### 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.
- The abelian Higgs model, Bogomol'nyi's topological energy bound, the Bogomol'nyi equations.
- Taubes's Theorem, construction of the moduli space.
- The geodesic approximation, definition of the L
^{2} metric.
- Samols's formula, vortex scattering.
- The case of compact domains, volume of moduli space and vortex gas
thermodynamics.

#### 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 L^{2} 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.
- Review of supersymmetry in classical field theory.
- Supersymmetry algebras and the Bogomol'nyi bound; examples
(kinks, lumps, monopoles).
- Supersymmetric quantum mechanics of solitons.
- Supersymmetry and the geometry of the monopole moduli space;
Sen's conjecture.

#### 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.
- Pure Yang-Mills theory, the self-dual Yang-Mills equations, instantons.
- The ADHM transform.
- The Nahm transform: monopoles and calorons.
- Fourier-Mukai and instantons on elliptic surfaces.