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Instructor: Matt Kerr Office: Cupples I, Room 114 e-mail: matkerr [at] math.wustl.edu Office Hours: Tu/Th 10-11, and by appointment Course Outline: I. Riemann mapping theorem and the Dirichlet problem II. Elliptic functions and some number theory III. Additional topics in one complex variable IV. Introduction to several complex variables This is the second half of a year-long course which forms the basis for the Ph.D. qualifying examination in complex variables. This semester we will cover the remainder of Ahlfors (3rd edition), including elliptic functions and the Riemann mapping theorem, as well as the big Picard theorem and prime number theorem. Time permitting, we will then give an introduction to modular forms, followed by several complex variables and the Bergman kernel. Prerequisites: Math 5021 or permission of instructor. Class Schedule: Lectures are on Tuesday and Thursday from 2:30-4 in Duncker Hall Room 101. First class is Tuesday Jan. 17 and last class is Thursday April 27, with two days off for spring break (Tues/Thurs March 14/16). Midterm (take-home) Exam: distributed April 4 in class; collected April 6 in class Final (Qualifying, 3 hour) Exam: Tuesday, May 9, 2:30-5:30, in Cupples I Rm. 199 The midterm will cover approximately the first two units above. The qualifying exam will cover material from Math 5021 and Math 5022, and everyone must take it at the time/date listed above. Assignments: Posted here each Thursday, collected 1 week later (at beginning of class on Thursday), and returned the following Tuesday. (If you prefer, you may also turn it in to Xiaoyu Dai's mailbox in the department mailroom by 4 PM on Friday.) Solutions will also be posted and will include students' work. Please feel free to come to office hours to discuss problem sets. Make sure you can do the problems I don't assign. Problem Set 1 / (solutions) Problem Set 2 / (solutions) Problem Set 3 / (solutions) Problem Set 4 / (solutions) Problem Set 5 / (solutions) Problem Set 6 / (solutions) Problem Set 7 / (solutions) Problem Set 8 / (solutions) Problem Set 9 / (solutions) Problem Set 10 / (solutions) Problem Set 11: not to hand in PROBLEM SESSION PROBLEMS Grader: Xiaoyu Dai Office: Cupples I, Room 213 e-mail: xiaoyu.dai [at] wustl.edu Lecture Notes: Will be scanned and posted here as I write them. The hope is that this makes taking notes optional. Here for your convenience is a link to the notes from last term. After some lectures I'll post "addenda" which are related (expository) journal articles, on history or recent developments. Some of these links may only be accessible on campus; if you can't access one (for free) at all, let me know. Lecture 1: Normal families Lecture 2: Riemann mapping theorem Lecture 3: Extension to the boundary Lecture 4: Explicit conformal mappings (addendum) Lecture 5: Harmonic functions revisited (addendum) Lecture 6: Subharmonic functions Lecture 7: The Dirichlet problem Lecture 8: Multiply connected regions Lecture 9: The Gamma function Lecture 10: Gamma and zeta Lecture 11: More on zeta functions Lecture 12: The prime number theorem Lecture 13: An application of the PNT (addendum) Lecture 14: Elliptic functions I Lecture 15: Elliptic functions II Lecture 16: Addition theorems and modular forms Lecture 17: More on modular forms Lecture 18: The Picard theorems Lecture 19: The Bloch and Landau constants Lecture 20: Schlict functions I Lecture 21: Schlict functions II (addendum) Lecture 22: Green's functions on Riemann surfaces Lecture 23: The Uniformization Theorem Lecture 24: Several complex variables Lecture 25: Poincare and Hartogs Lecture 26: The Bergman kernel Books: Lars Ahlfors, Complex Analysis (3rd Ed.); McGraw-Hill is the recommended textbook, which means I will follow it at least half the time and some of the problems I assign will come from it. Buy, check out, or borrow a copy. If you would like to read more adventurously than Ahlfors and/or my lecture notes, here are some suggestions. First, there are many other excellent standard texts, including John B. Conway, Functions of One Complex Variable; Springer Robert Greene and Steven Krantz, Function Theory of One Complex Variable; AMS and the second half of Walter Rudin, Real and Complex Analysis (3rd Ed.); McGraw-Hill. For a point of view based in formal and convergent power series (convenient for locally computing composition inverses and solutions of differential equations) you can consult Henri Cartan, Elementary Theory of Analytic Functions of One and Several Complex Variables; Addison-Wesley Serge Lang, Complex Analysis (3rd Ed.); Springer. For a view toward several complex variables there is Raghavan Narasimhan and Yves Nievergelt, Complex Analysis in One Variable; Birkhauser ; and the beautiful expository monograph Steven Krantz, Complex Analysis: the Geometric Viewpoint; MAA treats theorems in complex analysis through the prism of differential geometry. Finally, Harvey Cohn, Conformal Mapping on Riemann Surfaces; Dover leads (with lots of beautiful pictures and physical intuition) into Riemann surfaces and complex algebraic geometry. I will place one copy of each of these books on reserve at the Olin Library. Grading Policy: Your final grade for the semester is determined as follows: HW 40%, midterm 20%, final exam 40%. I will drop the lowest two grades you receive on homework. Note that the qualifying exam is three hours, and only part of it (approximately 2/3) is the final exam for this class. (The other part relates to 5021.) Homework and examination grades will be regularly updated on blackboard. |