Maurer-Cartan moduli and homotopy theory

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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.