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Counting surfaces. CRM Aisenstadt chair lectures. (English) Zbl 1338.81005

Progress in Mathematical Physics 70. Basel: Birkhäuser (ISBN 978-3-7643-8796-9/hbk; 978-3-7643-8797-6/ebook). xvii, 414 p. (2016).
In 1963 the Canadian mathematician William T. Tutte discovered a combinatoric recursive equation about counting the number of triangulations of the sphere. In 1974 Gerald t’Hooft showed that matrix integrals are naturally related to graphs drawn on surfaces. In his book, the author explains how matrix models and counting surfaces are related and aims at presenting to mathematicians and physicists the random matrix approach to quantum gravity. The book has eight chapters. Chapter 1 introduces discrete surfaces and generating functions for counting maps. Chapter 2 studies formal matrix integrals which turn out to be identical with the generating functions of maps. Chapter 3 provides solutions of Tutte’s equation. The goal of Chapter 4 is to discuss multicut examples in order to a better understanding of the 1-cut cases. In Chapter 5, the author argues that in quantum gravity surfaces with small polygons may very well approximate Riemann surfaces. Next, in Chapter 6, generating functions are constructed for directly counting Riemann surfaces. In physics, this approach is called topological gravity. Chapter 7 studies symplectic invariants of spectral curves and provides the solution to Tutte’s recursion equation for maps, while in Chapter 8 the goal is to put the Ising model of statistical physics on a random map. All amplitudes can now be computed using the topological recursion of Chapter 7. The book is an outstanding monograph of a recent research trend in surface theory.

MSC:

81-02 Research exposition (monographs, survey articles) pertaining to quantum theory
81V17 Gravitational interaction in quantum theory
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
81T99 Quantum field theory; related classical field theories
15B52 Random matrices (algebraic aspects)
83C45 Quantization of the gravitational field
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
32B25 Triangulation and topological properties of semi-analytic andsubanalytic sets, and related questions
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