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

Students who are admitted to study programmes at UiO must each semester register which courses and exams they wish to sign up for in Studentweb.

If you are not already enrolled as a student at UiO, please see our information about admission requirements and procedures.

Overlapping courses

10 credits overlap with MAT9595 – 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

1 mandatory assignment.

Final oral examination.

Examination support material

No examination support material is allowed.

Language of examination

You may write your examination