Introduction to Topological Recursion and Random Maps

Topological recursion and random map theory provide a powerful mathematical bridge connecting enumerative geometry, statistical mechanics, and modern field theory. This text-based course breaks down these theoretical concepts into clear, accessible written explanations designed for self-paced study. You will build a solid mathematical foundation in the enumeration of planar surfaces, explore how statistical models like the Ising and Potts models behave on random lattices, and understand the scaling limits that lead to continuous geometries. What you'll learn: - Understand core definitions of planar maps, ribbon graphs, and surface topologies - Apply topological recursion techniques to set up and solve loop equations - Analyze lattice statistical mechanics models including the Ising and Potts models on random maps - Explore large-scale continuum limits and their connection to modern 2D quantum gravity - Learn the conceptual links between matrix model asymptotics and Liouville conformal field theory The course begins with foundational vocabulary and combinatorial basics before walking you systematically through spectral curve properties, recursive relations, and scaling limits. This guide is suitable for curious students, researchers, and mathematics enthusiasts seeking a structured and readable entry point into random geometry and topological recursion. Start reading today to master the foundations of topological recursion and random planar maps.