MAT9520 – 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
- will present, on a scientific level, a short thesis on a chosen topic of relevance, selected in collaboration with the lecturer.
Admission to the course
PhD candidates from the Faculty of Mathematics and Natural Sciences at the University of Oslo should apply for classes and register for examinations through Studentweb.
If a course has limited intake capacity, priority will be given to PhD candidates who follow an individual education plan where this particular course is included. Some national researchers’ schools may have specific rules for ranking applicants for courses with limited intake capacity.
PhD candidates who have been admitted to another higher education institution must apply for a position as a visiting student within a given deadline.
Recommended previous knowledge
- MAT3500 – Topology / MAT4500 – Topology
- It may also be useful to have taken MAT4510 – Geometric Structures
Overlapping courses
- 10 credits overlap with MAT4520 – Manifolds.
- 10 credits overlap with MA152.
- 10 credits overlap with MA252.
- 10 credits overlap with MA352.
- 10 credits overlap with MA352.
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.
In addition, each PhD candidate is expected to give an oral presentation on a topic of relevance chosen in cooperation with the lecturer. The presentation has to be approved by the lecturer 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: MAT4520 – 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 submit your response in Norwegian, Swedish, Danish or English.
Grading scale
Grades are awarded on a pass/fail scale. 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.