We present an alternative approach to higher derived homotopy structures induced by the ’basic perturbation lemma’. This approach is motivated by physical considerations and makes use of operads and their representations. As an application we prove that the BVF-complex controls the formal deformations of coisotropic submanifolds – at least locally or under assumptions on the topology of the coi...

Operads and PROPs are presented, together with examples and applications to quantum physics suggesting the structure of Feynman categories/PROPs and the corresponding algebras.

It is clarified how cohomologies and Gerstenhaber algebras can be associated with linear pre-operads (comp algebras). Their relation to mechanics and operadic physics is concisely discussed.

Physicists have developed very advanced methods to compute such integrals. For example, the famous Feynman diagram technique allows us to compute the above integral for a rather particular function S(x), a small perturbation of a positive de nite quadratic form in R . Sometimes, elaborate techniques of physicists to compute simple things nd an amazing application to very general things in mathe...