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In recent years, statistical mechanics has been increasingly recognized as a central domain of mathematics. Major developments include the Schramm - Loewner evolution, which describes two-dimensional phase transitions, random matrix theory, renormalization group theory and the fluctuations of random surfaces described by dimers. The lectures contained in this volume present an introduction to recent mathematical progress in these fields. They are designed for graduate students in mathematics with a strong background in analysis and probability. This book will be of particular interest to graduate students and researchers interested in modern aspects of probability, conformal field theory, percolation, random matrices and stochastic differential equations.