6 EC
Semester 1, period 1, 2
5334TICA6Y
| Owner | Master Mathematics |
| Coordinator | prof. dr. Jan Wiegerinck |
| Part of | Master Mathematics, year 1 |
Topics in Complex analysis deals with the theory of analytic functions of several complex variables. In particular the differences between the theory of one and of several variables will be emphasised: While for every domain D in C there exist holomorphic functions f on D that do not extend analytically outside D, there are many domains G in C^n so that every holomorphic function f extends to a larger domain G. Understanding this phenomenon is a major goal of the course. There is a close relation with solvability of the inhomogeneous Cauchy-Riemann equations, which we will also discuss. These are always solvable in one variable, but in the several variable case solvability depends on the domain. Another major difference is that biholomorphic mappings are much more difficult to understand. We will see that there is no Riemann mapping theorem and that there are very many biholomorphic maps from C^n to (subsets of) C^n.
Krantz: Several Complex Variables AMS-Chelsea
Fritsche - Grauert:From Holomorphic Functions to Complex Manifolds GTM 213 Springer
At the end of the course, the student can solve problems involving Reinhardt domains, domains of holomorphy, and pseudoconvexity. She can identify and construct plurisubharmonic functions with prescribed properties, is able to solve relevant inhomogeneous Cauchy-Riemann equations, either by L^2 or by kernel methods. the student is able to read some of the literature in several complex variables. The students will be able to read as a group parts of Rosay and Rudin "Holomorphic maps from C^n to C^n". The successful student will be able to write a masters thesis in complex analysis or complex dynamics.
|
Activity |
Number of hours |
|
course |
28 |
|
self study |
140 |
The programme does not have requirements concerning attendance (OER-B).
Additional requirements for this course:
| Item and weight | Details |
|
Final grade | |
|
0.2 (20%) presentation | |
|
0.8 (80%) written exam |
The written exam will be based on the following (dynamical, I may delete or add some) list of exercises from the notes. I recommend that you work on these parallel to the course. I will not grade these exercises, but I will discuss them if asked for.
Chapter 1 no 8,9,12,18,20,24,29
Chapter 2 no 2,7,12,24,25,26
Chapter 3 no 6,14 (and develop from earlier exercises what you need), 24
Chapter 4 no 5,7,9,16,21,26,27,28
Chapter 5. no 12,13,20,21,30,33,
Chapter 6. no 3, 4,5,10,11.
Chapter 6 no 16, 21, 23,26
Chapter 7. no 6,10,12,17,24
Chapter 8. no 5,9,15,33,34
Chapter 9. no 2 5,6,7,11,12,13.
The 'Regulations governing fraud and plagiarism for UvA students' applies to this course. This will be monitored carefully. Upon suspicion of fraud or plagiarism the Examinations Board of the programme will be informed. For the 'Regulations governing fraud and plagiarism for UvA students' see: www.uva.nl/plagiarism
| Weeknummer | Onderwerpen | Studiestof |
| 1 | differences between one and several complex variables | notes chap 1 |
| 2 | Reinhardt domains | notes chap 2 |
| 3 | analytic continuation | notes chap 3 |
| 4 | analytic continuation structure of zero sets | notes chap 3,4 |
| 5 | structure of zero sets | notes chap 4 |
| 6 | holomorphic mappings | notes chap 5 |
| 7 | domains of holomorphy | notes chap 6 |
| 8 | domains of holomorphy, Cousin, d-bar and Levi problems | notes chap 6,7 |
| 9 | Cousin, d-bar and Levi problems | notes chap 7 |
| 10 | plurisubharmonic functions | notes chap 8 |
| 11 | pseudoconvexity | chap 9 |
| 12 | solving d-bar | chap 10-11-12 |
| 13 | presentations by students (from Rosay and Rudin) | |
| 14 | presentations by students (from Rosay and Rudin) | |
| 15 | ||
| 16 |
The schedule for this course is published on DataNose.