This project focuses on the theoretical description of low-energy structure of atomic nuclei and the applications to other finite quantum many-body systems. The primary emphasis is on collective excitation and shape phenomena in finite nuclei, and on the structure of exotic nuclei under extreme conditions. Microscopic energy density functionals currently provide an accurate global description of nuclear ground-state properties, while algebraic theories, such as the interacting boson model, are successful in calculating the low-energy structure of medium-heavy and heavy nuclei. To describe spectral properties of nuclei based on a global theory, this project proposes a robust framework constructed by combining algebraic with microscopic theories. This theoretical framework will be used in the study of outstanding open problems in nuclear physics: Microscopic realization of quantum shape phase transition, understanding of the microscopic mechanism behind proton-neutron mixed symmetry, role of pairing correlations in exotic nuclei, the importance of proton-neutron pairing in N~Z nuclei, emergence of symmetries in complex quantum systems, etc. The method will also be used to predict the spectroscopy of exotic nuclei which are extensively studied nowadays at rare-isotope beam facilities around the world, including the planned SPIRAL2 facility in GANIL. The proposed method is general and will be applied to other fields of physics such as the study of analogous excitation modes in molecules or phase structure and low-energy excitations of Bose-Einstein condensates with intrinsic spin. Therefore the project points to giving a comprehensive theory of nuclei under various extreme conditions as well as of general finite quantum systems. In addition, an important component of the project concerns applicant's training-through-research and teaching/outreach activities, mainly in the form of training of students and participation at and contribution to advanced summer schools.