Harmonic Measure and Geometric Function Theory Fundamentals

Harmonic measure is a fundamental concept bridging complex analysis, potential theory, and geometric measure theory. This text-based course offers an accessible entry point into the geometric structure of harmonic measures, helping you understand how analytic maps shape boundary distributions. You will build a solid intuition for conformal mappings from the unit disk, explore how boundary behavior reflects domain geometry, and work through rigorous definitions and properties with clarity. What you'll learn: Understand key definitions of harmonic measure and conformal mapping; Analyze the geometric properties of simply connected domains; Explore the boundary behavior of analytic functions and measure transport; Apply modern analytical techniques to investigate rectifiability and singularity on domain boundaries; Practice solving step-by-step theoretical problems in potential theory. The course begins with foundational terminology, measure theory concepts, and complex analysis prerequisites before moving into deeper geometric analysis and boundary estimates. Designed for students and self-learners with basic mathematical curiosity, this guide breaks down advanced concepts into digestible, self-paced reading. Start exploring the geometric foundations of harmonic measure today.