Abstract:
String backgrounds are described as purely geometric objects related to moduli spaces of Riemann surfaces, in the spirit of Segal's definition of a conformal field theory. Relations with conformal field theory, topological field theory and topological gravity are studied. For each field theory, an algebraic counterpart, the (homotopy) algebra satisfied by the tree level correlators, is constructed.

Abstract:
The purpose of this paper is to suggest the construction and study properties of semi-infinite induction, which relates to semi-infinite cohomology the same way induction relates to homology and coinduction to cohomology. We prove a version of the Shapiro Lemma, relating the semi-infinite cohomology of a module with that of the semi-infinitely induced module. A practical outcome of our construction is a simple construction of Wakimoto modules, highest-weight modules used in double-sided BGG resolutions of irreducible modules.

Abstract:
These are notes of a mini-course given at Dennisfest in June 2001. The goal of these notes is to give a self-contained survey of deformation quantization, operad theory, and graph homology. Some new results related to "String Topology" and cacti are announced in Section 2.7.

Abstract:
We show that for g > 2k+2 the k-rational homotopy type of the moduli space M_{g,n} of algebraic curves of genus g with n punctures is independent of g, and the space M_{g,n} is k-formal. This implies the existence of a limiting rational homotopy type of M_{g,n} as g goes to infinity and the formality of it.

Abstract:
The purpose of this paper is to complete Getzler-Jones' proof of Deligne's Conjecture, thereby establishing an explicit relationship between the geometry of configurations of points in the plane and the Hochschild complex of an associative algebra. More concretely, it is shown that the B_infty-operad, which is generated by multilinear operations known to act on the Hochschild complex, is a quotient of a certain operad associated to the compactified configuration spaces. Different notions of homotopy Gerstenhaber algebras are discussed: one of them is a B_infty-algebra, another, called a homotopy G-algebra, is a particular case of a B_infty-algebra, the others, a G_infty-algebra, an E^1-bar-algebra, and a weak G_infty-algebra, arise from the geometry of configuration spaces. Corrections to the paper math.QA/9602009 of Kimura, Zuckerman, and the author related to the use of a nonextant notion of a homotopy Gerstenhaber algebra are made.

Abstract:
This paper extends Kontsevich's ideas on quantizing Poisson manifolds. A new differential is added to the Hodge decomposition of the Hochschild complex, so that it becomes a bicomplex, even more similar to the classical Hodge theory for complex manifolds.

Abstract:
We introduce a new operad, which we call the Swiss-cheese operad. It mixes naturally the little disks and the little intervals operads. The Swiss-cheese operad is related to the configuration spaces of points on the upper half-plane and points on the real line, considered by Kontsevich for the sake of deformation quantization. This relation is similar to the relation between the little disks operad and the configuration spaces of points on the plane. The Swiss-cheese operad may also be regarded as a finite-dimensional model of the moduli space of genus-zero Riemann surfaces appearing in the open-closed string theory studied recently by Zwiebach. We describe algebras over the homology of the Swiss-cheese operad.

Abstract:
This paper reemphasizes the ubiquitous role of moduli spaces of algebraic curves in associative algebra and algebraic topology. The main results are: (1) the space of an operad with multiplication is a homotopy G- (i.e., homotopy graded Poisson) algebra; (2) the singular cochain complex is naturally an operad; (3) the operad of decorated moduli spaces acts naturally on the de Rham complex $\Omega^\bullet X$ of a K\"{a}hler manifold $X$, thereby yielding the most general type of homotopy G-algebra structure on $\Omega^\bullet X$. One of the reasons to put this operadic nonsense on the physics bulletin board is that we use a typical construction of supersymmetric sigma-model, the construction of Gromov-Witten invariants in Kontsevich's version.

Abstract:
The purpose of this paper is to introduce the cohomology of various algebras over an operad of moduli spaces including the cohomology of conformal field theories (CFT's) and vertex operator algebras (VOA's). This cohomology theory produces a number of invariants of CFT's and VOA's, one of which is the space of their infinitesimal deformations.

Abstract:
The goal of this work is to describe a categorical formalism for (Extended) Topological Quantum Field Theories (TQFTs) and present them as functors from a suitable category of cobordisms with corners to a linear category, generalizing 2d open-closed TQFTs to higher dimensions. The approach is based on the notion of an n-fold category by C. Ehresmann, weakened in the spirit of monoidal categories (associators, interchangers, Mac Lane's pentagons and hexagons), in contrast with the simplicial (weak Kan and complete Segal) approach of Jacob Lurie. We show how different Topological Quantum Field Theories, such as gauge, Chern-Simons, Yang-Mills, WZW, Seiberg-Witten, Rozansky-Witten, and AKSZ theories, as well as sigma model, may be described as functors from the pseudo n-fold category of cobordisms to a pseudo n-fold category of sets.