Quantum Geometry: From Mirror Symmetry to Non-perturbative Physics

Physical theories typically rely on a perturbative formulation where the relevant quantities are computed order by order in a small coupling parameter. However, even if the challenge of accessing higher orders can be overcome, the resulting series is divergent, highlighting the shortcoming of the formulation and understanding of the theory. Within quantum geometry, parameter spaces of physical theories become mathematical moduli spaces and physical theories which are related by dualities share an overarching geometric structure. I will describe, on the one hand, how the study of quantum geometry has led to groundbreaking new ideas and connections within pure mathematics such as mirror symmetry, which bridges complex and symplectic geometry. On the other hand, I will show that the geometric treatment of quantization offers deep new insights and exact results for the interplay between perturbative and non-perturbative structures in physics.