A large class of important problems concerning the symmetries of super=AD string theories and non=ADcommutative geometry will be studied. Superstring theory is a promising model proposed to unify all the forces in Nature. The model possesses remarkable symmetries, called dualities, and suggests an exciting new understanding of the structure of space in terms of newly discovered non=ADcommutative geometries. A non-commutative geometry is one for which a measurement of one coordinate (say x) introduces an uncertainty in another seemingly independent coordinate (say y). This is similar to but quite distinct from the usual situation in quantum physics for which measurement of a particle's location introduces an uncertainty in the particle's speed. Evidently, quantum gravity requires the use of non-commutative geometry for a complete understanding. It is very important to properly include time evolution in this framework. Our research will do this.
