Mathematical gauge theories investigate solution spaces of partial differential equations defined with the help of a principal bundle connection. These partial differential equations are generalizing (nowadays classical) equations introduced by physicists Yang and Mills in the realm of the strong interaction or, mathematically speaking, in the realm of principal SU(3)-bundles over four dimensional manifolds. After fundamental works by Simon Donaldson, Nathan Seiberg and Edward Witten, the gauge theory approach is extensively used in investigations of the (low dimensional) topology of manifolds. There is a strong relationship of these theories to symplectic geometry. The Floer construction is throwing light on this relationship; it can be applied either to define symplectic invariants of Lagrangian submanifolds or to define certain invariants of 3-manifolds. The Heegaard Floer homology is derived from an application of the Lagrangian Floer homology and it is conjecturally equivalent to the Seiberg-Witten theory. However, it is much more combinatorial and often more suitable for calculations.

These striking interplays between the low dimensional topology and symplectic geometry are the main subject of this book, which is freely based on lecture courses given at the Clay Mathematics Institute Summer School in Budapest, Hungary, in 2004. Preparatory and introductory parts contained in these proceedings are written in an intelligible way, often with proofs. Also, many examples and figures are included in the book. In this way it can serve for researchers active in the subject and related fields as well as for graduate students. Specific chapters of the book are devoted to the Heegaard Floer homology and knot theory, Floer homologies, contact structures, symplectic 4-manifold and Seiberg-Witten invariants. The authors are the following experts in the mentioned research field: P. Oszváth, Z. Szabó H. Goda, J. Etnyre, A. Stipsicz, P. Lisca, T. Ekholm, R. Fintushel, R. Stern, J. Park, Tian-Jun Li, D. Auroux and I. Smith.