Titlebook: Asymptotic Combinatorics with Application to Mathematical Physics; Vadim Malyshev,Anatoly Vershik Book 2002 Kluwer Academic Publishers 200

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Combinatorics and Probability of MapsWe give an introductory pedagogical review of rigorous mathematical results concerning combinatorial enumeration and probability distributions for maps on compact orientable surfaces, with an emphasize on applications to two-dimensional quantum gravity.

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Statistical Mechanics and Number TheoryRiemann zeta function is an important object of number theory. It was also used for description of disordered systems in statistical mechanics. We show that Riemann zeta function is also useful for the description of correlation functions in quantum integrable models.

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Quantization of Thermodynamics and the Bardeen-Cooper-Schriffer-Bogolyubov EquationNew quantization methods for quantum problems of many-bodies are presented. These methods are used for quantization of thermodynamics of Quantum Statistics. In particular, this leads to an explanation of the superconductivity and superfluidity for nonzero temperatures.

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Theta Hypergeometric SeriesWe formulate general principles of building hypergeometric type series from the Jacobi theta functions that generalize the plain and basic hypergeometric series. Single and multivariable elliptic hypergeometric series are considered in detail. A characterization theorem for a single variable totally elliptic hypergeometric series is proved.

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