The goal of this course is to outline the new epoch of classical complex analysis.
At the beginning, we will introduce the hyperbolic geometry in the upper half plane. After discussing the normal family, we will show Riemann's mapping theorem which has many applications in the complex analysis. We will explain Riemann surfaces. The theory of Riemann surfaces provides a new foundation for complex analysis on a higher level. As in elementary complex analysis, the subject matter is analytic functions. But the notion of an analytic function will have now a broader meaning as we show. Finally, an introductory lecture on elliptic functions and elliptic curves will be given.
By the end of this course, students will be able to:
1) understand the hyperbolic geometry.
2) obtain the notion of normal family and its applications.
3) know Riemann's mapping theorem and its applications.
4) understand Riemann surfaces.
Normal family, Riemann's mapping theorem, Riemann surface.
✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | ✔ Practical and/or problem-solving skills |
Standard lecture course.
Course schedule | Required learning | |
---|---|---|
Class 1 | Conformal mappings, especially on the upper half plane and the unit disk. | Details will be provided during each class session. |
Class 2 | The linear fractional transformations : classification and properties. | Details will be provided during each class session. |
Class 3 | The hyperbolic plane and the Poincare disk | Details will be provided during each class session. |
Class 4 | Normal family. | Details will be provided during each class session. |
Class 5 | Wierstrass theorem on normal families and its applications. | Details will be provided during each class session. |
Class 6 | Montel's theorem and its applications. | Details will be provided during each class session. |
Class 7 | Riemann's mapping theorem. | Details will be provided during each class session. |
Class 8 | Riemann's mapping theorem and its applications. | Details will be provided during each class session. |
Class 9 | Analytic continuation. | Details will be provided during each class session. |
Class 10 | The definition of Riemann surfaces and a construction. | Details will be provided during each class session. |
Class 11 | Functions on Riemann surfaces, degree and genus | Details will be provided during each class session. |
Class 12 | Riemann surfaces as quotient spaces | Details will be provided during each class session. |
Class 13 | Elliptic functions | Details will be provided during each class session. |
Class 14 | Picard's theorems | Details will be provided during each class session. |
Class 15 | Groups of Mobius transformations, comprehension check-up | Details will be provided during each class session. |
None.
J. Gilman, I. Kra and R. Rodriguez: Complex Analysis (Springer, GTM 245).
E. Freitag : Complex Analysis 2 (Springer, Universität).
Final exam (70%) and report (30%).
Students are expected to have passed [MTH.C301 : Complex Analysis I] and [MTH.C302 : Complex Analysis II].
None in particular.