Instructor: Fabrizio Bianchi E-mail: fbianchi at imperial.ac.uk Office: Huxley 614 Office hours: Thursday 11-12. Feel free to write me a mail to arrange a different appointment if this time is not convenient for you. Student representative for the course: tba (tab at ic.ac.uk) Lectures Tuesday 17:00-18:00, room Huxley 139 Wednesday 11:00-12:00, room Huxley 342 or 658 Thursday 9:00-10:00, room Huxley 140 Always check on Celcat the correct room! Description of the course Complex analysis is the study of the functions of complex numbers. It is employed in a wide range of topics, including dynamical systems, algebraic geometry, number theory, and quantum field theory, to name a few. On the other hand, as the separate real and imaginary parts of any analytic function satisfy the Laplace equation, complex analysis is widely employed in the study of two-dimensional problems in physics such as hydrodynamics, thermodynamics, Ferromagnetism, and percolations. While you become familiar with basics of functions of a complex variable in the complex analysis course, here we look at the subject from a more geometric viewpoint. We shall look at geometric notions associated with domains in the plane and their boundaries, and how they are transformed under holomorphic mappings. In turn, the behaviour of conformal maps is highly dependent on the shape of their domain of definition. Below is a rough guide to the syllabus. Topics we cover in the course Schwarz lemma, authomorphisms of the disk and the half plane, Riemann sphere and rational functions, normal families, Riemann mapping theorem, Schlicht mappings, growth and distortion estimates, complex dilatations, absolute continuity on lines, quasi-conformal mappings, Beltrami equation, measurable Riemann mapping theorem. Prerequisites It will be assumed that students have had a previous course in complex analysis (such as M2PM3, or a course similar to that). However, this course is not a straight continuation of M2PM3; we shall revisit the basic definitions from geometric point of view, and build up the course from there. If you need more information on whether the course is appropriate for you, feel free to write me an email, or stop by my office. Lecture notes We will basically follow the lecture notes prepared by Davoud Cheraghi for the class of last year. Chapter 1: Preliminaries from Complex Analysis Chapter 2: Schwarz Lemma and automorphisms of the disk Chapter 3: Riemann sphere and rational maps Chapter 4: Conformal geometry on the disk Chapter 5: Conformal mappings Chapter 6: Growth and distortion estimates Chapter 7: Quasi-conformal maps and Beltrami equation Hints and solutions to the exercises (including homeworks) The lectures are recorded and available on Panopto. Assessment You will be required to hand in two sets of homeworks. These will form 10 percent of the final mark for the module. The first problem sheet is posted here on Thursday (Jan 29). The due date is February 15. Please submit your work by 16:00 on February 15 to the Undergraduate office on floor 6 in the Huxley building. The second problem sheet is posted here on Thursday (Mar 1). The due date is March 15. Please submit your work by 16:00 on March 15 to the Undergraduate office on floor 6 in the Huxley building. The remaining 90 percent is determined by a two-hour written examination at the end of the year. Further exercises from last year Mock exam and solutions Final exam and solutions References for background/further reading Complex analysis, Lars V Ahlfors Complex analysis, Elias M. Stein and Rami Shakarchi Functions of one complex variable, John B. Conway Complex analysis, the geometric viewpoint, Steven G Krantz Univalent functions, Peter L. Duren Potential theory in modern function theory, M. Tsuji Harmonic measure, Garnet and Marshal Elliptic partial differential equation and quasi-conformal mappings, Astala, Iwaniec, Martin Quasi-conformal mappings in the plane, O. Lehto and K. Virtanen