Explore the geometric structures underlying four-dimensional quantum field theories with eight supercharges in this differential geometry and physics seminar. Delve into how supersymmetry creates rich algebraic and geometric frameworks associated with moduli spaces of vacua, which serve as powerful non-perturbative analysis tools for quantum field theories. Learn about the Coulomb branch as a distinguished submanifold of the moduli space and discover how it captures the low energy dynamics through algebraic completely integrable systems. Understand the ongoing classification program for N=2 theories based purely on Coulomb branch geometry, and examine how these geometric constraints can determine specific physical quantities within the theoretical framework. Gain insights into the intersection of advanced differential geometry and theoretical physics through concrete examples of how mathematical structures illuminate quantum field theory behavior.