Combinatorics of Planar Maps and Tutte Equations

Understanding the mathematics of surfaces and random lattices begins with enumerative combinatorics and map theory. This course provides a structured guide to analyzing planar rooted maps, establishing generating functions, and applying Tutte's algebraic methods. You will start with core topological graph concepts and map definitions before advancing to rigorous equation setups and asymptotic analysis. What you will learn: Understand the structural definitions of planar maps, roots, marked faces, and boundary vertices; Formulate Tutte equations systematically for map enumeration problems; Derive and manipulate algebraic generating functions for discrete surfaces; Apply the quadratic method to solve functional equations with catalytic variables; Connect map enumeration to contemporary random geometry and matrix model concepts. The reading modules progress from fundamental topological definitions to analytic solutions and combinatorial applications. This guide is designed for beginners in algebraic and enumerative combinatorics with no prior advanced topology background required. Start reading to master the foundational tools of planar map enumeration.