The objective of this proposal is to study the mathematical theory of recovering the internal parameters of a medium and physical properties from the incomplete measurements obtained on the boundary of the medium. The physical situation is modeled by partial differential equations and the goal is to determine the coefficients of the partial differential equations from some measurements of the solutions on the boundary. The topics of this research include the inverse problems for systems of partial differential equations (the Navier-Stokes equations and Schroedinger equations with Yang-Mills potentials) and the inverse problems on unbounded domains (an infinite slab and an infinite cylinder). The proposer pursues three specific aims: (1) to prove unique determination results; (2) to establish and improve stability estimates; (3) to develop analytical reconstruction formulas. The main approach is the new strategies on the construction, analyzing and control of the solutions. They are mathematically challenging because less information about the solutions on the boundary is available and a deep understanding of systems of differential operators and domains is required. Inverse boundary value problems consist of recovering internal parameters by boundary measurements. Such problems and their solutions are critical to scientific endeavors, because in many instances it simply is not possible or feasible to cut something open to see what is inside. One must use indirect methods to "see" inside objects and measure vital phenomena without harming the thing being analyzed, and the key to being able to that is inverse problems. However, collecting data from the whole boundary is sometimes either not possible or extremely expensive in practice. The proposed research provides new insights to inverse problems when the data obtained by boundary measurements is incomplete. The problems addressed in this proposal arise in medical imaging, oil exploration, diagnosis of human blood fluid, among others. The proposed research can potentially have long-term impacts in those application areas by providing the fundamentally mathematical analysis. The proposer will incorporate some of the ideas and techniques developed in this proposal into a graduate level class, and mentor graduate and undergraduate (minority) students.