The main purpose of this research project is to study the topology and geometry of low and higher dimensional manifolds. Particular attention will be paid to symmetries of manifolds. The problems which will be studied cover a large spectrum of questions, ranging from very specific problems concerning low- dimensional manifolds to general problems concerning the geometry of positively curved higher dimensional manifolds. The main techniques which will be employed in this research include: gauge theory, surgery, homotopy theory, transformation groups and differential geometry. Manifolds are very natural objects to study. They are the spaces which are locally like Euclidean spaces. For example, the space in which we live (ignoring time), has three dimensions, making it a three-dimensional manifold, locally, but not necessarily globally, like Euclidean three-space. Taking time into account, physical space becomes a four-dimensional manifold. Whether it is globally Euclidean four-space or some other, possibly bounded, four-dimensional manifold is a deep question of cosmology. Answering this question is not a matter for mathematics alone, but exploring the various possibilities is an important and properly mathematical matter.
