Skyrmions are interesting both from a mathematical point of view and as a way of gaining a better understanding of nuclear physics. There are many different mathematical models of topological solitons (such as monopoles and instantons) which are related to Skyrmions. Moreover, many different techniques can be applied to this model. One way is to solve the equations numerically with the help of a large computer. However, often more insight is gained by using approximations, so-called ansätze . The most successful ansatz is the rational map ansatz which has been introduced by Houghton, Manton and Sutcliffe in 1997 and explains the symmetries of Skyrmions. In the following, I will describe some problems related to Skyrmions which all use different mathematical tools.
The first problem is to understand the structure of Skyrmions, in particular the occurrence of folds and cusps which is predicted by the theory of singularities of differentiable maps. A fold can be thought of the line where a paper is folded over. Physically, folds correspond to tiny amounts of anti-matter. In order to tackle this problem I generalize the rational map ansatz . This new ansatz proves to be very successful. In particular, using group theory, I am able to show that there is anti-matter in nuclei for B = 3 but not for B = 4, [2].
In the second problem I consider Skyrmions on the 3-sphere. By using the rational map ansatz I obtain the lowest energy solution for B = 3,..., 9 known to date. I also consider phase transitions as a function of the radius of the 3-sphere. As a bonus I obtain a geometrical understanding why Skyrmions have a shell-like structure. Roughly speaking the Skyrme configurations that minimize the energy are as conformal as possible without changing the winding number, [1].
The third problem is about fermions coupled to Skyrmions which is interesting from a mathematical point of view. I solve the Dirac equation in the vacuum and in the background of an SO(4)-symmetric Skyrmion explicitly. The solution suggest a relationship between the topology of the Skyrmions and the spectrum of the Dirac operator of the fermions, [3, 5].
Finally, I want to comment on some work in progress. Recently, there have been advances in the understanding of classical Skyrmions up to B = 22. One major problem in the quantization of Skyrmions are the so-called Finkelstein-Rubinstein constraints, which guarantee that Skyrmions are quantized as fermions for odd B. Following ideas of N S Manton I have found a way to calculate these constraints with the help of the rational map ansatz, [4]. Using the simplest approximation, the so-called zero-mode quantization, I calculated the ground state for up to B=22. For even B there is good agreement with nuclear physics. However, for odd B the zero-mode approximation is not accurate enough. I am currently working on a more accurate approximation.