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"Knowledge of the renormalization group and field theory is a key part of physics, and is essential in condensed matter and particle physics. Written for advanced undergraduate and beginning graduate students, this textbook provides a concise introduction to this subject. The textbook deals directly with the loop-expansion of the free-energy, also known as the background field method. This is a powerful method, especially when dealing with symmetries, and statistical mechanics. In focussing on free-energy, the author avoids long developments on field theory techniques. The necessity of renormalization then follows"--

Phase transformations (Statistical physics) --- Field theory (Physics) --- Statistical mechanics. --- Field theory (Physics). --- Phase transformations (Statistical physics).

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Topological groups. Lie groups

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French language --- Proverbs. Phrases --- English language --- Proverbs, French --- Proverbs --- Proverbes français --- Proverbes --- Français (Langue) --- Anglais (Langue) --- Dictionaries --- English --- French --- Dictionnaires anglais --- Dictionnaires français --- Proverbes français --- Français (Langue) --- Dictionnaires français

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This is a first book to show that the theory of the Gaussian random matrix is essential to understand the universal correlations with random fluctuations and to demonstrate that it is useful to evaluate topological universal quantities. We consider Gaussian random matrix models in the presence of a deterministic matrix source. In such models the correlation functions are known exactly for an arbitrary source and for any size of the matrices. The freedom given by the external source allows for various tunings to different classes of universality. The main interest is to use this freedom to compute various topological invariants for surfaces such as the intersection numbers for curves drawn on a surface of given genus with marked points, Euler characteristics, and the Gromov–Witten invariants. A remarkable duality for the average of characteristic polynomials is essential for obtaining such topological invariants. The analysis is extended to nonorientable surfaces and to surfaces with boundaries.

Random matrices. --- Matrices, Random --- Matrices --- Statistical physics. --- Topological Groups. --- Mathematical Physics. --- Statistical Physics and Dynamical Systems. --- Topological Groups, Lie Groups. --- Particle and Nuclear Physics. --- Complex Systems. --- Groups, Topological --- Continuous groups --- Physics --- Mathematical statistics --- Statistical methods --- Mathematical physics. --- Topological groups. --- Lie groups. --- Nuclear physics. --- Dynamical systems. --- Dynamical systems --- Kinetics --- Mathematics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Statics --- Atomic nuclei --- Atoms, Nuclei of --- Nucleus of the atom --- Groups, Lie --- Lie algebras --- Symmetric spaces --- Topological groups --- Physical mathematics