Bachelor's Thesis in Mathematical Physics and Geometry
On this page you will find information specific for the course SA104X, with specialization towards Mathematical Physics and Geometry.
Course Structure
Prerequisites for writing a bachelor's thesis in Mathematical Physics are covered by the course SF2713. The course SF2713, Foundation of Analysis, is given during the autumn semester and on special request also during the spring term. It is recommended that the student has taken this course before taking the course SA104X.
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
MODELS OF THE UNIVERSE
Supervisor: H. Ringström
The cosmological principle is the basic starting point when modeling the universe. According to this assumption, one does (at a given a moment in time) not see any difference between two points in space (homogeneity) nor between different directions (isotropy). As a consequence, one obtains the standard models in cosmology. These models start with a big bang and then expand forever (or recollapse). Since the cosmological principle is not exactly fulfilled, it is, however, of interest to see what happens if it is relaxed. Does one obtain a big bang with arbitrarily strong gravitational fields? Are the standard models stable? It is quite hard to answer these questions in all generality, and it is thus natural to reduce the symmetry assumptions gradually. A natural first step is to study homogeneous (but not necessarily isotropic) models. Under these assumptions, Einstein's equations become a system of non-linear ODEs. In the easiest case, it is possible to study the solutions with methods from the course SF1629 (differentialekvationer och transformer). In the more difficult cases, the relevant systems of equations are objects of current research.
In short, the equations to be investigated in this course are systems of non-linear ODEs (which appear in the study of the different cosmological models in general relativity). The objective of the course is that the participants should learn to analyze the asymptotic behavior of the solutions.
THE DRUM AS AN INSTRUMENT
Supervisor: A. Laptev / T. Ekholm / F. Portmann
A popular subject in many physics courses in high school is the analysis of a "guitar string" and its corresponding scale. In this course we intend to do the same for an idealized drum, by which we mean a circular membrane fixed to a ring. This leads to the study of the Helmholtz equation, which can be solved with techniques from a standard course on differential equations.
LIEB-THIRRING INEQUALITIES IN LOW SPACE DIMENSIONS
Supervisor: A. Laptev / T. Ekholm / F. Portmann
A very important object in quantum mechanics is the Schrödinger operator, who's measurements correspond to the energy of the system in a given state. In this course we will study the negative eigenvalues of this operator (corresponding to the bound states of the system), and to simplify calculations we will limit ourselves operators on the real line. Estimates of the sums (or sums of the square roots) of these eigenvalues in terms of the potential of the system are known as Lieb-Thirring inequalities and play a very important part when proving stability of quantum mechanical systems.
The objective of this course will be to study Lieb-Thirring inequalities for a certain class of potentials. To be a bit more specific, to investigate how the eigenvalues depend on the potential and if possible we want to find the potential which maximizes the aforementioned sums.
Contact
