MM549: Topology and Complex Analysis
Comment
Entry requirements
Academic preconditions
Students taking the course are expected to know the content of the course Mathematical analysis in MM533.
Course introduction
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.
The course builds on the
knowledge acquired in the courses Calculus and Mathematical and
Numerical Analysis, and gives an academic basis for studying the topics
probability theory, measure and integration and Banach spaces and
Hilbert- and Banach Spaces, that are part of the degree.
In relation to the competence profile of the degree it is the explicit focus of the course to:
- have
a fundamental understanding of the theory of topological spaces,
complete metric spaces, function spaces, normal topological spaces and
its applications. - have a fundamental understanding of the theory of analytic functions and its applications
- be able to use the calculation of residues to compute important types of integrals
- be
able to expand the most important holomorphic functions into power
series and expand meromorphic functions into Laurent series
Expected learning outcome
The learning objective of the course is that the student demonstrates the ability to:
- in a written exam, apply the concepts and ideas, from the course syllabus, in concrete mathematical examples.
- formulate the written presentation in a mathematically correct way and argue for the validity of achieved results in stringent mathematical language.
Content
- 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
Examination regulations
Exam element a)
Timing
Tests
Mandatory assignments and written exam
EKA
Assessment
Grading
Identification
Language
Examination aids
A closer description of the exam rules will be posted under 'Course Information' on Blackboard.
ECTS value
Additional information
The compulsory assignments are a part of an overall assessment and constitute 1/3 of the final grade.
The examination form for re-examination may be different from the exam form at the regular exam.
Indicative number of lessons
Teaching Method
Lectures will introduce general concepts and theory and exercise sessions will be devoted to learning the material in depth.
Other study activities include studying the course material and preparing the weekly exercises, individually or through group work.