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Tentative programs of mini-courses:

F. BROGLIA & F. ACQUISTAPACE

"Géométrie Semi-analytique Globale"

Program:

Dans le cadre de la géométrie analytique on distingue en général l'aspect local (ou des germes) de l'aspect global (algèbre des fonctions définies sur un ouvert fixé).

Les ensembles semi-analytiques globaux sont les constructibles de l'algèebre des fonctions analytiques globales sur un ouvert Ω de Rn; si on se borne dans les definitions à utiliser seulement des égalités, on obtient les ensembles C-analytiques, c'est-à-dire les ensembles analytiques support d'un faisceau coherent.

Pour ce type d'ensembles on se pose les mêmes questions que pour les semi-algebriques ou les semi-analytiques dans le sens de Łojasiewicz, comme par example :

• Problèmes de finitude: chaque semi-analytique global ouvert est-il réunion d'un nombre fini de semi-analytiques globaux de base ouverts (même question pour les fermés) ?
• Les composantes connexes, la fermeture d'un semi-analytique global sont-elles encore un semi-analytique global ?

Pour les fonctions analytiques globales on a seulement des reponses partielles à ces problemes, ainsi que pour d'autres qui sont les analogues du Nullstellensatz, Positivstellensatz et 17è Problème de Hilbert dans le cas algéebrique.

Dans ce cours on donnera un cadre de l'etat de l'art de quelques uns parmi ces problèmes.

A. DU PLESSIS

"Tame geometry of mappings"

Program:

1. The Thom-Mather theory of stability.
2. The geometry of topological stability.
3. Mappings of finite codimension.

V. GRANDJEAN & A. PARUSIŃSKI

"Autour de  la conjecture du gradient de R. Thom"

Program:

The gradient conjecture of R. Thom says that any trajectory of the gradient vector field of a real analytic function has a tangent at its limit point. We show how the ideas of Lojasiewicz and Thom led to a recent proof of this conjecture. The result obtained by Kurdyka, Mostowski, & Parusinski is stronger and shows that the length of the trajectory projected from its limit point to the unit sphere is finite. We present the details of the proof and develop the necessary techniques.

1. Outline of the proof. We introduce such basic concepts as: - Lojasiewicz's proof of the finitness of length of the trajectory based his inequality for gradient - Thom's concept of control function - Thom's idea of the use of blowing-up. Proof for homogeneous functions. As a corollary we obtain Thom-Martinet and Thom-Kuiper theorems
2. The notion of characteristic exponents and the hierarchy of atractors. If we have time we discuss the characteristic exponents in the context of conormal geometry and the space of limits of direction of the gradient.
3. Asymptotic critical values. These are the critical values of meromorphic (or more generally definable) non-proper functions that arise from their asymptotic behaviour, classically at infinity but in our case at the origin.
This technique allows us to treat the problem in similarly to the homogeneus case by restricting to the areas of its constant asymptotic behaviour.
4. The of conclusion the proof. A discussion of open questions and problems.

Z. JELONEK

"Geometry of real polynomial mappings"

Program:

Lectures will contain:

•  Basic properties of real polynomial mappings
•  Asymptotic variety of real polynomial mapping
•  Real Jacobian Conjecture
•  Nash manifolds
•  Extension of Nash embeddings of Nash manifolds

J.-M. LION

"o-minimalité et théorème du complémentaire en géométrie pfaffienne"

Program:

L'objet de ces exposés est de dresser un panorama en géométrie pfaffienne qui va de la propriété de finitude uniforme jusqu'au théorème  du complémentaire pour les projections de pfaffiens emboités.

J.-P. ROLIN

"Quasi-analytic solutions of analytic ordinary differential equations  and o-minimal structures"

Program:

It is well known that the non-spiraling leaves of real analytic foliations of codimension 1 all belong to the same o-minimal structure. It is natural to ask for the same property for non-oscillating solutions of an "irregular singular" analytic differential equation. We give a positive answer under some hypothesis on the equation. However, we also construct such solutions which are not definable in any o-minimal structure. Moreover, we show that there exists a differential equation  satisfying our hypothesis, but such that two non-oscillating solutions cannot be definable in the same o-minimal structure.

