Homological algebra of Feynman graphs

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The area of the proposed research is at the junction of several branches of pure mathematics and mathematical physics. It follows the pattern of applying the physical intuition and ideas to solving mathematical problems which has been a characteristic feature of many groundbreaking developments in algebra in geometry in the last two decades.The project has two closely related themes. One purports to link two deep algebraic constructions which have been extensively studied in their own right. The first is the calculation of the Chevalley-Eilenberg cohomology of infinite matrices with values in an algebra and the second is the calculation of the Chevalley-Eilenberg cohomology of the infinite-dimensional algebra of noncommutative hamiltonians. One conjectural application is the construction of an algebraic version of chain level Gromov-Witten invariants.The second theme derives its motivation from the general problem of quantizing field theories defined classically, i.e. in terms of an action functional. More precisely, a classical field theory is modelled as a certain algebraic structure, called L-infinity algebra which is determined by a certain polynomial or power series function, and one studies other algebraic structures derived by integrating this function. The application will include topological theories of Chern-Simons type and Poisson sigma-models.