Topological quantum field theories may be defined as appropriate functors > from the cobordism category. The classical > and perturbative versions thereof have started to be investigated only > recently. We discuss topological field theories in the AKSZ formalism (this includes Chern-Simons and BF theories as well as the Poisson sigma model). In this case, to boundary components (objects) one can associate a symplectic manifold and a coisotropic submanifold thereof, whereas to cobordisms one associates canonical relations. This very natural construction a special case with more structure than the general one recently introduced by V. Fock), yields to new viewpoints on, say, the moduli space of flat connections or the symplectic groupoid of a Poisson manifold. This also constitutes the starting point for the perturbative quantization of these theories. The possibility of including boundaries of boundaries (and so on) naturally yields to a Lurie-type description. Eventually, one may hope to be able to reconstruct perturbative topological field theories by gluing simple pieces. This is based on joint work in progress with P. Mnev and N. Reshetikhin.

Supergeometry a generalization of differential geometry where some coordinate are allowed to anticommute with each other. Most concepts and theorems for ordinary manifolds (like, e.g., vector fields and differential forms, Frobenius theorem, integration) have a generalization to supermanifolds. This allows one to use geometric intuition to understand many algebraic objects of current use (like differential forms or multivector fields). Another advantage is that many interesting geometric structures (like Poisson, Courant, generalized complex ) may be reformulated in terms of super symplectic geometry (with a refinement of the super grading). Reduction methods for the above structure may be unified in terms of super symplectic reduction. Using integration and map spaces (AKSZ method), one can associate to these structures topological fields theories, one example of which is the Poisson sigma model.