MAT4520 – Manifolds
Course description
Schedule, syllabus and examination date
Course content
The course gives an introduction to smooth manifolds. It covers tangent bundles, vector fields and integral curves (ordinary differential equations), Lie groups, differential forms, and integration. This theory is fundamental to both modern geometry and theoretical physics.
Learning outcome
After completing the course you
- understand well the concepts smooth manifold, smoth map, and tangent space
- know how the inverse function theorem can be used to describe the local structure of immersions and submersions, and you know Sard?s theorem
- can work with submanifolds and know Whitney?s embedding theorem
- know fundamental results about vector fields, Lie brackets, and integral curves, and you know what it means for the flows of two vector fields to commute
- are familiar with the Lie algebra and exponential map of a Lie group
- can do calculations with differential forms and characterise the exterior derivative, and you know Stokes? theorem and understand how this generalises classical theorems in calculus.
Admission to the course
Students admitted at UiO must?apply for courses?in Studentweb. Students enrolled in other Master's Degree Programmes can, on application, be admitted to the course if this is cleared by their own study programme.
Nordic citizens and applicants residing in the Nordic countries may?apply to take this course as a single course student.
If you are not already enrolled as a student at UiO, please see our information about?admission requirements and procedures for international applicants.
Recommended previous knowledge
- MAT3500 – Topology / MAT4500 – Topology
- It may also be useful to have taken MAT4510 – Geometric Structures
Overlapping courses
- 10 credits overlap with MAT9520 – Manifolds.
- 10 credits overlap with MA252.
- 10 credits overlap with MA352.
- 10 credits overlap with MA352.
- 10 credits overlap with MA152.
Teaching
4 hours of lectures/exercises per week.
The course may be taught in Norwegian if the lecturer and all students at the first lecture agree to it.
Upon the attendance of three or fewer students, the lecturer may, in conjunction with the Head of Teaching, change the course to self-study with supervision.
Examination
Final oral exam which counts 100 % towards the final grade.
This course has 1 mandatory assignment that must be approved before you can sit the final exam.
It will also be counted as one of the three attempts to sit the exam for this course, if you sit the exam for one of the following courses: MAT9520 – Manifolds
Examination support material
No examination support material is allowed.
Language of examination
Courses taught in English will only offer the exam paper in English. You may write your examination paper in Norwegian, Swedish, Danish or English.
Grading scale
Grades are awarded on a scale from A to F, where A is the best grade and F is a fail. Read more about the grading system.
Resit an examination
This course offers both postponed and resit of examination. Read more:
More about examinations at UiO
- Use of sources and citations
- Special exam arrangements due to individual needs
- Withdrawal from an exam
- Illness at exams / postponed exams
- Explanation of grades and appeals
- Resitting an exam
- Cheating/attempted cheating
You will find further guides and resources at the web page on examinations at UiO.