On index of nonlocal elliptic equations associated with diffeomorphisms of manifolds with boundary

In the present talk we construct the topological index of nonlocal elliptic voundary value problems on the manifols with boundary. At first we consider the case of the isometric action of discrete group on the manifold with boundary. We obtain the index formula for the general case and also consider the numerical examples for the Euler operator.
Moreover we consider the case of nonisometric group action. We construct the topological index of nonlocal boundary value problems using the cyclic cohomologies. The Fredholm property of such problems is proven. We also consider the nonlocal problem associated with shear mappings of a finite cylinder and prove its ellipticity.
The result presented in the talk are published in the following papers: 1. Boltachev A. V., Savin A. Yu. Elliptic boundary value problems associated with isometric group actions // Journal of Pseudo-Differential Operators and Applications — 2021. — Vol. 12, no. 4. — P. 50. 2. Boltachev A. V., Savin A. Yu. Index of twisted elliptic boundary value problems associated with isometric group actions // Lobachevskii Journal of Mathematics — 2022. — Vol. 43, no. 10. — Pp. 2399–2410. 3. Boltachev A. V., Savin A. Yu. Periodic Cyclic Cocycles on the Boutet de Monvel Symbol Algebra // Russ. J. Math. Phys. — 2022. — Vol. 29. — Pp. 417–425. 4. Boltachev A. V., Savin A. Yu. Trajectory Symbols and the Fredholm Property of Boundary Value Problems for Differential Operators with Shifts // Russ. J. Math. Phys. — 2023. — Vol. 30, — Pp. 135–151. 5. Boltachev A. V., On ellipticity of operators with shear mappings, CMFD, 69, no. 4, PFUR, M., 2023, 565–577.