March  4, 2024

Thematic Program on o-minimal Structures and Real Analytic Geometry -- January-June 2009

Upcoming Mini-workshops at the Fields Institute

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These mini-workshops are intended to bring people together for two to three days to work on one particular project, while allowing other visitors to the program to participate. Each mini-workshop involves between five and ten core participants and typically focuses on the understanding of a specific problem or solution thereof.

The information below is tentative and subject to change.

Past Mini-Workshops

Title and dates
Abstract and participants

January 30-31, 2009
Mini-workshop on Expansions of the real field by multiplicative groups

Ayhan Gunaydin
Chris Miller
Lou van den Dries, Philipp Hieronymi and Michael Tychonievich

March 5-6, 2009
10:30 am
Mini-workshop on o-minimality for Certain Dulac Transition Maps
Room 210

Tobias Kaiser
Patrick Speissegger
We present the main ideas for proving o-minimality of the expansion of the real field generated by all Dulac transition maps near a non-resonant hyperbolic singularity of a planar analytic vector field. We also show how the existence of (non-explicit) uniform bounds on the number of limit cycles of certain (very special) families of analytic vector fields can be obtained from our approach.

Jean-Philippe Rolin, Dmitry Novikov, Sergei Yakovenko

March 16-20, 2009
Mini-workshop on the Infinitesimal Hilbert's 16th Problem

March 16 -3:30 pm
March 17 -3:30 pm
March 18 -3:30 pm
March 20 -3:30 pm


Dmitry Novikov
Sergei Yakovenko
Edward Bierstone, Andrei Gabrielov, Boris Khesin, Askold Khovanskii

March 23-25, 2009
Mini-workshop on New Perspectives in Valuation Theory
Room 210


Franz-Viktor Kuhlmann
Florian Pop
Bernard Teissier
In recent years new perspectives in valuation theory have begun to appear as well as unexpected applications. The two historical flows of valuation theory, namely the Henselian and the Zariskian, are merging like never before in the development of Berkovich geometry and the new approaches to resolution of singularities, which now extend to the singularities of vector fields. One begins to really be able to do analysis on spaces of of valuations, leading to important new results on complex analytic dynamical sytems stemming from a radically new point of view on the use of valuations of the ring of holomorphic functions. There is a new understanding of the structure of spaces of valuations with a given center, exemplified by the valuative tree of Favre-Jonsson, and also of the more global aspects for which tropical geometry gives useful hints. The purpose of the workshop is to gather experts who are contributing to this new perspective so that they can strengthen their common views and share problems and results.

Charles Favre, Mattias Jonsson, Daniel Panazzolo, Florian Pop, Mark Spivakovsky
March 23
3:30 pm - Bernard Teissier
Some recent developments in valuation theory
5:15 pm Askold Khovanskii
March 24
1:30 p.m. Charles Favre
Valuation Spaces
3:30 p.m. Mark Spivakovsky
Desingularization of 3-dimensional vector fields by blowing-up along non-singular centres
5:15 p.m. Franz-Viktor Kuhlmann
On local uniformization in positive characteristic
March 25

10:30 am - Mattias Jonsson
More on valuation spaces
1:30 pm - Salih Azgim
Extremal Fields
3:30 pm - Florian Pop
On the space of intertia elements

April 3-4, 2009
Miniworkshop on Differential Kaplansky Theory

Room 210

Salma Kuhlmann
Mickael Matusinski

Let (K,<,d) be an ordered diffenrential field, and v the natural valuation. We assume that d is compatible with v, i.e. that v is a differential valuation in the sense of M. Rosenlicht. Denote by k the residue field and by (G,Ψ) the induced asymptotic couple; i.e. G = v(K) is the value group endowed with the map Ψ(v(a)) := v(a′: =a).

The purpose of this workshop is to study a differential Kaplansky theory in this setting. We want to achieve progress on the following problem: Find necessary and su±cient conditions on (K; <, d) so that: (i) the data (G, Ψ) allows to define a derivation d on the field of generalized series k((G)); (ii) the induced asymptotic couple is precisely (G,Ψ); (iii) there is an order preserving di®erential embedding of (K,<,d) in (k((G)),<,d); (iv) the embedding may be chosen to be truncation closed; i.e. the image of the embedding is closed under the operation of taking initial segments of series. Partial progress has been achieved on this topic, for example regarding item (i), we have described the construction of "well-defined" derivations on k((G)). Regarding item (iii), J.M.Aroca and J. Del Blanco have considered the case of archimedean value group. Other approaches to this problem are described in the works of M. Aschenbrenner - L. v. D. Dries on H - fields, and the works of J. v. D. Hoeven on Transseries.

J. Del Blanco Marana, Lou van den Dries

April 3
Room 210

10:00 am - Salma Kuhlmann, Presentation of the workshop

10:30-12 pm - M. Matusinski
Hardy type derivations on generalized series fields
We consider an arbitrary Hahn group of monomials and the corresponding field of generalized series with real coefficients. First we show how to construct well-defined derivations on such fields. We then give a criterion for such a derivation to be of Hardy type, that is to verify the same properties as those in Hardy fields.

1:30-3 pm - L. van den Dries
What is an asymptotic differential field and when is it differentially henselian
I will define asymptotic differential fields, propose a notion of "differentially henselian" and indicate some of its good properties.

3:30-5 pm - discussion

April 4
Room 210
10:30-12 pm - discussion

3:00- 4:30 pm - J. del Blanco Marana
A differential Kaplansky immersion theorem on rank one valued fields with real residue field
I will present the following result:
Let (K, v, d) be a differential valued field with archimedean value group G = v(K -{0}) and residue field kv = R included in K.
The following two statements are equivalent:
(i) The derivation d is functional.
(ii) There exists a differential analytic morphism (K, v, d) ->
(R[[X^G]], ord, ?), where ? is some monomial derivation on the
generalised series field R[[X^G]].
A functional derivation carries an abstract version of properties that hold in the case of functions (continuity, L'Hospital's rule), for instance germs in a Hardy field. The notion of monomial derivation is a generalisation of the usual derivation for formal power series.

