An It\^o type formula for the additive stochastic heat equation

We use the theory of regularity structures to develop an It\^o formula for $u$, the solution of the one dimensional stochastic heat equation driven by space-time white noise with periodic boundary conditions. In particular for any smooth enough function $\varphi$ we can express the random distribution $(\partial_t-\partial_{xx})\varphi(u)$ and the random field $\varphi(u)$ in terms of the reconstruction of some modelled distributions. The resulting objects are then identified with some classical constructions of stochastic calculus

Questions on surface braid groups

We provide new group presentations for surface braid groups which are positive. We study some properties of such presentations and we solve the conjugacy problem in a particular case

Braids on surfaces and finite type invariants

We prove that there is no functorial universal finite type invariant for braids in $\Sigma\times I$ if the genus of $\Sigma$ is positive.Comment: 6p., revised version, C.R.Acad.Sci.Paris, to appea

Representations of Braid Groups and Generalisations

We define and study extensions of Artin's representation and braid monodromy representation to the case of topological and algebraical generalisations of braid groups. In particular we provide faithful representations of braid groups of oriented surfaces with boundary components as (outer) automorphisms of free groups. We give also similar representations for braid groups of non oriented surfaces with boundary components and we show a representation of braid groups of closed surfaces as outer automorphisms of free groups. Finally, we provide faithful representations of Artin-Tits groups of type $\mathcal{D}$ as automorphisms of free groups.Comment: Few misprints corrected, simpler proof for proposition 1