By Peter Ebenfelt, Norbert Hungerbühler, Joseph J. Kohn, Ngaiming Mok, Emil J. Straube

This quantity represents the lawsuits of a convention on a number of advanced Variables, PDE's, geometry, and their interactions, held July 7-11, 2008 on the collage of Fribourg, Switzerland, in honor of Linda Rothschild. The members are best specialists who have been invited plenary audio system on the convention, or who have been invited through the editors to give a contribution to this quantity.

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5. 1. n = 2 , f (x, y, t) = tx2 − y 3 . 2 may be applied in this example. 3 we obtain that the extension of X |f |2λ |t|2μ presents an oblique polar line of direction (3, 1) through (−5/6 − j, −1/2), for j 1 . In fact it follows from general facts that j = 2 is large enough because here X is a neighborhood of 0 in C3 . Proof. 1/2). We take therefore k = 2 and we have f˜(x, y, τ ) := τ 2 x2 − y 3 . Put ∗ ˜ S = S˜∗ ∪ S˜∗ with S˜∗ := {x = y = 0} ∩ {τ = 0}, S˜∗ := {τ = y = 0} ∩ {x = 0}. 1) and ω ˜ induces a nonzero element in the H 1 of the Milnor ﬁbre of f˜ at 0 because it induces on S˜1∗ the pullback of the multivalued section of H 1 (5/6) we started with.

1. An involutive system of ﬁrst-order partial diﬀerential equations of rank n on M is a closed C ∞ submanifold Σ of CJ 1 (M) satisfying the following properties: (i) the projection CJ 1 (M) → M maps Σ onto M; (ii) each point of Σ has a neighborhood O on which there are n C ∞ functions F1 (x, ζ0 , ζ), . . 6) hold. 2. 1. A C 1 function u on M is called a solution if its ﬁrst jet lies in Σ. 3. Let (M, V) be an involutive structure. Let V ⊥ ⊆ CT ∗ M denote the one-forms ω such that ω, L = 0 for every section L of V.

After [4], this current is an analytic nonzero functional supported in the origin in ˜ . There exists therefore w ˜ Ω ˜ n+1 ) such that X ˜ ∈ Γ(X, ˜ X P2 (λ = −m − u, ¯ df˜ ˜ ¯˜ = 0, ∧ Ωk−l ∧ χw) |f˜|2λ f˜−j ˜ f˜ X for any cutoﬀ χ equal to 1 near 0. The change of variable τ → ζτ shows that w ˜ may be replaced by its component w ˜k−l in the above relation. With w ∈ ∗ k−l ) such that p w = τ w ˜ Γ(X, Ωn+1 k−l we get X |f |2λ f¯−j |t|−2(k−l )/k P2 λ = −m − u, X df ∧ Ωk−l ∧ χw ¯ = 0. f 18 D. -M. 4.