Hello, neighbor! My name is Lark Song and I am a Chancellor’s Undergraduate Research Fellow. As I start my senior year this spring, I am working towards my undergraduate thesis under the supervision of Dr. Thomas Hales in the Mathematics Department. In my spare time, I enjoy the serenity that comes from reading and writing poems. I used to play the piano a lot, but not as much recently. I also love going on hikes, whether with friends or solo.

My project supported by this fellowship focuses on an isoperimetric problem derived from an old math puzzle known as the Kelvin problem. In 1887, Lord Kelvin asked one of the most difficult questions in geometry: What is the most efficient soap bubble foam in three-dimensional space? This question led to the discovery of an inherently related, simpler yet fundamental problem known as the Truncated Octahedral Conjecture. This conjecture asserts that among all parallelohedra – the only convex polyhedra capable of tiling space through translation alone – the semi-regular truncated octahedron, when compared to others of equal volume, has the smallest surface area. We investigate the five types of parallelohedra identified by Evgraf Fedorov in 1885. Our goal is to prove that the semi-regular truncated octahedron is not only a local minimum among truncated octahedra but is indeed the one with the smallest surface area globally. To date, we have determined local minima for the first two types of parallelohedra, namely parallelepipeds and hexagonal prisms, uncovering patterns of symmetry and beauty akin to spheres. For the remaining types, we are adopting a strategy similar to the one Dr. Hales used in his proof of the Kepler conjecture.

Addressing the full scope of this conjecture is challenging and requires sophisticated linear programming. Nonetheless, we anticipate producing notable partial findings at the end of the spring semester. The potential result of this research could mark a significant step forward in the field of isoperimetric problems in geometry, with potential implications for fields outside math such as particle physics, crystallography, and materials science.

After graduation, my plan is to pursue graduate studies in mathematics. My research interests are currently broad, spanning discrete geometry, algebraic geometry, number theory, mathematical physics, and possibly the Langlands program as I delve deeper into the math world. After that, I hope to build a career as a professional mathematician. I view this Chancellor’s fellowship as a crucial stepping stone towards achieving my goals. I am deeply grateful to the Frederick Honors College for its pivotal role in promoting undergraduate research within our community. I am excited to share what I learn and discover during this fellowship with all of you!