Conformal Mapping, Loewner Chains, and Hele-Shaw Flow

Understanding the mathematical behavior of moving boundaries and growing interfaces is essential across geometric function theory and theoretical physics. This text-based course offers a clear, structured path through the analytical foundations connecting fluid dynamics, complex analysis, and boundary evolution processes. You will build an intuitive and rigorous foundation in dynamic conformal mappings, moving step by step from basic principles to classic continuous models. You will explore how physical interface problems such as Hele-Shaw flow connect directly to complex differential equations and measure theory. What you'll learn: - Understand foundational complex analysis concepts, conformal maps, and boundary dynamics - Derive the Polubarinova-Galin equation and connect it to physical Hele-Shaw fluid interfaces - Study the Loewner differential equation, Loewner chains, and slit-mapping evolutions - Apply the Herglotz representation theorem to characterize driving measures and generating functions - Examine measure types including Dirac masses and absolutely continuous distributions - Explore regularization techniques and stochastic growth models such as Diffusion-Limited Aggregation The course begins with core terminology and underlying principles of geometric function theory before guiding you through analytical derivations, measure-theoretic connections, and modern interface dynamics. This reading-focused program is designed for students, researchers, and mathematically minded learners seeking an accessible entry point to dynamic Loewner theory without requiring previous specialist background in geometric flow. Start reading today to deepen your understanding of dynamic conformal systems and boundary evolutions.