M.-F. ROY

"Algorithms in real algebraic geometry: complexity results based on the critical point method"

Program:

While the cylindrical decomposition method give complexity bounds doubly exponential in the number of variables, the critical point method provides single exponential algorithms for deciding emptyness, making global optimization, counting connected components, deciding connectivity, describing connected components, and computing the first Betti numbers.

Moreover in the quadratic case, the critical point method provides polynomial time algorithms for testing emptyness and global optimization.

The course will be based on the book Algorithms in real algebraic geometry by Basu, Pollack and Roy, and recent papers by Basu Pollack Roy, Basu and Grigoriev Pasechnik.

Z. SZAFRANIEC

"Effective methods for computing the topological invariants of real algebraic sets"

Program:

If there is given either a real algebraic set, or a real polynomial mapping, then one may associate with it some topological invariants (the Euler characteristic, the number of branches of a curve emanating from a point, the topological degree,...). Several of those invariants can be computed in terms of the signature of an appropriate quadratic form.    Construction of the quadratic form usually requires techniques based on multivariate Bezoutians and the Kronecker symbol. After presenting those notions, I will talk about

• algebraic methods for calculating the topological degree
• the Eisenbud-Levine formula for the local topological degree
• the Khimshiasvili formula, and other formulea for the Euler characteristic  of real algebraic sets
• the Aoki-Fukuda-Nishimura formula for the number of branches of a real   algebraic curve

V. THILLIEZ

"Quasi-analytic local rings"

Program:

Quasi-analytic classes of functions are well known since several decades in classical analysis. Their study from the viewpoint of analytic geometry has begun much more recently and, to some extent, earned them a new interest. In the summer school, we will thus give an introduction to quasi-analytic classes emphasizing results which are clearly related to topics in local algebra and tame geometry. After having reviewed some basic facts, we will study, in particular, Weierstrass division problems and the role of hyperbolicity, as well as properties of ideals of quasi-analytic germs.

1. Ultradifferentiable classes: motivation and definition
2. Elementary properties, questions of stability
3. The Borel map: injectivity vs. surjectivity
4. Weierstrass division: obstructions and sufficient conditions
5. Ideals of germs and formal series
6. Some applications of resolution of singularities

D. TROTMAN

"Stratification in real geometry"

Program:

1. Stratifications.
It has been known for 40 (resp. 30, resp. 10) years that analytic varieties and semianalytic (resp. subanalytic, resp. o-minimally definable) sets admit Whitney stratifications. For these sets the regularity conditions of T.-C. Kuo and Verdier can be imposed. The stronger conditionsof Mostowski ensuring local Lipschitz triviality along strata may be obtained for subanalytic sets but not in general for definable sets in non polynomially bounded o-minimal structures. We define these regularity conditions, giving examples to confirm they are distinct, and state the corresponding existence theorems. It is often useful that the transversal intersection of two regularly stratified sets possesses the same regularity.

2. Equisingularity.
The result about Whitney's (b)-regular stratifications which makes them useful is Thom-Mather isotopy. This says that a stratum of a Whitney stratified set has a neighbourhood which is the total space of a locally trivial topological fibre bundle. The proof of Mather uses integrable controlled stratified vector fields whose stratified isotopies yield local topological trivialisations. The same works for the (c)-regular stratifications of K. Bekka; this is important when studying topological stability. The stronger conditions of Verdier and Mostowski give more information about the isotopies. Theorems for manifolds giving transversality after isotopy allow stratified generalisations. Classical Morse theory statements have stratified counterparts (Goresky and MacPherson), by the Thom-Mather theorem, for sufficiently general functions. For constructible sets the isotopy theorem has counterparts (M. Coste and M. Shiota).

3. Stratifications and determinacy.
A weak regularity condition (t) introduced by Thom in 1964 says that a submanifold transverse to a stratum remains transverse to nearby strata. Specifying that the submanifolds be C^k gives (t^k). It turns out that (t^k) and the Kuo-Verdier condition (w) are related. Generalising the blow-up of a point by considering planes rather than lines defines the Grassmann blow-up (Kuo-Trotman, 1987). The induced pullback of (t^k) is (t^k-1), while pullback of (t^1) is (w). Moreover pushforward of (t^k-1) is (t^k). Thus (t^1) implies top uniqueness of germs of transversal intersections at a point and characterisations of jet sufficiency (Trotman-Wilson 1999), recovering theorems of Kuo. Incidentally this led Thom to the introduction of stratification theory in his 1964 Bombay paper.

