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# Counting surfaces : CRM Aisenstadt chair lectures

Author: Bertrand Eynard Switzerland : Springer, [2016] ©2016 Progress in mathematical physics, v. 70. eBook : Document : EnglishView all editions and formats The problem of enumerating maps (a map is a set of polygonal "countries" on a world of a certain topology, not necessarily the plane or the sphere) is an important problem in mathematics and physics, and it has many applications ranging from statistical physics, geometry, particle physics, telecommunications, biology ... etc. This problem has been studied by many communities of researchers, mostly combinatorists, probabilists, and physicists. Since 1978, physicists have invented a method called "matrix models" to address that problem, and many results have been obtained. Besides, another important problem in mathematics and physics (in particular string theory), is to count Riemann surfaces. Riemann surfaces of a given topology are parametrized by a finite number of real parameters (called moduli), and the moduli space is a finite dimensional compact manifold or orbifold of complicated topology. The number of Riemann surfaces is the volume of that moduli space. More generally, an important problem in algebraic geometry is to characterize the moduli spaces, by computing not only their volumes, but also other characteristic numbers called intersection numbers. Witten's conjecture (which was first proved by Kontsevich), was the assertion that Riemann surfaces can be obtained as limits of polygonal surfaces (maps), made of a very large number of very small polygons. In other words, the number of maps in a certain limit, should give the intersection numbers of moduli spaces. In this book, we show how that limit takes place. The goal of this book is to explain the "matrix model" method, to show the main results obtained with it, and to compare it with methods used in combinatorics (bijective proofs, Tutte's equations), or algebraic geometry (Mirzakhani's recursions). The book intends to be self-contained and accessible to graduate students, and provides comprehensive proofs, several examples, and gives the general formula for the enumeration of maps on surfaces of any topology. In the end, the link with more general topics such as algebraic geometry, string theory, is discussed, and in particular a proof of the Witten-Kontsevich conjecture is provided.  Read more... (not yet rated) 0 with reviews - Be the first.

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Genre/Form: Electronic books Printed edition: Document, Internet resource Internet Resource, Computer File Bertrand Eynard Find more information about: Bertrand Eynard 9783764387976 3764387971 3764387963 9783764387969 945445120 1 online resource. I Maps and discrete surfaces -- II Formal matrix integrals -- III Solution of Tutte-loop equations -- IV Multicut case -- V Counting large maps -- VI Counting Riemann surfaces -- VII Topological recursion and symplectic invariants -- VIII Ising model -- Index -- Bibliography. Progress in mathematical physics, v. 70. Bertrand Eynard.

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This book explains the "matrix model" method developed by physicists to address the problem of enumerating maps and compares it with other methods. It includes proofs, examples and a general formula  Read more...

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"This book brings together details of topological recursion from many different papers and organizes them in an accessible way. ... this book will be an invaluable resource for mathematicians Read more...

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