ИСТИНА

New standpoints for the project are recent progress in noncommutative harmonic analysis(symmetries of Plancherel decompositions and explicit constructions of separations of spectra) and geometric descriptions of categories of double cosets for infinite-dimensional groups; The research will follow several lines: 1. to develop operational calculus for Fourier transforms on groups and for Plancherel decompositions of unitary representations; 2. to investigate various problems related to separation of spectra into uniform pieces in noncommutative harmonic analysis; 3. to continue my works on categories of double cosets on infinite dimensional-groups (trains), their actions om measure spaces, applications for finite and finite-dimensional groups; 4. to examine infinite-dimensional groups over finite fields and to solve some classification problems; 5. to investigate some spectral problems of ergodic theory; 6. to examine combinatorial counterpart of the group of diffeomorphisms of the circle and Thompson groups; 7. to examine some explicit spectral problems for commuting differential operators and the corresponding integral transforms;