The focus of this proposal is on homogeneous continua. Progress in classifying homogeneous continua has been slow but fertile with new ideas. The results concerning the pseudo-arc and the Menger curve are the most spectacular studies, which have opened new areas in topology. We are still far from having a complete classification of homogeneous continua in 3-space, and may expect further surprising results. New results in this area can have significant impact on continuum theory and geometric topology. New examples of homogeneous continua may be important as geometric objects, admit interesting compact group actions, and play an essential role in topological dynamics. A significant part of the project concerns the study of homogeneous 1-dimensional continua, which is a well established area. It is intended to explore the main stream questions asked in the past by J. T. Rogers, P. Minc, D. P. Bellamy, W. Lewis and others. The following two questions, among others, are also asked: (1) Is each 2-dimensional locally connected homogeneous continuum in 3-space either a surface or the Pontryagin sphere? (2) Is 2-sphere the only simply connected, nondegenerate homogeneous continuum in 3-space?
Throughout mathematics and its applications the idea of symmetry plays a special role. The most important objects, such as special functions, curves, equations, polygons, polyhedra, manifolds and other geometric or topological spaces, frequently have strong symmetric properties. Generally, a space has ``strong symmetry" if it admits a large collection of selftransformations that preserve all relevant properties. From the topological view point a (topological) space has ``strong symmetry" if it admits a rich collection of 1-to-1 continuous self-transformations whose inverses are also continuous. Such transformations are called homeomorphisms. The most studied topological spaces, such as manifolds, satisfy this description. If each point of a space can be mapped to every other one by some homeomorphism, the space is called homogeneous. The circle and 2-sphere are the simplest examples of such spaces embeddable in 3-space. The purpose of this project is to contribute to progress in classifying homogeneous compact connected spaces different from manifolds with particular stress on those contained in 3-space.
