Combinatorics of quantum field theory

量子场论为构成宇宙的基本粒子提供了已知最精确的描述。微扰量子场论是一种将量子场理论函数展开为费曼积分的无限和的方法，费曼积分是由图索引的某些积分。为了改变人类对场论的理解，我们需要新的数学思想来揭示其中隐藏的模式。我们需要新的技术来积分Feynman积分，了解给出微扰展开的发散级数的结构，并为强大的启发式(即路径积分)奠定坚实的数学基础。最后，我们需要学习如何结合非微扰物理。这个研究项目将使用组合工具来建立这些新的数学思想，从而对量子场论有更深的理解。目标有三个，为了更好地理解量子场论，从而更好地理解我们宇宙的基本组成部分，为量子场论提供更严格的基础，并推进离散结构的数学。朝着这些目标将实现三个具体目标：理解跨级数和复现的组合学，以便具体理解如何从微扰中获得非微扰，发展弦图展开的一般理论，以便建立量子场论中微扰展开的新框架，证明了Feynman图的c_2不变量的完备性猜想，以便更全面地理解几何与量子场论之间的联系。世界上没有其他研究人员以这种方式将各个领域和方法结合在一起，或者用这种工具的组合来攻击问题--代数组合学、图论、量子物理、代数几何、复苏理论和微分方程，以及数论。这项拟议的工作意义重大，质量出众，因为它既回答了物理学中的深层次问题--为什么我们有基本粒子，为什么物质粘在一起而不是分裂，除了我们已知的粒子之外还有什么存在--也在纯数学中建立了新的基本知识。

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.