This book explores conformal mappings in the complex plane, linking them to fractals and chaos. It offers a thorough study of the Mandelbrot set and aims to provide students with organized material and proofs on complex dynamics. The revised notes are based on lectures from UCLA, enhanced by contributions from colleagues and students.
Targeted at upper division undergraduates and PhD candidates, this introduction to complex analysis covers essential topics like Julia and Mandelbrot sets, Dirichlet series, and the uniformization theorem for Riemann surfaces. The text emphasizes the three geometries: spherical, Euclidean, and hyperbolic. With sixteen chapters, it includes a range of exercises from simple to challenging, ensuring comprehensive preparation for qualifying exams. The material is drawn from the author's extensive teaching experience at various prestigious universities worldwide.
Focusing on the intersection of functional analysis and function theory, these lecture notes explore contemporary research topics in this dynamic field. They aim to provide insights into advanced concepts and methodologies, making them a valuable resource for scholars and students interested in the latest developments and applications in mathematical analysis.