Koroteeva, Olesya2025-01-282025-01-282024-08https://hdl.handle.net/11299/269653University of Minnesota Ph.D. dissertation. August 2024. Major: Physics. Advisor: Arkady Vainshtein. 1 computer file (PDF); vii, 71 pages.This thesis presents a collection of results that interconnect recent advancements in supersymmetric gauge theories, string theory, algebraic geometry, and integrable systems. We commence by introducing a novel geometric object, the q-oper, and demonstrate its emergence in the analysis of various Seiberg-like duality frames within supersymmetric quiver gauge theories featuring four supercharges. Utilizing the oper approach, we investigate the geometry underlying the moduli space of vacua of these theories and uncover deeper structures within the associated dual integrable systems, specifically the XXZ spin chain and the trigonometric Ruijsenaars-Schneider model. These systems have previously been linked to supersymmetry through the Bethe/Gauge correspondence.Subsequently, we delve into the diverse array of integrable models that emerge from the Bethe/Gauge correspondence. At the pinnacle of the hierarchy of models derived from N = 2 theories with adjoint matter is the DELL (double-elliptic) model, originating from a six-dimensional theory compactified on a two-torus. We examine a certain double-scaling (Inosemtsev) limit where the theory with adjoint matter transitions into a pure Yang-Mills theory – and discuss its implications for both physics and integrability. Lastly, we explore the physics and geometry of many-body integrable models as the number of particles increases significantly. In this regime, a continuous hydrodynamic description of the degrees of freedom becomes advantageous. We analyze the Inosemtsev limit within such hydrodynamic systems.enThesis or Dissertation