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### OVERVIEW

The main focus of the program is to extend local resolution of singularities
techniques in order to establish the o-minimality of certain expansions
of the real field, such as those generated by the functions studied
by Ilyashenko and Ecalle in his proof of Dulac’s problem. Over
the last twenty years, the notion of o-minimal structure has become
increasingly useful in the fields of real algebraic and real analytic
geometry. Discovered by Van den Dries in the early 1980s and developed
in its present model-theoretic generality soon after by Knight, Pillay
and Steinhorn this notion provides a unifying framework for what is
sometimes loosely referred to as tame real geometry. Since then, the
development of o-minimality has been strongly influenced by real analytic
geometry; this is apparent in the adaptation of methods of real analytic
geometry to the o-minimal setting and in the motivation to find new
and mathematically interesting examples of o-minimal structures. Conversely,
model-theoretic methods available through the o-minimal point of view
have led to new insights into real analytic geometry.

Many recent developments in the intersection of o-minimality and
real analytic geometry use resolution of singularities in crucial
ways, and they can in turn be viewed as extending the notion of
resolution of singularities in the sense of the preparation theorems
mentioned above. There is good reason to believe that extending
resolution algorithms to certain classes of functions involving
exponential scales may help shed new light on various interesting
problems in real analytic geometry, coming from Pfaffian geometry
and dynamical systems. Examples of particular interest to us are
the classes of multisummable and of resurgent functions. The program's
focus and main activities, two week-long workshops and three graduate
courses, will be centered around the topics described above. There
are of course many other developments both in o-minimality and in
real anaytic geometry, and only the future will tell which of them
may be relevant in addressing the questions discussed here. We intend
to explore such developments in some of the mini-workshops.

### Scientific Activities

**May 25- 27, 2009 **

Jean-Christophe Yoccoz, Collège de France

**Week-long workshops**

Organized by Fernando Sanz and Patrick Speissegger

This workshop will consist of regular invited one-hour lectures
and will be held at the conclusion of the proposed program.

**Mini-workshops**

A key activity will be the mini-workshops, 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.

**January 30-31, 2009 **

Expansions of the
real field by multiplicative groups

**Organizers:** Ayhan Gunaydin, Chris Miller

**March 5-6, 2009**

O-minimality for certain
Dulac transition maps

**Organizers**: Tobias Kaiser, Patrick Speissegger

**March 16-20, 2009**

The Infinitesimal
Hilbert's 16th Problem

**Organizers**: Dmitry Novikov, Sergei Yakovenko

**March 23-25, 2009**

Workshop on new perspectives
in Valuation theory

**Organizers**: Franz-Viktor Kuhlmann , Bernard Teissier

**April 3-4, 2009 **

Differential Kaplansky Theory

Organizers: Salma Kuhlmann, Mickael Matusinski

**May 6-8, 2009**

Mini-Workshop on
(Co)Homology and sheaves in O-minimal and Related Settings

Organizers: M. Edmundo, A. Piekosz and L. Prelli

**June 5-6, 2009 **

Decidability in
analytic situations

Organizer: Gareth O. Jones

**June 8-10, 2009**

Finiteness theorems
for certain quasi-regular algebras and Hilbert's 16th problem

**Organizer: **Abderaouf Mourtada

Three semester-long graduate courses will take a more detailed look
at the topics of the programme and will serve both students looking
for a research problem and established researchers hoping to learn
more about a particular subject. We plan to teach the three courses
in parallel; each course in turn will be split into three modules.
Each of these modules is four weeks long, with three hours of lectures
per week. The courses are:

Course on Topics in o-minimality
Module 1: *o-minimality and Hardy fields* (C. Miller)

Module 2: *Construction of o-minimal structures from quasianalytic
classes* (J.-P. Rolin)

Module 3: *Pfaffian closure* (P. Speissegger)

**Course on Multisummability and Quasianalyticity**

Module 1: *Basic multisummability* (R. Schäfke)

Module 2: *Resurgent functions* (D. Sauzin)

Module 3: *Non-oscillatory trajectories* (F. Sanz)

**Course on Resolution of Singularities**

Module 1: *Resolution of singularities for functions*
(E. Bierstone)

Module 2: *Resolution of singularities
for foliations* (Felipe Cano)

Module 3: *Resolution of singularities of real analytic vector
fields* (D. Panazzolo)

**Apply to the Program:**

All scientific events are open to the mathematical sciences community.
Visitors who are interested in **office space or funding** are
requested to apply by filling out the **application
form** Additional support is available (pending NSF funding)
to support junior US visitors to this program. Fields scientific
programs are devoted to research in the mathematical sciences, and
enhanced graduate and post-doctoral training opportunities. Part
of the mandate of the Institute is to broaden and enlarge the community,
and to encourage the participation of women and members of visible
minority groups in our scientific programs.

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