Conference "Singularity Theory and Regular Stratifications" (On the occasion of the retirement of David Trotman)

COFFEE

L. WILSON (via ZOOM) : On the $t^r$ condition : 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.

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 $k=1$.

COFFEE

LE DUNG TRANG (à la Frumam) : Condition de Thom pour une stratification : Dans le cas d'un morphisme entre espaces analytiques complexes, on considère le cas de morphismes stratifiés. Pour morphismes stratifiés d'espaces analytiques complexes, Thom a introduit une condition appelée par Thom de morphismes sans éclatement et qu'on appelle maintenant la condition de Thom. Nous donnons la définition de la condition de Thom et nous montrons l'importance de cette condition dans l'étude de la Topologie locale d'un morphisme stratifié.

LUNCH

N. DUTERTRE (à la FRUMAM) : Principal kinematic formulas for germs of closed definable sets : 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 $SO(n) \times \mathbb{S}^{n-1}$.

G. COMTE (à la Frumam) : Invariants additifs en géométrie définissable Résumé : Je présenterai certains invariants additifs en théorie des singularités, dans le cadre de la géométrie modérée, et décrirai les correspondances qu'ils entretiennent. Dans le cas particulier de la géométrie non archimédienne définissable, je présenterai quelques pistes qui montrent que l'on peut encore s'attendre à des résultats ayant un contenu géométrique.

COFFEE

A.&G. VALETTE (ONLINE):"On Sobolev spaces of bounded subanalytic sets »