In this talk, I will examine the geometry of moduli spaces of stable bundles on manifolds that do not admit Kaehler metrics. In particular, I will show that, in the case of Hopf surfaces, these moduli spaces admit interesting geometric structures such as hypercomplex structures and strong HKT-metrics, as well as algebraic completely integrable systems.

In this talk, I will examine the geometry of moduli spaces of stable bundles on Kodaira surfaces, which are non-Kaehler compact surfaces that can be realised as torus fibrations over elliptic curves. These moduli spaces are interesting examples of holomorphic symplectic manifolds whose geometry is similar to the geometry of Mukai's moduli spaces on K3 and abelian surfaces.

I will discuss the construction of certain nonself-adjoint operator algebras from a discrete dynamical system (namely a space X and one or more maps of X into itself), and discuss how the algebra encodes the system, and how it can be recovered from information about the algebra.

I will describe a result by Eliashberg and Polterovich that allows to construct a family $J_n$ of complex structures on $T^2\times D^2$ with strictly pseudoconvex boundary which are biholomorphically equivalent and homotopic through complex structures but not homotopic through complex structures with strictly pseudoconvex boundary. Note: $T^2\times D^2$ denotes the product of the 2-torus and the closed unit disk in $R^2$. The proof is based on the Lagrangian surgery method of K. Luttinger.