Oka theory is a very geometric part of research in the area of Several Complex Variables. It is both a very traditional and a very active area of research. In the new 2020 AMS classifiation the new classification number 32Q56 "Oka principle and Oka manifolds" was introduced, witnessing about the importance of this field in modern Mathematics and its active development.In this project we will consider naturally given natural mathematical problems and work out whether for their solution a certain Oka principle can be applied, similar as we did in our solution to the Gromov-Vaserstein problem. On one hand these are applications to K-theory of rings, where we always work with concrete examples of rings, namely rings of holomorphic functions on Stein spaces. We also will expand our studies to rings of quaternionic holomorphic functions, an area which has been successfully developing in the last years and gives a wealth of wonderful examples of non-commutative rings of very natural origin to be studied in the realm of K-theory.On the other hand we will try to apply our methods developed for holomorphic and continuous matrix factorization to problems in theoretical robotics. A natural question is the localization and description of singularities of a kinematic chain depending on the joint variables. Another crucial problem is the transformation of the movement of the final tip of a robot into a movement of the robot’s joints, together with continuous dependence on the movement. We try to investigate under which conditions a continuous/smooth path described inside the workspace by the end manipulator can be achieved by a continuous/smooth movement of the joints. We also plan to develop a complexified version of the kinematic chain and prove approximation by holomorphic or partially holomorphic (Cauchy Riemann-) movements. Similar PDE-techniques as used in the D-bar problem could lead to find energy minimizing movements of the robot’s joints.CR geometry studies the geometry of real manifolds in connection with a complex structure. An important situation when CR geometry comes into play is when one considers smoothly bounded domains in complex spaces. For example, if a Stein space is realized as such, then the boundary of the domain must be pseudoconvex. In general, many CR geometric properties of a boundary imply interesting properties of the domain it bounds.In this project, we consider problems in CR geometry of a boundary having implications to the analysis and geometry of the domain. We shall work with weakly pseudoconvex boundaries and seek mild conditions them which implies, e.g., the regularity of the d-bar problems on the domain. In the strictly pseudoconvex case, there are finer boundary invariants that are important for geometry of the domain. Obstruction tensor, for example, is a boundary invariant that control the local extendability to the boundary of Kahler-Einstein potential of the Cheng-Yau metric. It has been studied recently with several important results, yet there are still interesting open problems need to be resolved.The D-bar problem is one of the most intensively studied PDEs in several complex variables.Different from classical PDEs — the most tractable problems are elliptic — the D-bar problem is a non-elliptic problem. However, the pioneering work of J. Kohn, L.