Humphreys, Reece

Symmetries are a driving force in the universe and continue to reveal themselves
the deeper we look. While many undergraduate mathematics students
are introduced to symmetries with group theory, many practical applications
are often overlooked. The application of symmetries provides
deep insights into various problems and can frequently simplify complex
mathematics. Applying symmetries typically requires high-level mathematics
that can be prohibitive for people within other disciplines. As such,
this paper explores how Lie groups, a mathematical structure for representing
symmetries, can be utilized in computing, physics, and control theory
to solve practical problems from within these elds. It also serves as an
introduction to the mathematics necessary to utilize Lie groups.

The amount of orbital debris has rapidly grown due to humanity's de-
sire to work in, explore, and utilize space. However, compared to most
types of pollution that people experience daily, it is unlikely that the aver-
age person will ever encounter space pollution. Yet, it poses a signi cant
threat to many technologies we depend on daily such as GPS and weather
prediction satellites. Orbital debris are the remnants of orbital collisions,
weapons tests, decommissioned satellites, and spent rocket stages that are
passing over our heads faster than bullets and containing similar energy
to hand grenades. This paper explores the existing models of orbital de-
bris generation, how clouds of debris evolve with respect to time, and their
rami cations.