Year 2019-2021

The project aims to explore a number of interconnected challenges of modern theory of elliptic operators on manifolds with the actions of groups. The ellipticity of such problems and their homotopic properties are investigated, index formulas are established.

Non-commutative elliptic theory is studied in the following geometric situations: (1) problems of relative elliptic theory (Sobolev problems), (2) problems on non-compact manifolds with group action. As part of this project, analytical, geometric and topological methods, both classical and those related to non-commutative geometry (A. Connes, A. Moscovici, R. Nest, B. Tsygan and others), as well as to coarse geometry (N. Higson, G.Yu., J. Roe) are developed.

- Savin Anton, Schrohe Elmar. Analytic and algebraic indices of elliptic operators associated with discrete groups of quantized canonical transformations. Journal of Functional Analysis, 2020, 278 - 5, 108400, IPF 1.637
- Nazaikinskii V. E., Savin A. Yu., Sipailo P. A.. Sobolev Problems with Spherical Mean Conditions and Traces of Quantized Canonical Transformations. Russian Journal of Mathematical Physics, 2019, 26 - 4, 483-498, IPF 0.874
- Sipailo P. A.. Traces of Quantized Canonical Transformations on Submanifolds and Their Applications to Sobolev Problems with Nonlocal Conditions. Russian Journal of Mathematical Physics, 2019, 26 - 1, 135-138, IPF 0.874
- Manuilov V. M.. Roe Bimodules as Morphisms of Discrete Metric Spaces. Russian Journal of Mathematical Physics, 2019, 26 - 4, 470-478, IPF 0.874

- The main goal of the project is to construct a noncommutative elliptic theory for a new class of operators, which we call G - operators associated with the representation of a group G by quantized canonical transformations on different manifolds. The operators under consideration are represented in three situations: 1) operators on closed manifolds, 2) relative (noncommutative) elliptic theory, and 3) noncommutative elliptic theory on manifolds with singularities. In all cases, the Fredholm property of the studied operators is studied and the index formulas for them are presented.

Anton Savin

- The results of the project have applications in the theory of inverse problems for hyperbolic equations, the theory of dynamical systems, as well as in noncommutative geometry.