Noncommutative elliptic theory

Year 2018

**Area of science:**

Mathematics

**Scientific direction:**

Partial differential equations

The main goal of the project is to create a noncommutative elliptic theory for a new class of operators associated with the representation of a group by quantized canonical transformations on different varieties. We study the Fredholm property of the studied operators and index formulas are presented for them.

**Key publications on the project:**

- A. Y. Savin, E. Schrohe, B. Y. Sternin Elliptic operators associated with groups of quantized canonical transformations. Successes of Mathematical Sciences. Vol. 73, 2018.
- A. Savin, E. Schrohe, B. Sternin Elliptic operators associated with groups of quantized canonical transformations. Bull. Sci. Math. 2018, 20p. (in print)
- P.A. Sipailo, Traces of quantized canonical transformations centered on a finite set of points. Differential Equations, 54, n. 4, 2018
- P.A. Sipailo, On traces of integral Fourier operators on subvarieties, Mathematical Notes, 2018 (in print)
- A. Y. Savin, B. Y. Sternin, “Elliptic problems with extensions-contractions on manifolds with boundary. C* - theory”, Differential Equations, 52 (2016), n. 10, 1383-1392.
- A. Y. Savin, B. Y. Sternin, Elliptic differential problems with extensions-contractions on manifolds with boundary. Differential Equations, 53 (2017), n. 5, 672-683
- A. Y. Savin, B. Y. Sternin, “Homotopy classification of elliptic problems, associated with actions of discrete groups on manifolds with boundary”, Ufa Mathematical Journal, vol. 8, No. 3, 2016, 126-134.
- A. Y. Savin, On homotopy classification of elliptic boundary value problems and problems with conjugation terms. Contemporary Mathematical Fundamental Directions. 2018, 8p.
- A. Y. Savin, B. Y. Sternin, “Elliptic G-operators on varieties with isolated singularities”, Contemporary Mathematics. Fundamental Directions, 59 (2016), 173-191.
- V. M. Manuilov. Pairs of operators with equal deviation from unitarity and their relative index. Matematicheskii Sbornik, 208 (2017), № 10, 113-125.
- A. Y. Savin, B. Y. Sternin, “On traces of operators associated with the actions of compact Lie groups”, Fundamental and Applied Mathematics. 2016. Vol. 21, n. 5.
- D.A. Loshchenova, On the traces of G-operators, focusing on submanifolds. Works of MIPT, Vol. 9, № 2(39), 2017, 85-96.
- N. R. Izvarina. On the symbol of nonlocal operators associated with a parabolic diffeomorphism. Eurasian Math. Journal, v.9, 2018, 11p.

Project goals

Project leader
All participants

Anton Savin

Doctor of Sciences, Professor, participant

Project results

The index problem for elliptic G-operators associated with the action of a discrete group of quantized canonical transformations is studied.

The relative index for balanced pairs of pseudo-differential operators whose symbols do not have to be elliptic and whose operators do not have to be Fredholm is studied.

Boundary value problems with nonlocal conditions are studied, in particular, problems associated with strains-compressions. Examples - problems for the perturbation of the Dirichlet problem for the Laplace operator in a ball by stretching and compression operators were analyzed. The problem of stable homotopic classification of elliptic operators with strains and compressions was also analyzed. This classification was obtained in the case where the compression acts identically on its attracting set.

Traces of G-operators associated with the action of a compact Lie group are studied for the case where the fixed points of the group action form a submanifold of arbitrary dimension embedded in a given submanifold. A complete description of such traces in terms of psedvo-differential operators with operator-valued symbols was obtained, the terms of their ellipticity were formulated, and the situation was reviwed when these traces are not Fredholm operators, but nevertheless they can be completed to Fredholm ones. Traces of quantized canonical transformations and, more generally, integral Fourier operators were also studied. The terms were formulated under which such traces are again integral Fourier operators.

The dependence of Fredholm solvability of parabolic diffeomorphisms of spheres on the smoothness index s in the scale of Sobolev spaces is studied. It is shown that the Fredholm solvability does not depend on s.

Application area