

Tentative programs of minicourses:F. BROGLIA & F. ACQUISTAPACE "Géométrie Semianalytique 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 semianalytiques globaux sont les constructibles de l'algèebre des fonctions analytiques globales sur un ouvert Ω de R^{n}; si on se borne dans les definitions à utiliser seulement des égalités, on obtient les ensembles Canalytiques, c'estàdire les ensembles analytiques support d'un faisceau coherent. Pour ce type d'ensembles on se pose les mêmes questions que pour les semialgebriques ou les semianalytiques dans le sens de Łojasiewicz, comme par example :
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:
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.
Z. JELONEK "Geometry of real polynomial mappings" Program: Lectures will contain:
J.M. LION "ominimalité 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 "Quasianalytic solutions of analytic ordinary differential equations and ominimal structures" Program: It is well known that the nonspiraling leaves of real analytic foliations of codimension 1 all belong to the same ominimal structure. It is natural to ask for the same property for nonoscillating 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 ominimal structure. Moreover, we show that there exists a differential equation satisfying our hypothesis, but such that two nonoscillating solutions cannot be definable in the same ominimal 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
V. THILLIEZ "Quasianalytic local rings" Program: Quasianalytic 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 quasianalytic 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 quasianalytic germs.
D. TROTMAN "Stratification in real geometry" Program:
1. Stratifications. Y. YOMDIN "Singularities and image processing" Program: There are several aspects of the Image Analysis, Representation, and Processing where Singularity Theory and Semialgebraic 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 threedimensional objects is well known to incorporate various elements and structures of Singularity Theory and Semialgebraic geometry. Third, the interaction of the two Imaging aspects, mentioned above, poses certain nontrivial 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(T_{1},T_{2})+Cte'.K.T_{1}.T_{2}, where T_{1}, T_{2} 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 ominimale 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 R^{2n}. Two germs of symplectic forms on R^{2n} 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 quasihomogeneous 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 ADE, S_{5} 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. A. ELKHADIRI " Weierstrass division Theorem in definable germs in a polynomially bounded ominimal structure" (joint work with Hassan SFOULI) Abstract: We give some examples of polynomially bounded ominimal 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 R^{n}, of definable functions. G. FICHOU "Zeta functions and BlowNash equivalence" Abstract: We propose a refinement of the notion of blowNash equivalence between Nash function germs, which is an analog in the Nash setting of the blowanalytic 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 ominimales polynomialement bornées" Abstract: On étend le théorème du complémentaire explicite de Gabrielov à des familles de fonctions qui engendrent des structures ominimales polynomialement bornées. Ainsi, si F est une algèbre différentielle d'applications définissables dans une structure ominimale 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 R^{2} belonging to a quasianalytic DenjoyCarleman class is a sum of two squares of function germs that are in a DenjoyCarleman 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 ominimale" Abstract: F. SANZ "Gradient vector fields can not generate twister dynamics" Abstract: Thom's Gradient Conjecture, proved by KurdykaMostowskyParusinski (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 nonoscillating). 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 nonoscillating. The proof is based on a result of CanoMoussuSanz (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 ominimal structures. By"metric type" we mean that we study the sets with the induced metric of R^n, up to biLipschitz equivalence. 