Topics in Geometry

Course content

We will generally be concerned with various advanced topics from modern and classical differential geometry, geometric analysis, Riemannian geometry and topology - close to active areas of research - such as: Minimal surfaces, prescribed curvature problems, variational problems in geometry, curvature flows, geometric analytical methods, with the precise contents varying each year and depending on the interests of the participants.

Fall 2023: A tentative plan for this year is to give an introduction to the modern theory of minimal 2-surfaces in 3-manifolds, via excerpts from Colding-Minicozzi's book A Course in Minimal Surfaces (AMS, 2011) plus other relevant notes/articles. Time permitting we will also include some applications to other areas of geometry and topology.

Contents in past years:
Fall 2021: Themed "Interactions Between Topology & Geometric Analysis", the main topics we covered were:
(1) Proof of Hopf's Theorem on uniqueness of the round sphere as the only genus 0 constant mean curvature 2-dimensional closed surface in R3, i.e. "(sometimes) soap bubbles are round" (via the Poincaré-Hopf index of vector fields combined with complex analysis, among other things).
(2) Proof of existence of non-trivial closed geodesic curves on any smooth closed n-dimensional manifold, via so-called min-max variational methods, which make use of some basic algebraic topology (homotopy groups) combined with L2-theory for differential (Euler-Lagrange) equations.

Fall 2020: This year, the course ran informally, and we discussed advanced topics in Mean Curvature Flow, such as gluing problems and uniqueness questions for singularity models in the flow.


MSc Programme in Mathematics
MSc Programme in Mathematics with a minor subject

Learning outcome
  • Knowledge: To display knowledge of the course topics and content, at the level of a beginning researcher.
  • Skills: To be able to use the acquired knowledge to perform computations.
  • Competencies: To be able to produce independent proofs in extension of
    the acquired knowledge.

4 hours of lectures and 3 hours of exercises for 9 weeks.

Please note that to prepare lectures as a participant in this topics course is a substantial time commitment. So participants should plan their time accordingly.

MSc students, who have taken Geometry 2 and preferably Riemannian Geometry.

Academic qualifications equivalent to a BSc degree is recommended.

Continuous feedback during the course of the semester
7,5 ECTS
Type of assessment
Continuous assessment
Type of assessment details
Weekly written assignments (7 such in total) combined with two in-class oral presentations:
- One lecture (45 minutes) about a relevant topic (to be decided together with the lecturer or teaching assistant).
- One short presentation of a homework solution (to be decided together with the lecturer or teaching assistant).
The written assignments together are weighted 50%, the in-class oral presentations together are weighted 50%, and the course performance is assessed as a whole.
All aids allowed
Marking scale
7-point grading scale
Censorship form
No external censorship
One internal examiner
Criteria for exam assessment

The student should convincingly and accurately demonstrate the knowledge, skills and competences described under Intended learning outcome.

Single subject courses (day)

  • Category
  • Hours
  • Lectures
  • 36
  • Preparation
  • 143
  • Theory exercises
  • 27
  • English
  • 206


Course number
7,5 ECTS
Programme level
Full Degree Master

1 block

Block 1
No limit
The number of seats may be reduced in the late registration period
Study Board of Mathematics and Computer Science
Contracting department
  • Department of Mathematical Sciences
Contracting faculty
  • Faculty of Science
Course Coordinator
  • Niels Martin Møller   (7-504f716e6e6774426f63766a306d7730666d)
Saved on the 28-02-2023

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