9424344 Lesniewski It is proposed to continue and expand in new directions an ongoing program of studying quantized spaces and superspaces by means of the techniques of operator algebras and non-commutative differential geometry. The goals of the program are: (1) To understand the mathematics of quantization of systems (involving both finitely and infinitely many degrees of freedom) with complicated geometry of phase space; (2) To study the non-commutative geometric principles underlying the physics of the microworld, and in particular possible implications for quantum gravity; (3) To explore new mathematical phenomena relating analysis to other fields of mathematics. Non-commutative geometry is a new direction of mathematics created in the eighties. Its prime objective is to study objects, called non-commutative spaces or manifolds, which are not ordinary geometric objects but still have a geometric flavor. Examples of structures of this sort arise naturally e.g. in physical theories of subatomic phenomena. Methods of non-commutative geometry became a powerful tool in studying various problems in modern analysis and understanding the fundamentals of the physics of the microworld. It is proposed to study examples of non-commutative manifolds with emphasis put on applications to quantum physics. Also, it is proposed to relate some of the ideas of quantum theory to problems in number theory and modern analysis. ***
