New algebraic structures inspired by quantum field theory

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Quantum field theory (QFT) is the conceptual basis of particle physics and has provided a challenge and an inspiration for mathematicians for more than 6 decades and yet, despite constant efforts, its mathematical foundations are not completely understood. On the other hand, formal computations done in particle physics agree with experiment to an enormous accuracy. This leaves mathematicians with a great challenge and the current proposal is a part of a larger research program, meant to address it.The aim of this project is to investigate certain algebraic structures that recently were postulated as a robust conceptual basis for QFT and to use the examples provided by QFT models to prove new results concerning these structures. Subsequently, these new mathematical tools will be used to understand the formal computations done in physics (with the use of perturbation theory). The long term goal is to get enough control on the algebraic structures in question to enable one to go beyond perturbation theory and obtain rigorous results.