Combinatorial approaches to quantum field theory

量子场论以非常高的精度描述了我们的宇宙，覆盖了一个巨大的数量级范围。它也充满了美丽的图案和结构。这项研究计划的目标是通过加深我们对费曼图、费曼积分及其组合成有物理意义的量的基本模式的组合理解来增加对量子场论的理解。**尤其是，这一建议首先通过费曼积分给出图和超验值之间的联系的图论理解，其次理解费曼图递归分解为子图的过程，以及这种分解的解析版本，即戴森-施温格方程。**这项工作的结果将是更好地理解构成我们宇宙的基本理论以及丰富和美丽的数学对象和结果。这项提议将包括对各级学生进行纯粹技能和应用技能的培训。这项工作是协作的，增强了我与其他研究人员的优势，以产生创新的结果。**这项工作作为纯数学和数学物理都具有价值，并使加拿大在量子场论的这一前沿方法中占有一席之地。

Quantum field theory describes our universe over an enormous range of orders of magnitude and at very high precision. It is also full of beautiful patterns and structures. The goal of this research program is increase understanding of quantum field theory by furthering our combinatorial understanding of the underlying patterns in Feynman graphs, Feynman integrals, and their combinations into physically meaningful quantities.**In particular, this proposal seeks first to give a graph theoretical understanding of the connection between graphs and transcendental values through their Feynman integrals, and second to understand the recursive decomposition of Feynman graphs into subgraphs along with the analytic version of this decomposition, the Dyson-Schwinger equations.**The outcome of this work will be a better understanding of the fundamental theories which make up our universe as well as rich and beautiful mathematical objects and results. This proposal will incorporate the training of students at all levels in both pure and applied skills. The work is collaborative, enhancing my own strengths with those of other researchers, to yield innovative results.**This work is of value both as pure mathematics and as mathematical physics, and gives Canada a place in this cutting edge approach to quantum field theory.