Mathematical papers

Vector Fields on the Sphere (20th of December, 2001)

Master thesis, supervisor: Jesper Michael Møller.
English: [PS-format] [PDF-format]

The existence and non-existence of vectorfields on the sphere is a classical problem arising in the theory of fibre bundles and homotopy theory with implications in many areas of mathematics.

In this thesis we proove the negative statement, that is, there exists at most ρ(n)-1 linearly independent vector fields on the n-1-sphere, where for n=(2a+1) 2^c+4d, ρ(n)=2^c+8d. This was proven by J. F. Adams in his article in 1962, and we follow his approach. The converse also holds, this was proven by Beno Eckmann in 1942.

The Kerr solution (20th of November, 2000)

Paper, supervisor: Bergfinnur Durhuus.
Danish: [PS-format] [PDF-format]
English: [PS-format] [PDF-format]

In this paper we determine a solution to the Einstein field equation of general relativity:

R_μν-½Rg_μν+Λg_μν = 8πT_μν

namely the solution describing a static axial symetric space-time manifold. This solution was first determined by Roy Kerr in 1963 and is also known as the Kerr solution.

Compact Lie groups (22nd of February, 2000)

Paper, supervisor: Jesper Michael Møller.
Danish: [PS-format] [PDF-format]
English: [PS-format] [PDF-format]

Lie groups are a classical subject from analisys, and the results we will prove here are also classical and well known form the Lie group theory. As such, it is not the results in them selves, which makes this project interesting but rather the method applied in the proofs.

Rather than the usual analytical approach, we apply a homotopy theoretical point of view, and through this try to expose some of the geometrical structures which characterizes Lie groups. We prove existence and uniqueness of the maximal torus and use this result to determine the center and the Weyl group of compact Lie groups.

The project is based on the article by Dwyer and Wilkerson, The Elementary Geometric Structure of Compact Lie Groups.

Spherical Harmonic Functions (5th of May, 1997)

Bachelor thesis, supervisor: Henrik Schlichtkrull.
Danish: [PS-format] [PDF-format]

The Laplace equation appears in many places in physics, for example the wave equation and the Schrödinger equation. On certain bounded sets the associated Dirichlet problem can be solved by separation of variables. On the sphere in three dimensions this is known as the Legendre problem which has the Legendre polynomials as eigen functions.

Probability and statistics reference (May, 1997)

Danish: Not available for download

Quick reference for probability and statistics. This reference was written for a course in probability and statistics held autumn 1996 by Flemming Topsøe. The reference does not replace the notes as it does not give the foundation for understanding, but is usefull for solving exercises.

Printing

All publications are available in PostScript and Portable Document Format. Both can be viewed using programs such as Acrobat Reader or GSView, both available for free for many platforms.

All documents has been prepared for A4 paper, which is ideal as it allows printing booklets (4 pages per sheet, doublesided printing). Please consider printing booklets rather than 1 page per sheet.

On Unix and Unix-like systems, printing booklets is rather easy:

   psbook <file.ps> | psnup -2 | lpr -P<printer>

On Windows this is almost as easy: Simply replace your Windows with a modern operating system and follow the instructions above.