Discrete Subgroups of Lie Groups

Symmetry describes the structures of the world: One symmetry governs the progress of time, while another governs the quantum state of an electron. Continuous symmetries concern things such as motion through space and the rotation of a sphere. Unlike a cube, which needs discrete 90-degree rotations to look like itself again, a sphere can rotate fluidly without breaking its core structure. These symmetries are described by fundamental mathematical objects called Lie groups.

Such mathematical structures are found in everything from quantum physics to general relativity to robotics. Better understanding Lie groups, therefore, could yield powerful new tools for studying the universe.

The new collaboration will investigate this area via three interconnected subfields: randomness, representation theory, and geometry and dynamics.