"The project's main goal is composed of a scientific and an academic training component, and it is to be implemented in the Department of Mathematics at the Technical University of Munich in the research group of Prof. Michael Wolf.The first scientific aim is to extend the mathematical framework underlying the ''dissipative paradigm'' of quantum computing, placing special emphasis on continuous-time evolutions that may be controllable to a certain extent and may include non-Markovian elements. Providing such a rigorous toolbox and extending vital theorems from the Hamiltonian case to the dissipative paradigm forms the first two scientific objectives. Two further objectives use these tools to achieve the project's aim to incorporate controllable dissipative evolution into central parts of Quantum Information Theory. Namely, we aim to establish a Quantum-Shannon-Theory for the controllable time-continuous case - relevant for quantum memories - and explore its basic features. Further, we aim to investigate the computational complexity of dissipative state preparation in quantum many-body systems, contrasting it to Hamiltonian Complexity. These objectives will be achieved mainly through analytic work, also employing computer simulations.Beyond research, the academic training component aims to prepare the fellow for a future career as an independent researcher. As specific objectives to achieve this, the fellow will mentor students and (co-)supervise research projects jointly with Prof. Wolf during the fellowship. Further, the fellow will (co-)teach a specialized lecture course and a seminar. Through active involvment in the research group and grants, the fellow aims to acquire essential leadership and management skills.The project objectives are of high relevance to the Work Programme PEOPLE since they strengthen the human potential in research in Europe by supporting the fellow in attaining a leading independent position as professor at a European university."