Topological quantum field theory and algebraic structures

These notes are from lectures given at the Quantum field theory and noncommutative geometry workshop at Tohoku University in Sendai, Japan from November 24-30, 2002. We give a survey of operads and their relationship to topological quantum field theories (TQFT). We give simple examples of operads, particularly those arising as moduli spaces of decorated oriented 2-spheres, describe the notion of algebras over them, and we study their "higher loop" generalizations. We then focus upon the example of the moduli space of stable curves and its relationship to cohomological field theories, in the sense of Kontsevich-Manin. The latter can be regarded as a generalization of a TQFT which is relevant to quantum cohomology and to higher KdV integrable hierarchies.