Talks in beamer

Vous trouverez ici les exposés beamer du colloque:

- N. DUTERTRE: Principal kinematic formulas for germs of closed definable sets

Abstract: We prove two principal kinematic formulas for germs of closed definable sets in $\mathbb{R}^n$, that generalize the Cauchy-Crofton formula for the density due to Comte and the infinitesimal linear kinematic formula due to the author. In this setting, we do not integrate on the space of euclidian motions $SO(n) \ltimes \mathbb{R}^n$, but on the manifold $SO(n) \times \mathbb{S}^{n-1}$.

Dutertre

- H. HAMM (via ZOOM) : Holomorphic mappings and Milnor fibrations Holomorphic functions, i.e. holomorphic mappings $f:X\to\mathbb{C}$,  $(X,0)$ being a complex analytic subgerm of $\mathbb{C}^N,0)$, lead to a Milnor fibration without the hypothesis of having an isolated singularity - this is due to the existence of stratifications of $X$ with Thom's $a_f$ condition. The situation changes radically, of course, in the case of mappings to $f:X\to \mathbb{C}^k, k>1$. A typical example is a blowing-up: $f:\mathbb{C}^2\to\mathbb{C}^2, (z_1,z_2) mapsto (z_1,z_1z_2)$. For instance, the image of $f|B_\epsilon\cap f^{-1}(D_\alpha)$, $0<\alpha\ll\epsilon$, depends on the radius $\epsilon$ and is merely semianalytic. It turns out that these two phenomena are not independent. Here it is helpful to have some evaluative criterion for complex constructibility, applied to a direct image sheaf in our case. Altogether, heavy restrictions on $f$ are necessary in order to have a similar situation as in the case $k=1$.

Hamm

- C. McCRORY: The intrinsic stratification of a semialgebraic set :

Shiota’s semialgebraic Hauptvermutung and methods of PL topology are used to investigate semialgebraically locally trivial stratifications.

Mccrory

- PARUSINSKI : Algebraic Stratified General Position and Transversality

Abstract: We use the method of Whitney interpolation to construct, for any real or complex projective algebraic variety, a stratified submersive family of self-maps that yields stratified general position and transversality theorems for semialgebraic chains. This theorem can be used to define an intersection pairing for real intersection homology, an analog of intersection homology for real algebraic varieties. Based on joint works with Clint McCrory and Laurentiu Paunescu.

Parusinski

- L. PAUNESCU: ARC-WISE ANALYTIC STRATIFICATION, WHITNEY FIBERING CONJECTURE AND ZARISKI EQUISINGULARITY (with Adam Parusinski) 

Abstract: For a given germ of complex or real analytic set, we show the existence of a stratification satisfying a strong (real arc-analytic with respect to all variables and analytic with respect to the parameter space) trivialization property along each stratum. We call such a trivialization arc-wise analytic and we show that it can be constructed under the classical Zariski algebro-geometric equisingularity assumptions. Using a slightly stronger version of the Zariski equisingularity, we show the existence of Whitney’s stratified fibration, satisfying the conditions (b) of Whitney and (w) of Verdier. Our construction is based on the Puiseux with parameter theorem and a generalization of Whitney’s interpolation. For algebraic sets our construction gives a global stratification.

Paunescu

- M. RUAS : On the multiplicities of families of non-isolated hypersurface singularities.

Abstract: We discuss in this lecture the Zariski multiplicity conjecture for two classes of families of non-isolated hypersurface singularities.

Ruas

Teissier

- S. TRIVEDI : Transversality and its relation to regular stratifications in smooth, complex and o-minimal setting 

Abstract: Trotman in his Ph.D. thesis proved that the stability of transversality to a stratification implies Whitney (a)-regularity of the stratification. I will talk about a generalization of this result and also its analogues in the complex case and in o-minimal structures.

Trivedi

- A.&G. VALETTE :"On Sobolev spaces of bounded subanalytic sets »

Valette

- L. WILSON :  On the $t^r$ condition

Abstract: This talk will be a survey of the $t^r$ stratification condition and its relation to results on sufficiency of jets. After discussing some of the results of the 60ís through the 80ís, it will focus on the 1999 Proc. London Math Society paper of Trotman and Wilson, which tied together and generalized many of the earlier results. Then it will cover some of the outgrowths from that paper to integral closure, bi-lipschitz sufficiency and sufficiency of weighted jets.

Wilson