Syllabus

Course 1: The geometry of soliton moduli spaces

Moduli space methods are particularly well-developed in the case of abelian Higgs vortices, encompassing not only classical vortex scattering, but also the quantum mechanics of vortices and the statistical mechanics of a vortex gas. This course will introduce the fundamental notions of moduli space methods in the context of the abelian Higss model.

Course 2: Supersymmetry and solitons

Soliton bearing theories of Bogomol'nyi type have a remarkable unified structure, defined in the language of supersymmetry. The quantum dynamics of solitons is particularly elegant in the supersymmetric formulation, with strong connexions to the L2 cohomology of the moduli space. In fact, many of the interesting geometric properties of soliton moduli spacecs (e.g. the hyperkaehler structure for monopoles) have a natural explanation in terms of supersymmetry.

Course 3: ADHM, Nahm and Fourier-Mukai transforms

In many examples of interest, the PDEs satisfied by static solitons turn out to be integrable, and the problem of constructing soliton moduli spaces has motivated many influential developments in integrable systems theory. The third lecture course will be concerned with two of these, highly nontrivial analogues of the classical Fourier transform which have applications to gauge theory, called the ADHM and Nahm transforms. These provide a powerful method of studying various Euclidean instantons, monopoles, and calorons. Their main advantage is that they convert the difficult problem of solving nonlinear PDEs into a simpler problem involving only ODEs or just vector spaces and maps between them. More recently, the so-called Fourier-Mukai transform has generalized the Nahm transform and brought it into the realm of algebraic geometry and derived categories.