D-modules on Noncommutative Spaces. Noncommutative Local Algebra and Representations. Noncommutative Smooth Spaces

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The project is devoted to the following topics of noncommutative geometry: a) Theory of D-modules and differential operators on noncommutative locally affine spaces and schemes. b) Noncommutative local algebra and its applications to representation theory of quantum enveloping algebras and related algebras of mathematical physics. c) Noncommutative smooth locally affine spaces and related structures. Main purposes of the project: Studying D-modules on noncommutative spaces, in particular on quantum flag varieties and noncommutative smooth spaces. Developing an analogue of the crystallin approach to D-modules in the case of noncommutative spaces and schemes. Combining methods of D-module theory and noncommutative local algebra, study the spectrum and representations of quantum enveloping algebras and other important algebras of mathematical physics. Studying properties and important examples of noncommutative smooth spaces and their applications. Results should have impacts on noncommutative geometry, noncommutative algebra,representation theory, deformation theory and some other topics ofmathematical physics.Noncommutative geometry is a relatively new field of mathematics which is now becoming one of important tools, or ruther ways of thinking, in many areas of mathematics and theoretical physics. It takes roots in quantum mechanics and representation theory. But the main motivations come from recent amazing developments in mathematical physics (quantum groups and related quantized 'spaces') and from physics. In the recently proposed M-theory (which is nowadays regarded as a candidate for the theory describing all interactions existing in the Nature), the geometry of physical space-time is noncommutative. A considerable part of the project, the one concerned with smooth noncommutative spaces (which the investigator studies together with M. Kontsevich), is naturally related to M-theory on curved spaces. The inverstigator and his collegues expect that the language and new intuition of noncommutative spaces will be used not only in 'pure' mathematics, but also in modern physical theories.