  If $M$ is a symplectic manifold then the space of smooth loops $\mathrm
C^{\infty}(\mathrm S^1,M)$ inherits of a quasi-symplectic form. We will focus
in this thesis on an algebraic analogue of that result. Kapranov and Vasserot
introduced and studied the formal loop space of a scheme $X$. It is an
algebraic version of the space of smooth loops in a differentiable manifold.
  We generalize their construction to higher dimensional loops. To any scheme
$X$ -- not necessarily smooth -- we associate $\mathcal L^d(X)$, the space of
loops of dimension $d$. We prove it has a structure of (derived) Tate scheme --
ie its tangent is a Tate module: it is infinite dimensional but behaves nicely
enough regarding duality. We also define the bubble space $\mathcal B^d(X)$, a
variation of the loop space. We prove that $\mathcal B^d(X)$ is endowed with a
natural symplectic form as soon as $X$ has one.
  To prove our results, we develop a theory of Tate objects in a stable
$(\infty,1)$-category $\mathcal C$. We also prove that the non-connective
K-theory of $\mathbf{Tate}(\mathcal C)$ is the suspension of that of $\mathcal
C$.
  The last chapter is aimed at a different problem: we study there the
existence of a Lie structure on the tangent of a derived Artin stack. This in
particular applies to not necessarily smooth schemes.
  Throughout this thesis, we will use the tools of $(\infty,1)$-categories and
symplectic derived algebraic geometry.
