Homological algebra of Feynman graphs

拟议的研究领域处于纯数学和数学物理的几个分支的交界处。它遵循应用物理直觉和思想来解决数学问题的模式，这是过去20年来几何学中许多突破性发展的特征。该项目有两个密切相关的主题。人们声称要将两个已被广泛研究的深代数结构联系起来。第一个是计算具有代数中值的无限矩阵的Chvalley-Eilenberg上同调，第二个是计算无限维非对易哈密尔顿代数的Chvalley-Eilenberg上同调。一个猜想的应用是构造链级Gromov-Witten不变量的代数版本。第二个主题的动机来自于量子化经典定义的场论的一般问题，即根据动作泛函。更确切地说，经典场论被建模为由某个多项式或幂函数确定的某种代数结构，称为L无穷代数，并研究了通过积分该函数得到的其他代数结构。应用将包括Chern-Simons型和Poisson Sigma模型的拓扑理论。

The area of the proposed research is at the junction of several branches of pure mathematics and mathematical physics. It follows the pattern of applying the physical intuition and ideas to solving mathematical problems which has been a characteristic feature of many groundbreaking developments in algebra in geometry in the last two decades.The project has two closely related themes. One purports to link two deep algebraic constructions which have been extensively studied in their own right. The first is the calculation of the Chevalley-Eilenberg cohomology of infinite matrices with values in an algebra and the second is the calculation of the Chevalley-Eilenberg cohomology of the infinite-dimensional algebra of noncommutative hamiltonians. One conjectural application is the construction of an algebraic version of chain level Gromov-Witten invariants.The second theme derives its motivation from the general problem of quantizing field theories defined classically, i.e. in terms of an action functional. More precisely, a classical field theory is modelled as a certain algebraic structure, called L-infinity algebra which is determined by a certain polynomial or power series function, and one studies other algebraic structures derived by integrating this function. The application will include topological theories of Chern-Simons type and Poisson sigma-models.