In a variety of real-world applications, eigenvalues of linear partial differential operators describe physical phenomena of interest, e.g., light propagation, mechanical vibrations, and liquid sloshing. It is of practical and fundamental interest to study the dependence of an eigenvalue on a control variable, such as the material coefficient or the domain shape, and to engineer/design/optimize control variables to enhance relevant spectral properties. This project will develop and analyze new computational methods for solving extremal eigenvalue problems, especially involving challenging geometric constraints. The research activities will advance discovery and understanding in computational mathematics and mathematical physics, as well as more general areas of science and engineering through applications. Educational activities are integrated with research activities in four specific ways: (i) training of students (including K-12, undergraduate, and graduate students across different schools) and junior researchers at different levels, (ii) encouraging participation of researchers in the area (iii) dissemination and sharing of research results publicly, and (iv) organization of international workshops on proposed research topics. Due to the collaborative nature of this proposal, students will engage in activities across R1 and primarily undergraduate institutions. The aim of this project is to tackle two canonical extremal eigenvalue problems from the mathematical and en