Radio-frequency (RF) resonators are essential hardware building blocks for many electronic systems that we frequently use almost every day, including wireless phones, satellite Global Positioning System (GPS), microwave ovens, and automobile radars. They are responsible for selecting the frequencies over which the system operates or for crafting its antenna radiation pattern. As the systems evolve from one generation to next, these RF resonators face the demand to become more agile, with reconfigurability to tune their frequencies on-the-fly during operation. Making them agile and efficient, however, is challenging under the current state of the art and the traditional design approach, which suffer from having a limited set of advanced mathematical tools at the RF design engineer's disposal. Because of this limitation, the RF design engineers have only been able to explore, optimize, and create design solutions from a relatively small subset out of the vast set of all possible design solutions. Specifically, the resonator’s geometric shape or topology is often the key for controlling its operation, and yet the traditional methods have only explored a few elementary shapes, which are intuitive to understand and model with existing commercial tools. This project aims at leveraging advanced tools from applied mathematics and theoretical physics to explore and utilized more complex shapes which have thus-far been untapped for designing novel RF resonators. It will fundamentally