May 6-8, 2009
Mini-Workshop on (Co)Homology and sheaves in O-minimal and Related Settings

Room 210

M. Edmundo, A. Piekosz and L. Prelli The workshop aims to introduce everyone to the subject. Talk 1 will include an introduction to sheaves with focus on sheaves on sub-analytic site and applications. Talk 2 illustrates the use of sheaves to construct locally definable spaces and the more sophisticated weakly definable spaces, which when considered over an o-minimal expansion of a field, have a well developed homotopy theory giving in particular different kinds of homology and cohomology. Talk 3 focus on sheaves on o-minimal site with applications to the theory of definable groups.

M. Edmundo, P. Eleftheriou, A. Piekosz, L. Prelli, S. Starchenko.

May 6
Room 210
10:30 am - Luca Prelli
Sheaves on subanalytic sites and D-modules
May 7
Room 210
10:30 am - Artur Piekosz
Locally definable and weakly definable spaces
Abstract: As they are defined using structure sheaves, locally definable spaces and more sophisticated weakly definable spaces are a natural generalization of both the subanalytic and the definable o- minimal settings. If considered over an o-minimal expansion of a field, they have a well developed homotopy theory, which in particular gives different kinds of homology and cohomology. The recommended reading for my lecture is the preprint arXiv:0904.4896 (
May 8
Room 210
10:30 am - Mario Edmundo
O-minimal sheaves and applications

June 5-6, 2009
Mini-workshop on decidability in analytic situations

Room 230

Gareth O. Jones The workshop aims to understand the work of Macintyre and Wilkie on the real exponential field, and the more recent work of Macintyre on Weierstrass functions. The relation with the constructive results of Gabrielov and Vorobjov would also be investigated. The hope is that after careful study of these papers, we would be able to prove further constructive model completeness results for theories related to those above. If this goes to plan, we would then combine the constructive model completeness with recent work around Schanuel?s conjecture, with the aim of proving unconditional decidability results for certain analytic expansions of the real field.

Tamara Servi, Dan Miller, Andrei Gabrielov
June 5
Room 230
10:00-11:30 Tamara Servi
On the decidability of the real field with a generic power function part I
(joint work with G. Jones) In recent work we proved that, if A is a real number not zero-definable in the real exponential field, then the theory of the real field with the power function x^A is decidable, relatively to an oracle for A. I will prove this statement, and give a proof of the existence of a computable generic real number.

13:30-15:00 Tamara Servi
On the decidability of the real field with a generic power function part II
June 6
Room 230
10:00-11:30 Dan Miller

13:130-15:00 Andrei Gabrielov
Multiplicity of a Noetherian intersection and degree of nonholonomy
A differential ring of analytic functions in several complex variables is called a ring of Noetherian functions if it is finitely generated as a ring and contains the ring of all polynomials. The multiplicity of an isolated solution of a system of $n$ equations $f_i=0$, where $f_i$ belong to a ring of Noetherian functions in $n$ complex variables, can be expressed in terms of the Euler characteristics of the generalized Milnor fibers associated with this system. This provides an effective upper bound on this multiplicity. In combination with constructive resolution of singularities over the fields of characteristic zero, this allows one to obtain an effective upper bound on the complexity of the resolution of singularities defined by Noetherian functions. For $n=1$, Noetherian functions are soultions of a system of algebraic ordinary differential equations. The upper bound on their multiplicity implies an effective upper bound for degree of nonholonomy of a system of algebraic vector fields, an important problem in control theory.

June 8-10, 2009
Mini-workshop on Finiteness theorems for certain quasi-regular algebras and Hilbert's 16th problem

Room 210

Abderaouf Mourtada

On introduit une algèbre de germes de fonctions, dite algèbre quasi- régulière d'Hilbert: ces éléments sont quasi-analytiques et possèdent une structure asymptotique de "type Hilbert". Cette algèbre contient les compositions des déploiements analytiques d'applications de Dulac pour les singularités heyperboliques de champs de vecteurs du plan. L'étude de la cyclicité des polycycles hyperboliques du plan se ramène a l'étude de l'action sur cette algèbre d'une certaine classe de dérivations dites "Dérivations d'Hilbert". La désingularisation de telles dérivations (dans l'algèbre quasi-régulière d'Hilbert!!) fournit des dérivations irréductibles qui sont hyperboliques, linéaires et diagonales. Les théorèmes de finitude exposés sont relatifs a ces dérivations irréductibles. Une première application globale de ces résultats, dans le cadre du 16eme problème d'Hilbert est la suivante: la cyclicité d'un cycle singulier d'un champ Hamiltonien du plan, ne dépend que du degré de l'Hamiltonien et de "la multiplicité algébrique" de l'intégrale abélienne associée, ce qui constitue la généralisation naturelle du théorème de Khovanski- Varchenko.

Jean-Philippe Rolin, Patrick Speissegger

June 8
Room 210
10:30 am- Introducion des algebres et derivation d'Hilbert.

3:30 pm- A la demande pour plus de details.

June 9
Room 210
10:30 am- Les theoremes de finitude et une application localedans le cadre du 16eme probleme d'Hilbert.

3:30 pm- A la demande pour plus de details.

June 10
Room 210
10:30 am- Une application globale dans le cadre du 16eme probleme d'Hilbert.

3:30 pm- A la demande pour plus de details.


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