Maurer-Cartan moduli and homotopy theory

建议的研究领域包括同调代数和拓扑学的那些方面，它们使用了其他数学领域的方法，如代数和微分几何，以及来自理论物理的想法和直觉。毛雷尔-卡坦元和毛雷尔-卡坦方程(在物理学中也称为主方程)的概念是经典的，实际上，可以追溯到100多年前毛雷尔和卡坦写的经典论文；这是微分几何中的标准工具。人们已经注意到，毛雷尔-卡尔坦结构出现在数学和理论物理的各个分支中，从特征层理到保形场理论。许多作者从不同的角度考虑了这一概念，但到目前为止还没有一个普遍的处理方法。本文的目的是从有理同伦理论的角度为Maurer-Cartan元及其模空间理论奠定基础。所研制的仪器将应用于形变理论和图形同调。

The area of the proposed research comprises those aspects of homological algebra and topology which use methods from other areas of mathematics such as algebraic and differential geometry as well as ideas and intuition from theoretical physics.The notions of a Maurer-Cartan element an Maurer-Cartan equation (also known in physics as a master equation) are classical and indeed, go back to the classical papers of Maurer and Cartan written over 100 years ago; this is a standard tool in differential geomtery. It has been noticed that the Maurer-Cartan structures appear in various branches of mathematics and theoretical physics, from characteristic classes of foliations to conformal field theory. This notion has been considered by many authors from different points of view but until now a general treatment has been lacking.The present proposal intends to lay the foundations for the theory of Maurer-Cartan elements and their moduli spaces from the standpoint of rational homotopy theory. The developed apparatus will be applied to deformation theory and graph homology.