4. Metric and tangential properties.>
To generalise the tangent space of a differentiable manifold to a stratified space one has tangent cone and normal cone. Whitney and Hironaka proved properties of analytic Whitney stratifications, for example equimultiplicity in the complex case and openness of the projection associated to its normal cone. These can be extended to differentiable stratifications (Orro-Trotman, 2002), and provide criteria for approximation of a subanalytic set by a normal cone (Ferrarotti-Fortuna-Wilson, 2004). Hironaka's equimultiplicity has a real counterpart in the continuity of the density proved by G. Comte for Verdier stratifications (2000) and G. Valette for Whitney stratifications (2004). A generalisation of multiplicity to real functions is the Fukui invariant (1995), introduced as a way of testing for blow-analytic equivalence (Kuo).

Y. YOMDIN

"Singularities and image processing"

Program:

There are several aspects of the Image Analysis, Representation, and Processing where Singularity Theory and Semi-algebraic geometry come into play.

First of all, this concerns detection of singularities of images, like edges and ridges, and their capturing by the appropriate "Normal Forms" and "Fitting transformations".

Secondly, representation of three-dimensional objects is well known to incorporate various elements and structures of Singularity Theory and Semi-algebraic geometry.

Third, the interaction of the two Imaging aspects, mentioned above, poses certain non-trivial problems of "recognition of singularities" with a noisy data and finite accuracy computations.

We plan to present these topics, both their theoretical and implementation sides.

Abstracts of talks:

G. COMTE

"Rotations de trajectoires de champs de vecteurs"

Abstract:

A trjectory of a smooth vector field may have in finite time an infinite rotation around a given (non integral) line. This phenomena does not occur for two trajectories of a lipschitz vector fields: we may bound their mutual rotation by Ctemax(T1,T2)+Cte'.K.T1.T2, where T1, T2 are the time intervals for each trajectory and K is the Lipschitz constant of the field.

M. COSTE

"Comment obtenir (presque) gratuitement des résultats d'uniformité"

Abstract:

Je veux expliquer dans cet exposé comment des démonstrations "élémentaires" (au sens logique) dans une structure o-minimale permettent d'obtenir des résultats uniformes pour les familles. Ce principe sera illustré notamment par la transformation de divers résultats de triangulation en résultats de trivialisation.

D. D'ACUNTO

"Effective Łojasiewicz inequalities"

Abstract:

The method of ridge and valley lines in the semialgebraic context developed together with K. Kurdyka appears to be efficient when one wants to give a bound for the Łojasiewicz exponent in the gradient inequality. When the function is a real polynomial in $n$ variables of degree $d$, we deduce an explicit bound in terms of $n$ and $d$. This bound holds even if the singular level of the function is not an isolated point. We then deduce a bound for the Łojasiewicz exponent appearing in the inequality that compares the absolute value of a polynomial with the distance to its set of zeros.

W. DOMITRZ

"Algebraic restrictions of symplectic form to singular sets and its application" (joint work with S. Janeczko and M. Zhitomirskii)

Abstract:

Let N be the germ of a singular subset of R2n. Two germs of symplectic forms on R2n have the same algebraic restriction to N if their difference is equal to β+dα where β, α are germs of forms vanishing on N. We prove that if N is quasi-homogeneous set and two germs of symplectic have the same algebraic restriction to N then there exists a germ of diffeomorphism which is identity on N and maps one germ of symplectic form to the other. In the talk I explain how to apply this theorem for classification problem of singular curves on a symplectic manifold. I show how to calculate the set of algebraic restrictions of symplectic forms to a singular variety. I use this method to described local symplectic invariants of planar curves A-D-E, S5 singularity and a union of germs smooth submanifolds with clean intersection of the stata.

N. DUTERTRE

"Degree formulas and signature formulas for the Euler characteristic of algebraic sets"

Abstract:

We will describe formulas, due to different authors, for the computation of the Euler characteristics of real algebraic sets.

" Weierstrass division Theorem in definable germs in a polynomially bounded o-minimal structure" (joint work with Hassan SFOULI)

Abstract:

We give some examples of polynomially bounded o-minimal expansion $\mathcal{R}$ of the ordered field of real numbers where the Weierstrass Division Theorem does not hold in the ring of germs, at the origin of Rn, of definable functions.

