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Let k be an algebraically closed field of characteristic p>0 and C a connected nonsingular projective curve over k with genus g>1. Let G be a p-subgroup of the k-automorphism group of C such that |G| > 2pg/(p-1). Then, C -->C/G is an étale cover of the affine line Spec k[X] totally ramified at infinity. To study such actions, we focus on the second ramification group G_2 of G at infinity, knowing that G_2 actually coincides with the derived group of G. We first display realizations of such actions with G_2 abelian of arbitrary large exponent .

The talk will be devoted to the problem of classification of all natural operations acting on the Hochschild cohomology of an associative algebra, by which one usually means the understanding of the homotopy type of the operad B formed by these operations. We will explain what a "natural operation" is and present two equivalent definitions of completely different natures. We then describe the homotopy type of B and show how is this descrition related to the Deligne Hochschild cohomology conjecture.