Bachelor's Thesis in Mathematical Physics and Geometry
On this page you will find information specific for the courses SA114X and SA120X, with specialization towards Mathematical Physics and Geometry.
Course Structure
Recommended background knowledge for writing a bachelor's thesis in Mathematical Physics is the course SF1677. The course SF1677, Foundation of Analysis, is given during the spring semester. It is recommended that the student has taken this course before taking the course SA114X or SA120X. If you are interested in writing a bachelor's thesis in Mathematical Physics and Geometry you are encouraged to contact the supervisor as soon as possible to discuss possible projects and get advice on needed background material for their particular project.
General information about Bachelor's thesis
Information about specialization within mathematics
Suggestions for projects
The geometry of curves
Curves in Euclidean space can be completely described by an initial point, an initial velocity and a set of n-1 parameters where n is the dimension of the Euclidean space. In the Euclidean plane, there is only one parameter, called the curvature of the curve. In three-dimensional Euclidean space, there are two parameters, called curvature and torsion. The fundamental theorem of the theory of curves makes this description precise.
The many models of hyperbolic space
Hyperbolic geometry was discovered independently by Lobachevsky and Bolyai in the 19th century by proving the negation of Euclids 5th postulate. There are many equivalent ways to realize hyperbolic geometry, for example through the Poincaré half-plane model, the Poincaré disk model or the hyperboloid model. In all these models, the distance-shortening curves, called geodesics are explicitly known. Many interesting geometric statements (for example the hyperbolic law of cosines for hyperbolic triangles) can be discussed.
The Gauss-Bonnet theorem
One of the main features of the non-Euclidean geometry of curved surfaces is that the sum of angles of (geodesic) triangles is not longer 180 degrees. In fact, the deviation from 180 degrees equals an integral of the Gauss curvature over the interior of the triangle. This assertion is known as the Gauss-Bonnet formula. Covering a given compact surface by triangles, one is able to conclude the famous Gauss-Bonnet theorem which asserts that the integral of the Gauss curvature over the whole surface is up to a constant equal to a topological invariant, known as the Euler Characteristic. This is the prototypical example of a theorem relating local properties of a surface (curvature) to global ones (topology).
The geometry of Minkowski space and de-Sitter space
Minkowski space models time and space in the flat universe of special relativity. Minkowski space is equipped with the Minkowski metric which allows to distinguish space and time directions and to describe relativistic phenomena as time dilation and length contraction. The de-Sitter space is a prominent cosmological model which can be realized as the unit sphere in Minkowski space with respect to the Minkowski metric. It has many intersting features, for example it contains observes which are not able to communicate with each other due to the accalerated expansion of the space.
Singular ODEs from general relativity
A classical question in general relativity is whether there could exist other types of black holes in the universe than the ones we already know about. This turns out to be a very difficult mathematical problem, which is still far from being solved. One reason is that the partial differential equation (PDE) describing the gravitational field has singular coefficients at the event horizon of the black hole. In this project, we study a simplified problem, where the PDE is replaced by an ODE. Similar to the PDEs describing black holes, the type of ODEs we study will have coefficients that are singular at one point. The goal is to answer mathematical questions for the ODEs similar to the ones researchers are currently trying to answer for the black hole PDEs.
Your own idea
It is of course also possible to pursue a project not listed here. Maybe you have an idea of your own? The only requirements are that the project is of an adequate level of difficulty and relevance and that your KEX advisor feels competent to supervise it.