Elliptic partial differential equations in hyperkähler geometry and gauge theory: Moduli spaces of solutions and geometric invariants.

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The study of nonlinear elliptic partial differential equations on manifolds is a central topic in Riemannian and symplectic geometry as well as in gauge theory. In many cases one considers moduli spaces of solutions to such equations with the aim of finding invariants of the underlying geometries. One particular instance are pseudoholomorphic curves (which are described by first order elliptic equations), resulting in Gromov-Witten invariants and Floer homology groups. This has lead to completely new insights into the geometry of symplectic manifolds throughout the past decades. Recently some of these elliptic methods have successfully been extended to the study of other geometries, like hyperkähler and quaternion kähler manifolds. Nonetheless, this approach raises many analytical issues which at present are only poorly understood. Two new types of first order elliptic partial differential equations with additional gauge symmetry, which naturally appear in hyperkähler geometry and in low dimensional gauge theory, shall in-depth be studied in this research endeavor. One the one hand, we aim to characterize the structure of solution spaces of these equations, like e.g. compactness and blow-up behaviour. On the other hand, through an investigation of the resulting moduli spaces, new geometric invariants shall be found and further be studied.