Algebraic Operads: An Algorithmic Companion presents a systematic treatment of Groebner bases in several contexts. The book builds up to the theory of Groebner bases for operads due to the second author and Khoroshkin as well as various applications of the corresponding diamond lemmas in algebra. The authors present a variety of topics including: noncommutative Groebner bases and their applications to the construction of universal enveloping algebras; Groebner bases for shuffle algebras which can be used to solve questions about combinatorics of permutations; and operadic Groebner bases, important for applications to algebraic topology, and homological and homotopical algebra. The last chapters of the book combine classical commutative Groebner bases with operadic ones to approach some classification problems for operads. Throughout the book, both the mathematical theory and computational methods are emphasized and numerous algorithms, examples, and exercises are provided to clarify and illustrate the concrete meaning of abstract theory.

Fractal Structure in 4d Gravity.- A One Dimensional Ideal Gas of Spinons, or Some Exact Results on the XXX Spin Chain with Long Range Interaction.- Kodaira-Spencer Theory of Gravity.- 3d Gravity and Gauge Theories.- On the W-Gravity Spectrum and its G-Structure.- Light-Cone Quantization of Matrix Models at c>1.- Multicritical Points of 2-Matrix Models.- The Super Self-Dual Matreoshka.- The Phenomenology of Strings and Clusters in the 3-d Ising Model.- Conformai Field Theory Techniques in Large N Yang-Mills Theory.- to Differential W-Geometry.- Topological Strings and QCD in Two Dimensions.- Continuum QCD2 in Terms of Discrete Random Surfaces with Local Weights.- Strings and Causality.- Loop Equation and Area Law in Turbulence.- The Two-Dimensional String as a Topological Field Theory.- Linear Systems for 2d Poincare Supergravities.- Quantization of Mirror Symmetry.- Integrable Qft2 Encoded on Products of Dynkin Diagrams.- Remarks on Topological String Theories.- Hamiltonian Reduction of the BRST Complex and N=2 SUSY.- Lattice Models and N=2 Supersymmetry.- Canonical Construction of Liouville Field Operators with Arbitrary Spin.- Bethe Ansatz for the Bloch Particle in Magnetic Field.

The Cargese Workshop "Quantum Field Theory and String Theory" was held from May 10 to May 21, 1993. The broad spectrum of the work presented at the Workshop was the reflec tion of a time of intensive search for new ways of solving some of the most fun damental problems in string theory, quantum gravity and non-perturbative field theory. A number of talks indicated the emergence of new promising domains of investigation. It is this very diversity of topics which, in our opinion, represents one of the most attractive features of the present volume which we hope will provide a good orientation in the abundant flow of ideas and publications in modern quantum field theory. Many contributions to the present proceedings are concerned with two di mensional quantum field theory. The continuous advances in the domain of two dimensional integrable theories on the lattice as well as in the continuum, including conformal field theories, Liouville field theory and matrix models of two dimensional quantum gravity are very well represented. Other papers address physically realistic (and therefore very complicated) problems like de veloped turbulence, the Hofstadter problem, higher dimensional gravity and phenomenological strings. A new elegant class of topological field theories is presented. New ideas in the string representation of multicolor quantum chromo dynamics were widely discussed at the Workshop, more particularly the example of the exactly solvable two dimensional case.