This book takes a historical approach to Einstein's General Theory of Relativity and shows the importance that geometry has to the theory. Starting from simpler and more general considerations, it goes on to detail the latest developments in the field and considers several cutting-edge research areas. It discusses Einstein's theory from a geometrical and a field theoretic viewpoint, before moving on to address gravitational waves, black holes and cosmology.

This book describes applications of the PDE methods to the construction and study of Ricci-flat metrics with special holonomy. Particular attention is paid to Ricci-flat Kahler (Calabi-Yau) structures on complex manifolds and hyper-Kahler structures on K3 surfaces. Complex manifolds are also an object of study in algebraic geometry and special consideration is given to the interplay between some well-known algebraic varieties (K3 surfaces, Fano threefolds, for example) and differential-geometric structures of special holonomy. The interplay between the gluing techniques, Calabi-Yau theory and algebraic geometry is further illuminated by the connected sum construction of compact 7-dimensional manifolds with holonomy G2.