Modular operads and topological field theories

这是一项纯数学的研究，它处于几个数学学科和理论物理的交界处。它涉及拓扑学和代数几何的某些方面，它使用了其他数学和数学物理领域的方法，如表示论、量子场论和组合学。费曼图最初出现在理论物理学家研究微扰量子场论中出现的路径积分的工作中。在过去的10-15年里，由于康采维奇和威腾等人的努力，费曼图可以很好地用于解决纯数学中的问题。所提出的研究的核心概念是模运算符，其基本结构由费曼图的组合学决定。这个算符的概念最初出现在代数拓扑学中，但最近在数学物理中得到了重要的应用。模运算符上的代数是具有相容内积的结合代数概念的抽象；这样的代数自然出现在各种物理理论中。了解这些结构可能会为曲线的模空间的交集理论带来新的曙光，这是现代代数几何中最重要和最深刻的问题之一。该项目将在莱斯特大学进行。它需要博士后研究员和来自英国和国外其他大学的专家的协助。

This is a research in pure mathematics which is at the junction of several mathematical disciplines and theoretical physics. It concerns certain aspects of topology and algebraic geometry which uses methods of other areas of mathematics and mathematical physics such as representation theory, quantum field theory and combinatorics.Feynman graphs originally appeared in the work of theoretical physicists studying path integrals appearing in perturbative quantum field theory. During the last 10-15 years, due to efforts of such people as Kontsevich and Witten it became clear thatFeynman diagrams could be used to great effect to solve problems in pure mathematics.The notion central to the proposed research is that of a modular operad, whose underlying structure is governed by the combinatorics of Feynman graphs. This notion of an operad originally appeared in algebraic topology but recently found important applications in mathematical physics. An algebra over a modular operad is an abstraction of the notion of an associative algebra with a compatible inner product; such algebras appear naturally in various physical theories. Understanding these structures will likely shed new light on the intersection theory on the moduli spaces of curves, one of the most important and deep problems of modern algebraic geometry. This project will be carried out at the University of Leicester. It requires assistance of a post-doctoral fellow and experts from other universities in the UK and abroad.