MM816: Geometry of Surfaces (5 ECTS)

STADS: 13005901

Level
Master's level course

Teaching period
The course is offered in the spring semester.
3rd quarter.

Teacher responsible
Email: svensson@imada.sdu.dk

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

Comment:
Ubegrænset deltagerantal. 3. kvartal. Fælles undervisning med MM516.

Prerequisites:
None

Academic preconditions:
Basic knowledge of point set topology as well as curves and surfaces is assumed to be known. Basic knowledge of complex function theory is recommended.

Course introduction
The course introduces some important aspects of the interplay between geometry and topology for surfaces in 3-space, in particular the Gauâ-Bonnet Theorem.

Expected learning outcome
At the end of the course the student will be able to:

• reproduce the definitions within the scope of the course’s syllabus in a precise mathematical language
• reproduce results, together with complete proofs, within the scope of the course’s syllabus
• relate the results within the scope of the course’s syllabus to each other
• apply the theory to solve problems within the scope of the course's syllabus
• combine concepts from differential geometry, topology and complex analysis to explain results within the scope of the course's syllabus
• prove the inverse function theorem and apply this theorem to describe the structure of smooth surfaces and smooth maps between them.

Subject overview

• Surfaces in 3-space.
  
• The Inverse Function Theorem.
  
• Change of coordinates for surfaces.
  
• Level curves and level surfaces.
  
• Orientation, area and isometries.
  
• Minimal surfaces and the Weierstrass representation.
  
• Theorema Egregium.
  
• The Gauss-Bonnet Theorem and the classification of compact surfaces.

Literature

Meddeles ved kursets start.

Syllabus
See syllabus.

Website
This course uses e-learn (blackboard).

Prerequisites for participating in the exam
None

Assessment and marking:
a) Mandatory assignments prepared and presented during the course. The mandatory assignments are evaluated collectively, pass/fail, internal censorship by the lecturer. The mandatory assignments must be passed for the student to take the oral exam.
b) Oral examination, grades according to the Danish 7-point scale and external censorship.

Reexamination after 4th quarter. The exam form at the re-exam can be different than at the ordinary exam.

Expected working hours
The teaching method is based on three phase model.

Forelæsninger: 28 timer
Eksaminatorietimer/opgaveregning: 18 timer
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.