May 27, 2024

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

Graduate Courses held at the Fields Institute
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.
Tentative dates for these modules (in each course) are:
Module 1: Jan 19 - Feb 13
Module 2: Feb 23 - Mar 20
Module 3: Mar 23 - Apr 17
(The University of Toronto has a break during the week of February 16-20.)
All lectures will take place at the Fields Institute Rm. 230 unless notified otherwise.

Module 1: (Jan 19 - Feb 13)
o-minimality and Hardy fields (C. Miller)
Monday & Wednesday
10:30- 12 pm

Module 2: (Feb 23 - Mar 20 )
Construction of o-minimal Structures from Quasianalytic Classes
(J.-P. Rolin)

Monday & Wednesday
10:30- 12 pm

Module 3: (Mar 23 - Apr 17)
Pfaffian Closure
Lecture notes
(P. Speissegger)
Monday & Wednesday
10:30- 12 pm

Module 1: (Jan 19 - Feb 13)
Basic multisummability
(F.Sanz, J.P. Rolin, P. Speissegger)
Monday & Wednesday
1:30-3 pm

Module 2: (Feb 23 - Mar 20 )
Resurgent Functions
(D. Sauzin)
Monday & Wednesday
1:30-3 pm

Module 3: (Mar 23 - Apr 17)
Non-oscillatory Trajectories
(F. Sanz)
Monday & Wednesday
1:30-3 pm

Module 1: (Jan 19 - Feb 13)
Resolution of singularities for functions
(E. Bierstone)
Tuesday & Thursday
10:30- 12 pm

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

Note: This module is taught Jan. 19- Feb. 5 on Tuesday and Thursday 1:30- 3 pm

Module 3: (Mar 23 - Apr 17)
Resolution of Singularities of Real Analytic Vector Fields
(D. Panazzolo)
Tuesday & Thursday
10:30- 12 pm

Course on Topics in o-minimality

Module 1: O-minimality and Hardy fields
Instructor: C. Miller

Primary material: Hardy field theory as it relates to o-minimality; the growth dichotomy; and basic properties of the polynomially bounded case. More advanced topics and applications will be presented, as time permits, based on interests and preparation of the participants.

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

In a first part, we recall the definition of quasianalytic algebras of functions or germs of functions and give several examples of such algebras. Then we prove that convenient quasianalytic algebras generate o-minimal expansions of the real field. One important ingredient of the proof is the technique of resolution of singularities as discussed by Edward Bierstone in his course.

Module 3: Pfaffian closure and model completeness results for Pfaffian chains
Instructor: P. Speissegger
(Lecture notes)

The goal of my lectures is to outline a proof of Gareth O. Jones'recent result that the expansion of the real field by a Pfaffian chain and the exponential function is model complete. The proof combines both Wilkie's model-theoretic and Lion and Speissegger's geometric approaches to proving model completeness. Prerequisites for this course are basic differential geometry, basic model theory and the material covered by Chris Miller in his course.

Course on Multisummability and Quasianalyticity

Module 1: Basic multisummability
Instructor: F.Sanz, J.P. Rolin, P. Speissegger

- Gevrey asymptotics, Borel and Laplace transforms, k-summability.
- Cauchy-Heine transforms and decomposition theorems.
- Multisummability, iterated Borel and Laplace transforms, singular directions.
- Application: strong analytic transcendence from multisummability.
- Braaksma's theorem for nonlinear meromorphic ODEs.

Module 2: Resurgent functions
Instructor: D. Sauzin

- Reminders on Gevrey asymptotics and Borel-Laplace transform (cf. Schäfke's course). The Nevanlinna theorem. Examples (Airy, Ei(z), Erf (z), Stirling).

- The definition of "resurgence". Analytic continuation in the Borel plane and stability by convolution. Application to non-linear dynamics. The example of the saddle-node.

- Ecalle's "Alien calculus". The definition of "alien derivations". The "bridge equation".=

Module 3: Non-oscillatory trajectories
Instructor: F. Sanz

The course deals with the qualitative study of oscillatory and non- oscillatory trajectories of real analytic vector fields, mainly in dimension three. In the first part of the course, we describe several kinds of asymptotic behaviour that such transcendental objects can have: axial spiraling, asymptotic linking, separation by projection. In the second part, we will study non-oscillatory trajectories that belonging to new o-minimal structures, an application of the contents of the courses given by J.-P. Rolin, R. Schäfke and F. Cano.

Course on Resolution of Singularities

Module 1: Resolution of singularities for functions
Instructor: E. Bierstone

Background: examples, blowing-up and strict transform. Crucial exercises on transformation of differential operators by blowing up, semicontinuity of order of vanishing, normal crossings. Desingularization of spaces vs. desingularization of ideals. Motivating examples, marked ideals. Elementary proof of resolution of singularities.

Module 2: Resolution of singularities for foliations
Instructor: Felipe Cano
Note: This module is taught Jan. 19- Feb. 5 on Tuesday and Thursday 1:30- 3 pm

Bloc I (1 week): Basic concepts on singular foliations and reduction of singularities. a. Singular foliations and vector fields in dimension two. b. Blowing-up vector fields. simple singulaities. c. Separatrices and integral curves. Briot-Bouquet Theorem. d. Seidenberg's result on desingularization of vector fields. e. Camacho-Sad theorem of existence of separatrices.

Bloc II (1 week): Some applications of the reduction of singularities of codimension one foliations. a. Simple singularities in codimension one. Behavior under blow-up. b. The statement of reduction of singularities in dimension three. Consequences on the existence of invariant hypersurfaces. c. About the dicriticalness. d. Singular Frobenius I. e. Singualr Frobenius II.

Bloc III (1 week): Technics for the reduction of singularities. a. The reduction of the singularities of surfaces as a model for low dimensional problems. b. The main invariants used in the control: Multiplicities, Newton polygons and resonancies. c. Reduction of the singularities of codimension one foliations in dimension three. d. The valuative aproach for vector fields. The birational problem of reduction of singularities. Globalization. e. Local Uniformization of Vector Fields.

Module 3: Resolution of singularities of real analytic vector fields
Instructor: D. Panazzolo

Main Topics: Normal forms and classification of singularities of vector fields. Parametrized normal forms and bifurcations of limit cycles in analytic families of planar vector fields. The Newton polyhedron and the blowing-up of a vector field. Various recent results on resolution of singularities for vector fields: one-parameter familes in dimension two and real analytic three dimensional vector fields.