G. FICHOU

"Zeta functions and Blow-Nash equivalence"

Abstract:

We propose a refinement of the notion of blow-Nash equivalence between Nash function germs, which is an analog in the Nash setting of the blow-analytic equivalence defined by T.-C. Kuo. The new definition is more natural and geometric. We will discuss about invariants.

P. GOLDSTEIN

" Some remarks on gradients of holomorphic functions " (joint work with Marek GROCHOWSKI)

Abstract:

We present some results and remarks on gradients of holomorphic functions in C^n, in particular that only a finite number of trajectories enter a generic point on a critical level set.

O. LE GAL

"Un théorème du complémentaire pour les structures o-minimales polynomialement bornées"

Abstract:

On étend le théorème du complémentaire explicite de Gabrielov à des familles de fonctions qui engendrent des structures o-minimales polynomialement bornées. Ainsi, si F est une algèbre différentielle d'applications définissables dans une structure o-minimale polynomialement bornée, on montre que les ensembles obtenus par projection des zéros des applications de F sont stables par passage au complémentaire.

M. MERLE

"Cycles évanescents d'une fonction composée"

Abstract:

On décrit un calcul des cycles évanescents d'une fonction obtenue par composition avec un polynôme non dégénéré pour son polygone de Newton. (Travail commun avec G. Guibert et F. Loeser)

R. MOUSSU

"Quelques propriétés des courbes intégrales d'un gradient"

Abstract:

Soit f une fonction analytique sur une variété M munie d'une métrique riemannienne g, et soit γ:R+--> M une courbe intégrale relativement compacte de ∇f, le gradient de f pour g. Grace à sa célèbre inégalité, S. Łojasiewicz a prouvé que |γ| a une longueur finie; en particulier, γ possède un unique point limite ω(γ) qui est un point singulier de f. R. Thom a alors proposé sa fameuse conjecture: |γ| possède une tangente en ω(γ). Par une étude minutieuse de f au voisinage de ω(γ), K. Kurdyka, T. Mostowki et A. Parusiński l'ont prouvée récemment. Un argument classique de A. Khovanskii permet alors d'affirmer que |γ| est non oscillante si dim M =2. On peut espérer que cette propriété reste vraie en dimension 3. Cet exposé a pour but d'exposer quelques arguments qui justifient cet espoir.

F. PIERONI

"Sums of squares in quasianalytic rings"

Abstract:

We prove that any nonnegative function germ at the origin of R2 belonging to a quasianalytic Denjoy-Carleman class is a sum of two squares of function germs that are in a Denjoy-Carleman class again. If the germ is elliptic the class is the same, in the general case a loss of regularity is possible.

S. RANDRIAMBOLOLONA

"Des ensembles de petite arité d'une structure o-minimale"

Abstract:

F. SANZ

"Gradient vector fields can not generate twister dynamics"

Abstract:

Thom's Gradient Conjecture, proved by Kurdyka-Mostowsky-Parusinski (2000), states that a solution of an analytic gradient vector field approaching to a singular point has a tangent at . A stronger version of this conjecture asserts that meets any analytic hypersurface only finitely many times ( is non-oscillating). An intermediate version is that has all iterated tangents. We prove, in dimension (and for isolated singularity) that the strong conjecture is reduced to this last weaker form: if has all iterated tangents then is non-oscillating. The proof is based on a result of Cano-Moussu-Sanz (2000), asserting that an oscillating solution of an analytic vector field which has all iterated tangents must spiral around an analytic curve in a precise sense. Here we prove that such spiraling axis do not exists for gradients vector fields.

A. STASICA

"On asymptotic variety for real polynomial mappings"

Abstract:

We will show how one can detect an asymptotic variety for polynomial mappings $\R^2\to\R^2$ counting the numbers of points of the fibers.

M. TIBAR

"Regularity at infinity of polynomial maps"

Abstract:

J'expliquerai quelques conditions de régularité et leurs relations dans le cadre des applications , .

G. VALETTE

"Metric types in tame geometry"

Abstract:

We present some results about the metric types of subanalytic sets and more generally about definable sets in polynomially bounded o-minimal structures. By"metric type" we mean that we study the sets with the induced metric of R^n, up to bi-Lipschitz equivalence.