MM549: Topology and Complex Analysis (10 ECTS)

STADS: 13014901

Level
Bachelor course

Teaching period
The course is offered in the spring semester.

Teacher responsible
Email: qin@imada.sdu.dk

Timetable
Show entire timetable
Show personal time table for this course.

Comment:
Ubegrænset deltagerantal.

Prerequisites:
This course cannot be taken if students have taken MM511 or MM535.

Academic preconditions:
The content of the course MM533 Mathematical and Numerical Analysis is assumed known.

Course introduction
Topological properties play an important role in most branches of mathematics. The first purpose of the course is to develop further the concepts, which are introduced in Mathematical and Numerical Analysis, in more advanced settings. The second objective of the course is to give the students a fundamental knowledge of the theory of analytic functions, which will enable them to use this important theory in other areas of Mathematics and Applied Mathematics.

Qualifications
Having completed the course successfully the students are expected:

• to have a fundamental understanding of the theory of topological spaces, complete metric spaces, function spaces, normal topological spaces and its applications.
• to have a fundamental understanding of the theory of analytic functions and its applications.
• to be able to use the calculation of residues to compute important types of integrals.
• to be able to expand the most important holomorphic functions into power series and expand meromorphic functions into Laurent series.

Expected learning outcome
After having followed the course the student should be able to


• 
give an oral presentation of the statement and proofs related to any subject on a previously given list of topics within the course's syllabus 
• formulate the oral or written presentation in a mathematically correct way 
• answer supplementary questions from the teacher and the censor on definitions and results from the course syllabus

Subject overview

• Topological spaces, including construction methods and concepts of continuity,  compactness and connectedness. 
• Complete metric spaces. 
• Function spaces. 
• Normal topological spaces.
• Power series
• Analytic functions.
• Cauchy's integral theorem and integral formulas.
• The fundamental theorem of algebra.
• Taylor- and Laurent series of analytic functions.
• Poles and zeroes. The residue theorem and its applications to compute definite integrals.

Literature

Meddeles ved kursets start.

Website
This course uses e-learn (blackboard).

Prerequisites for participating in the exam
None

Assessment and marking:

1. Mandatory assignments. Evaluated by Danish 7-mark scale, internal censorship. 5 ECTS
2. Oral examination. Evaluated by Danish 7-mark scale, external censorship. 5 ECTS

Expected working hours
The teaching method is based on three phase model.
Intro phase: 56 hours
Skills training phase: 44 hours, hereof:
 - Tutorials: 44 hours

Educational activities

Language
This course is taught in Danish or English, depending on the lecturer. However, if international students participate, the teaching language will always be English.

Course enrollment
See deadline of enrolment.

Tuition fees for single courses
See fees for single courses.