Modular operads and topological field theories

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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.