2.1 To obtain an explicit expression for the Skyrmion from a multi-Taub-NUT instanton with a general integer parameter N.
2.2 To find the topological degree of a multi-Taub-NUT Skyrmion and the instanton number of the associated Yang-Mills instanton, and establish their dependence on the parameter N.
2.3 To establish a relationship or an identification between the Euler characteristic of a self-dual gravitational instanton and the instanton number of the associated Yang-Mills instanton.
2.4 To investigate a possible interpretation of the lepton number as a topological invariant, which may involve the topological degree of the Skyrmion, and the Euler characteristic and the signature of the gravitational instanton.
2.5 To understand the geometry of the space where a multi-Taub-NUT Skyrmion lives, obtain an explicit expression for the metric on the space and identify its corresponding solution to the SU(?)-Toda field equation.
2.6 To obtain sufficient conditions for a scalar-flat K?hler metric to be conformally related to a hyper-K?hler metric, in terms of conditions on solutions of the SU(?)-Toda field equation, and obtain more examples.
