Combinatorial structures in quantum field theory

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Quantum field theories (QFTs) give the most precise descriptions known of the fundamental particles which make up the universe. Perturbative QFT is the approach whereby field theoretical quantities are expanded as infinite sums of Feynman integrals. However, each individual integral is difficult; there are many that we have no way to evaluate, even numerically. Furthermore, the infinite series of Feynman integrals are typically divergent. To extract meaning from them we need to understand the recursive structure of perturbation theory. The overarching goal of my research program is to understand these matters using a combinatorial aesthetic and toolbox, and to develop new combinatorial and physical tools that lead to new insights. There are two large scale objectives within this overarching goal. First is to understand patterns in the values of Feynman graphs using properties of the graphs themselves. Second is to understand the recursive structure of QFT, particularly the Dyson-Schwinger equations (the quantum equations of motion), so as to rigorously and effectively resum graphs using combinatorial insights. These large scale objectives will be approached via a number of parallel short term objectives with the participation of students and collaborators. I have an excellent track record in this area and have trained 18 students who have gone on both in industry and academia. There remain many accessible yet fundamental questions that we will tackle in the coming years in order to advance the state of the art in both discrete mathematics and mathematical physics. Students will benefit by learning and developing new mathematics and working interdisciplinarily. Success in this program will produce both exciting mathematical results arising from the physics and useful physical results achieved using mathematics. The beauty and power of the program come from the two-way interaction between physics and pure mathematics. Achieving the first objective would shortcut the intricate computation of Feynman integrals and thus reveal currently inaccessible consequences of physical theories, including those which describe our world. Partial progress will help us understand the structure of QFT, and hence of our world, and also enrich our understanding of related mathematical objects, such as multiple zeta values, as well as feeding back into graph theory with new problems and new results. Achieving the second objective would give a rigorous underpinning to perturbative QFT and a rigorous link to the non-perturbative world. Partial results let us obtain better approximations to physically meaningful quantities and functions, and show us qualitative properties of our systems, as well as yielding rich combinatorial problems. The time is right for this program because related areas, such as resurgence theory, are maturing to the point that they are ready to be woven in, increasing the power of my program to answer fundamental questions.