Advances in Complexity Theory Stephen Cook (University of Toronto), Arvind Gupta (Simon Fraser University), Russell Impagliazzo (University of California, San Diego), Valentine Kabanets (Simon Fraser University), Madhu Sudan (M.I.T.), Avi Wigderson (Institute for Advanced Study, Princeton) July 4–8, 2004 Computational Complexity Theory is the field that studies the efficiency of computation. Its major goals are to find efficient algorithms for natural problems in natural computational models, or to show that no efficient solutions exist. The famed ”P versus NP” problem (one of the seven open problems of the Clay Institute) is the central problem of this field. In the last two decades, our understanding of efficient computation has improved significantly through a number of concepts, techniques and results, including: • Discovery of efficient ways of converting computational hardness into computational random- ness (hardness-randomness tradeoffs), and other techniques for eliminating or reducing ran- domness use in probabilistic algorithms. • Classification of hardness of approximation algorithms for a number of optimization problems, using the concept of Probabilistically Checkable Proofs (PCP). • Connections of both items above to old and new problems in coding and information theory, which fertilized both fields. • Investigations of the complexity of proofs, and their connections to limits on circuit lower bounds on the one hand, and to the complexity of search heuristics on the other. • Use of quantum computation to get efficient algorithms for classically difficult problems (such as factoring), as well as using quantum arguments to obtain complexity results in the classical model of computation. Many new developments in these areas were presented by the participants of the workshop. These new results will be described in the following sections of this report, grouped by topic. For each topic, we give a brief summary of the presented results, followed by the abstracts of the talks. 1 Probabilistically Checkable Proofs The area of Probabilistically Checkable Proofs (PCPs) and Hardness of Approximation continues to be one of the most active research directions in complexity. The talk by Irit Dinur discussed how to make the original algebraic proof of the PCP Theorem [AS98, ALM + 98] more combinatorial (and 1