03C64, 32C05, 54E52.
Antongiulio Fornasiero, Tamara Servi
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