We intend to study G2 manifolds and their relations with Calabi-Yau 3-folds. These are geometric objects in 6 and 7 dimensions that have been attracting much attention recently from both mathematicians working in differential geometry and from theoretical physicists studying super-string theory, super-gravity, and M-theory. We have a very clear objective in mind, to construct many new explicit examples of compact G2 manifolds starting from a Calabi-Yau 3-fold N possessing an anti-holomorphic involution. Such a structure determines a special Lagrangian submanifold L of N. There is then a canonical way to construct a 7-dimensional orbifold M by taking a quotient of N x S1 by an action defined from the given involution and a natural involution on the S1 circle factor. The singular set of this orbifold is the union of two copies of the special Lagrangian submanifold L. We plan to resolve these singularities and obtain smooth, compact 7-manifolds with holonomy G2.The proof of the existence of a parallel G2-structure on the resulting smooth 7-manifold will involve techniques of non-linear analysis similar to those developed by the proposed scientist in charge, Professor Dominic Joyce when he constructed the first examples of smooth compact G2 manifolds in 1993-4. Such constructions are important both for differential geometry and for super-string and super-gravity theories. The achievement of this goal would be an important step in the evolution of our understanding of these physical theories, and their close relations to differential geometry. Much of the recent spectacular progress in research in differential geometry has been inspired by similar questions arising from theoretical physics, and the cross-disciplinary benefits to both sides have been unquestionably significant. An Incoming International Fellowship would be a great opportunity for the proposed researcher, Dr Spiros Karigiannis, to further develop his training and advance his career.