Mathematics

Projects

Ph.D. project and Current Research

My research interests lie in the intersection of geometry and topology, more specifically the application of algebraic topology to solving problems that arise in geometry. I am particularly interested in problems involving closed geodesics. The topic of closed geodesics on manifolds is a classical one, initiated by the likes of Morse and Lusternik and Schnirelman who were interested in proving the existence of closed geodesics. A significant step in proving the existence of infinitely many closed geodesics was taken by Gromoll and Meyer who settled the cases where the rational cohomology ring of the manifold is not generated by one element. Except for the two sphere the remaining cases are open. Another interesting problem is to study the geometry of manifolds all of whose geodesics are closed. This is the topic of my current research. I work on a conjecture by Marcel Berger, which states that on a simply connected manifold all of whose geodesics are closed, the geodesics all have the same least period. The assumption that the manifold is simply connected is necessary, since on the lens spaces all geodesics are closed, but not of the same least period.

Bott and Samelson showed that a simply connected Riemannian manifold with this property has integral cohomology ring generated by one element. The spheres and projective spaces with the standard metric together with the so-called Zoll metrics on the spheres have the property that all geodesics are closed with the same least period. The weaker statement that there exists a common period is a special case of a theorem about periodic orbits of a Hamiltonian flow, and is due to Wadsley.

Gromoll and Grove proved the conjecture for metrics on the two sphere. Their proof is two dimensional in nature and the argument does not carry over to other dimensions. Recently Wilking proved the conjecture for the spheres of dimension greater than three. Concretely, I work on the conjecture for metrics on the three sphere.

One possible way to approach the Berger Conjecture is to use Morse Theory on the free loop space. I my thesis I investigate the Morse Theory for metrics on the three sphere all of whose geodesics are closed. For a short introduction to the methods as well as a summary of the results, see my research statement below.

You will find my thesis here: pdf.

The results of my thesis have been published by the Annals of Global Analysis and Geometry. Here is a link to the article online: SpringerLink.

If you are interested in my teaching statement you will find it here: pdf.

Masters Thesis and Bachelor Project

You can download my Master of Science thesis about topologigal properties of positively curved manifolds here: ps, pdf.

You can also download my bachelor project about the Schwarzschild metric (in Danish) here: ps, pdf.

Notes

From September 2002 to June 2004 I was responsible for organizing IMF's high school visiting programme. We gave lectures and organized problem sessions for high school students. Our main aim was to get gifted students interested in mathematics by giving talks about advanced mathematical topics. You can see the webpage (in Danish) "Besøgsservice".
Below you can download the notes I have written.

Kurver i planen og rummet, ps, pdf.
Komplekse tal og rækker, ps, pdf.
Komplekse tal og polynomier, ps, pdf.

Emil Hedevang Lohse, Erik Olsen and I have typed some lecture notes by Johan Dupont on curvature and characteristic classes in LaTeX. You can download the notes here:
"Fibrebundles and Chern-Weil Theory": report format, A4: ps, pdf.
"Fibrebundles and Chern-Weil Theory": scrbook, A5: ps, pdf.