Counting Surfaces

1. Front Matter

   Pages i-xvii

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2. Maps and Discrete Surfaces

   • Bertrand Eynard

   Pages 1-24

3. Formal Matrix Integrals

   • Bertrand Eynard

   Pages 25-51

4. Solution of Tutte-Loop Equations

   • Bertrand Eynard

   Pages 53-143

5. Multicut Case

   • Bertrand Eynard

   Pages 145-168

6. Counting Large Maps

   • Bertrand Eynard

   Pages 169-236

7. Counting Riemann Surfaces

   • Bertrand Eynard

   Pages 237-333

8. Topological Recursion and Symplectic Invariants

   • Bertrand Eynard

   Pages 335-363

9. Ising Model

   • Bertrand Eynard

   Pages 365-407

10. Back Matter

    Pages 409-414

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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. Mor

e 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 give

s 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.

• Algebraic geometry
• Combinatorics
• Field theory
• Integrability
• Matrix model
• Moduli Spaces
• Riemann surfaces
• String theory
• Topology
• Witten-Kontsevich Conjecture

“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 learning about topological recursion.” (Daniel D. Moskovich, Mathematical Reviews, February, 2017) 

“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 is an outstanding monograph of a recent research trend in surface theory.” (Gert Roepstorff, zbMATH 1338.81005, 2016)

• CEA Saclay Institut de Physique Théoriqu, Gif sur Yvette, France

  Bertrand Eynard

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