The project is based on the transfer of knowledge in the following Tasks:

- Semigroups and differential equations
- 1.a Asymptotic properties of semigroup orbits
- 1.b Functional calculi, semigroups and partial differential equations
- 2. Fine spectral theory of differential and difference operators
- 2.a Fredholm properties of differential and difference operators: an evolution semigroup approach
- 2.b Jacobi operators

- 3.Operator theory methods for numerical analysis of differential equations
- 3.a Exact estimates of quantities relevant to computation: operator theory approach
- 3.b Convergence of iterative methods.

Methods and techniques of the project include (but not reduce to):

- Resolvent language: boundary behavior of local resolvents, resolvent estimates;
- Spectral and perturbation theory of linear operators;
- Spaces of analytic, harmonic and subharmonic functions and associated operators on them;
- Concepts of R-boundedness and maximal regularity;
- Functional calculi for partial differential operators;
- Banach algebras technique;
- Evolution semigroups technique in the study of Fredholm properties;
- Exponential dichotomies, index theory, spectral flow, Evans and spectral shift functions theories;
- Numerical ranges methodology;
- Markov semigroups, kernel operators, positive semigroups, ergodic theory technique;
- Nonlinear analysis approaches (degree theory for nonlinear Fredholm maps, holomorphic maps and their geometric properties).

The progress will be based upon importing tools new at IM PAN and related to complex function theory, theory of partial differential equations, topology, numerical analysis and mathematical physics. We expect that new methods will also be worked out just during the phase of Transfer of Knowledge.