Combinatorics of quantum field theory

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Quantum field theories provide the most precise descriptions known of the fundamental particles that make up the universe. Perturbative quantum field theory is the approach whereby quantum field theoretical functions are expanded as infinite sums of Feynman integrals, certain integrals indexed by graphs. To transform humanity's understanding of field theories we need new mathematical ideas to uncover hidden patterns within them. We need novel techniques to integrate Feynman integrals, an understanding of the structure of the divergent series that give the perturbative expansion, and a firm mathematical foundation for the powerful heuristic that is the path integral. Finally, we need to learn how to incorporate non-perturbative physics.This research program will use combinatorial tools to build these new mathematical ideas and thence gain a deeper understanding of quantum field theory. The aims are threefold, to better understand quantum field theory and hence better understand the fundamental building blocks of our universe, to give more rigorous foundations to quantum field theory, and to advance the mathematics of discrete structures.Three concrete objectives will be accomplished towards these aims: understanding the combinatorics of transseries and resurgence in order to understand concretely how the non-perturbative can be obtained from the perturbative, developing a general theory of chord diagram expansions in order to build a new framework for perturbative expansions in quantum field theory, and proving the completion conjecture for the c_2 invariant of Feynman graphs in order to more fully understand the connections between geometry and quantum field theory.The methods of the program are mathematical.The research approach is highly innovative; no other researcher in the world combines the areas and approaches in this way or attacks problems with this combination of tools -- algebraic combinatorics, graph theory, quantum physics, algebraic geometry, resurgence theory and differential equations, and number theory. The proposed work is highly significant and of exceptional quality as it both answers deep questions in physics -- why do we have the fundamental particles that we have, why does matter stick together rather than fall apart, what exists beyond the particles we know -- and also builds new fundamental knowledge in pure mathematics.