THEMATIC PROGRAMS

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

Organizing Committee:	Scientific Committee:
David Marker (UIChicago) Chris Miller (Ohio State) Jean-Philippe Rolin (Bourgogne) Patrick Speissegger (McMaster) Carol Wood (Wesleyan)	Edward Bierstone (Toronto) Lou van den Dries (UIUC) Robert Moussu (Bourgogne) Alex Wilkie (Oxford)
Supported by

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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

Distinguished Lecture Series

May 25- 27, 2009
Jean-Christophe Yoccoz, Collège de France

Geometry and Model Theory Seminar

Week-long workshops

January 12 - 16, 2009
Winter School in o-minimal Geometry
Organizer: Matthias Aschenbrenner

June 22 - 26, 2009
Workshop on Finiteness Problems in Dynamical Systems

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

 

Graduate courses Jan 19 - April 17, 2009

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)

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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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