In the past half century, cutting-edge discoveries in mathematics have occurred at the interface of three major disciplines: number theory (the study of prime numbers), representation theory (the study of symmetries using linear algebra), and geometry (the study of solution sets of polynomial equations). The interactions between these subjects has been particularly influential in the context of the Langlands program, arguably the most expansive single project in modern mathematical research. My research advances geometric techniques in representation theory, so that innate symmetries in arithmetic conjectures can be realized through geometry. By using geometry to realize representation-theoretic phenomena, my work allows one to synthesize present-day understanding of “depth-zero” phenomena with outstanding “positive-depth” Langlands conjectures.