MAT9595 – Geometry and analysis
Course description
Schedule, syllabus and examination date
Course content
In the first part of the course we will discuss connections in vector bundles and principal bundles, Chern-Weil theory, the classification of flat connections in terms of representations of the fundamental group, Dirac operators and vanishing theorems. In the second part of the course we will prove fundamental results about elliptic operators on compact manifolds using Fourier series. One application of this theory is the Hodge theorem about harmonic differential forms.
Learning outcome
After completing the course you:
- are familiar with the definitions and basic properties of connections in vector bundles and principal bundles;
- can use Frobenius' theorem to show that flat connections are locally trivial;
- know how the Chern classes of a complex vector bundle can be expressed in terms of the curvature of a connection in the bundle;
- know how Dirac operators are constructed and can deduce the Bochner formula;
- know the definitions and basic examples of elliptic operators and elliptic complexes;
- can outline how to prove the fundamental results on elliptic operators on compact manifolds using Fourier series.
Admission
PhD candidates from 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.
Prerequisites
Recommended previous knowledge
MAT4520 – Manifolds/MAT9520 – Manifolds and MAT3400 – Linear Analysis with Applications/MAT4400 – Linear Analysis with Applications
Overlapping courses
10 credits overlap with MAT4595 – Geometry and analysis
*The information about overlaps for discontinued courses may not be complete. If you have questions, please contact the Department.
Teaching
4 hours of lectures/exercises per week.
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 examination.
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 for the student to be admitted to the final exam.
Examination support material
No examination support material is allowed.
Language of examination
You may write your examination paper in Norwegian, Swedish, Danish or English.
Grading scale
Grades are awarded on a pass/fail scale. Read more about the grading system.
Explanations and appeals
Resit an examination
Students who can document a valid reason for absence from the regular examination are offered a postponed examination at the beginning of the next semester.
Re-scheduled examinations are not offered to students who withdraw during, or did not pass the original examination.
Withdrawal from an examination
It is possible to take the exam up to 3 times. If you withdraw from the exam after the deadline or during the exam, this will be counted as an examination attempt.
Special examination arrangements
Application form, deadline and requirements for special examination arrangements.
Evaluation
The course is subject to continuous evaluation. At regular intervals we also ask students to participate in a more comprehensive evaluation.