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## Multi angle  de Rham theorem in non-Archimedean analytic geometry Berkovich, Vladimir (Auteur de la Conférence) | CIRM (Editeur )

In my work in progress on complex analytic vanishing cycles for formal schemes, I have defined integral "etale" cohomology groups of a compact strictly analytic space over the field of Laurent power series with complex coefficients. These are finitely generated abelian groups provided with a quasi-unipotent action of the fundamental group of the punctured complex plane, and they give rise to all $l$-adic etale cohomology groups of the space. After a short survey of this work, I will explain a theorem which, in the case when the space is rig-smooth, compares those groups and the de Rham cohomology groups of the space. The latter are provided with the Gauss-Manin connection and an additional structure which allow one to recover from them the "etale" cohomology groups with complex coefficients. In my work in progress on complex analytic vanishing cycles for formal schemes, I have defined integral "etale" cohomology groups of a compact strictly analytic space over the field of Laurent power series with complex coefficients. These are finitely generated abelian groups provided with a quasi-unipotent action of the fundamental group of the punctured complex plane, and they give rise to all $l$-adic etale cohomology groups of the space. ...

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## Multi angle  Equisingularity of map germs from a surface to the plane Nuño-Ballesteros, Juan José (Auteur de la Conférence) | CIRM (Editeur )

Let $(X,0)$ be an ICIS of dimension $2$ and let $f :(X,0)\to\mathbb{C} ^2$ be a map germ with an isolated instability. Given $F : (\mathcal{X} , 0) \to (\mathbb{C} \times \mathbb{C}^2, 0)$ a stable unfolding of $f$, we look to the invariants related to the family $f_s$ and we find relations between them. We obtain necessary and sufficient conditions for $F$ to be Whitney equisingular. (Joint work with B. Orfice-Okamoto and J. N. Tomazella)

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## Multi angle  The intersection form, logarithmic vector fields, and the Severi strata in the discriminant of a plane curve singularity Mond, David (Auteur de la Conférence) | CIRM (Editeur )

This talk will describe ongoing joint work with Paul Cadman and Duco van Straten, based on the PhD thesis of the former. Givental and Varchenko used the period mapping to pull back the intersection form on the Milnor fibre of an irreducible plane curve singularity $C$, and thereby define a symplectic structure on the base space of a miniversal deformation. We show how to combine this with a symmetric basis for the module of vector fields tangent to the discriminant, to produce involutive ideals $I_k$ which define the strata of parameter values $u$ such that $\delta(C_u)\leq k$. In the process we find an unexpected Lie algebra and a still mysterious canonical deformation of the module structure of the critical space over the discriminant. Much of this work is experimental - a crucial gap in understanding still needs bridging. This talk will describe ongoing joint work with Paul Cadman and Duco van Straten, based on the PhD thesis of the former. Givental and Varchenko used the period mapping to pull back the intersection form on the Milnor fibre of an irreducible plane curve singularity $C$, and thereby define a symplectic structure on the base space of a miniversal deformation. We show how to combine this with a symmetric basis for the module of vector fields tangent ...

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