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Instructor: Matt Kerr Office: Cupples I, Room 114 e-mail: matkerr [at] math.wustl.edu Office Hours: 8-9 W (on Zoom), 4-5 F (in office) 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. The material roughly correlates to the second half 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 also 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 Monday, Wednesday and Friday from 1-1:50 in Lopata 201. First class is Wednesday Jan. 18 and last class is Friday April 87, with three days off for spring break (M/W/F March 13/15/17). Midterm (take-home) Exam: TBA, possibly take-home Final (Qualifying) Exam: Wednesday, May 10, 1-3pm, in Lopata 201 The midterm will cover approximately the first two units above. The qualifying exam will cover material from the whole course, and everyone must take it at the time/date listed above. Assignments: Posted here by Friday, due the following Thursday by 6PM on Gradescope, and returned by the following Wednesday. Solutions will be posted on Canvas. Please feel free to come to office hours to discuss problem sets. Problem Set 1 (due Thursday Jan. 26) Problem Set 2 (due Thursday Feb. 2) [tex] Problem Set 3 (due Thursday Feb. 9) [tex] Problem Set 4 (due Thursday Feb. 16) [tex] Problem Set 5 (due Thursday Feb. 23) [tex] Problem Set 6 (due Thursday Mar. 2) [tex] Problem Set 7 (due Thursday Mar. 9) [tex] Problem Set 8 (due Thursday Mar. 23) [tex] Problem Set 9 (due Thursday Apr. 6) [tex] Problem Set 10 (due Thursday Apr. 13) [tex] Problem Set 11 (due Thursday Apr. 20) [tex] Problem Set 12 (not to hand in) Grader: RJ Acuna Office: Cupples I, Room 213 e-mail: rjacuna [at] wustl.edu Lecture Notes: Will be posted here the same day as the lecture. The hope is that this makes notetaking optional. Lecture 1: Normal families Lecture 2: Riemann mapping theorem Lecture 3: Extension to the boundary Lecture 4: From boundary to interior Lecture 5: Explicit conformal mappings Lecture 6: Harmonic functions revisited Lecture 7: More on harmonic functions Lecture 8: Subharmonic functions I Lecture 9: Subharmonic functions II Lecture 10: The Dirichlet problem Lecture 11: Applications of Dirichlet Lecture 12: Multiply connected regions I Lecture 13: Multiply connected regions II Lecture 14: Multiply connected regions III Lecture 15: The Gamma function Lecture 16: Gamma and zeta Lecture 17: More on zeta Lecture 18: Primes and zeta zeroes Lecture 19: The prime number theorem Lecture 20: Apery's theorem Lecture 21: Lattices in C Lecture 22: Abel's theorem Lecture 23: Elliptic functions and elliptic curves Lecture 24: Elliptic addition theorems Lecture 25: Introduction to modular forms Lecture 26: More on modular forms Lecture 27: Modular forms of higher level Lecture 28: The Picard theorems Lecture 29: Geometric function theory Lecture 30: The Bloch and Landau constants Lecture 31: Schlicht functions I Lecture 32: Schlicht functions II Lecture 33: Green's functions on Riemann surfaces I Lecture 34: Green's functions on Riemann surfaces II Lecture 35: The Uniformization Theorem Lecture 36: Several complex variables I Lecture 37: Several complex variables II Lecture 38: Several complex variables III Lecture 39: Several complex variables IV Lecture 40: The Bergman kernel Books: Lars Ahlfors, Complex Analysis (3rd Ed.); McGraw-Hill is the recommended textbook, which means I will follow it some of the time and some of the problems I assign will come from it. I assume most of you have a copy from Math 5021; if not, then 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. Homework and examination grades will be regularly updated on Canvas. Grades are typically curved in a course like this but will never be less than the following scale:
If you are a graduate student, a letter grade of B is required to pass; if you are an undergraduate taking this class Pass/Fail, you must earn a C- to pass. The Washington University academic integrity policies are here. All work submitted under your name is expected to be your own; please make sure to document any ideas that come from another source. |