2017/2018
Накрытия и теория Галуа
Статус:
Дисциплина общефакультетского пула
Кто читает:
Факультет математики
Где читается:
Факультет математики
Когда читается:
3, 4 модуль
Преподаватели:
Дунин-Барковский Петр Игоревич
Язык:
английский
Кредиты:
5
Контактные часы:
76
Course Syllabus
Abstract
This course is an introduction to the theory of coverings of Riemann surfaces and to the surprising analogy between the classification of coverings and the main theorem of the Galois theory. We will consider both unramified and ramified coverings. It is remarkable that the connection between the theory of coverings and the algebraic Galois theory works in both ways: Galois theory allows us to understand the geometry of coverings better, and, at the same time geometric considerations lead to some purely algebraic results. In particular, we will see how to obtain the complete classification of all finite algebraic extensions of the field of rational functions of a single variable geometrically in a natural way
Learning Objectives
- The objective of this course is to teach students basic facts about the relation of algebraic Galois theory and the theory of coverings. In this course both ordinary (unramified) coverings and ramified coverings are considered. The main objectives regarding unramified coverings are the theorem about the bijection between subgroups of the automorphism group of a covering and the set of isomorphism classes of subcoverings, and the theorem on the existence of the universal covering. The main objective regarding ramified coverings is the connection between Galois groups of fields of meromorphic functions on subcoverings of a branched Galois covering treated as extensions of the field of meromorphic functions on the base and automorphism groups of respective coverings
Expected Learning Outcomes
- After completing this course the student has to have a firm understanding of the following notions and theorems: coverings of topological spaces; coverings of complex manifolds; the existence of the universal covering for a given covering of a topological space; the connection between the normal subgroups of the Galois groups of the covering and the Galois subcoverings of this covering
Course Contents
- Basic facts from algebraic Galois theory
- Coverings of topological spaces. Classification of coverings with marked points (through fundamental groups)
- Riemann surfaces/algebraic curves. Riemann’s existence theorem (without proof). Coverings and ramified coverings of Riemann surfaces
- Analogy between the classification of intermediate subcoverings of a given normal covering and the classification of subfields of a given Galois extension
- Fields of meromorphic functions on Riemann surfaces and their algebraic extensions, fields of germs
- Riemann surface of an algebraic equation over the field of meromorphic functions
- Application of Galois theory to the fields of germs on Riemann surfaces
- Geometric description of all finite algebraic extensions of the field of rational functions of a single variable
- Coverings and ramified coverings of Riemann surfaces
Interim Assessment
- Interim assessment (4 module)0.25 * Exam + 0.25 * Execise sheet + 0.25 * solving problems in class + 0.25 * Talk
Bibliography
Recommended Core Bibliography
- James, I. M. Handbook of Algebraic Topology: North Holland: p.1324 , 1995. - ISBN 978-0-444-81779-2
Recommended Additional Bibliography
- Emil Artin. (2007). Algebra with Galois Theory. [N.p.]: AMS. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=1495050