
THEMATIC PROGRAMS 

October 4, 2015  
Thematic Program on ominimal Structures and Real Analytic Geometry, JanuaryJune 2009Graduate Courses held at the Fields Institute

Three semesterlong 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
1620.)
All lectures will take place at the Fields Institute Rm. 230
unless notified otherwise.
Module
1: (Jan 19  Feb 13) ominimality and Hardy fields (C. Miller) Monday & Wednesday 10:30 12 pm Module 2: (Feb 23  Mar 20 ) Construction of ominimal 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) Module 2: (Feb 23  Mar 20 ) Resurgent Functions (D. Sauzin) Monday & Wednesday 1:303 pm Module 3: (Mar 23  Apr 17) Nonoscillatory Trajectories (F. Sanz) Monday & Wednesday 1:303 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 
Module 1: Ominimality and Hardy fields
Instructor: C. MillerPrimary material: Hardy field theory as it relates to ominimality; 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 ominimal 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 ominimal 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 modeltheoretic 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.
Module 1: Basic multisummability
Instructor: F.Sanz, J.P. Rolin, P. Speissegger Gevrey asymptotics, Borel and Laplace transforms, ksummability.
 CauchyHeine 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 BorelLaplace 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 nonlinear dynamics. The example of the saddlenode.
 Ecalle's "Alien calculus". The definition of "alien derivations". The "bridge equation".=
Module 3: Nonoscillatory 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 nonoscillatory trajectories that belonging to new ominimal structures, an application of the contents of the courses given by J.P. Rolin, R. Schäfke and F. Cano.
Module 1: Resolution of singularities for functions
Instructor: E. BierstoneBackground: examples, blowingup 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 pmBloc I (1 week): Basic concepts on singular foliations and reduction of singularities. a. Singular foliations and vector fields in dimension two. b. Blowingup vector fields. simple singulaities. c. Separatrices and integral curves. BriotBouquet Theorem. d. Seidenberg's result on desingularization of vector fields. e. CamachoSad 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 blowup. 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. PanazzoloMain 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 blowingup of a vector field. Various recent results on resolution of singularities for vector fields: oneparameter familes in dimension two and real analytic three dimensional vector fields.