Lectures On Complex Lagrangians, Integrable Systems And Quantization

This volume offers a pedagogical introduction to the rich interplay between complex Lagrangians, integrable systems, and quantization. Developed from the 2023 Oxford workshop and school connected to the National Science Foundation Focused Research Group on Complex Lagrangians, Integrable Systems, and Quantization, the book serves both as an entry point for newcomers and as a reference for researchers seeking a clearer path into the technical language of the field.Its central theme is the two-way bridge between classical geometry, especially holomorphic symplectic geometry, Lagrangian fibrations, Higgs bundles, and spectral data, and quantum structures such as spectral and quantum curves, exact WKB analysis, Stokes phenomena, wall-crossing, and recursion formalisms. The volume emphasizes concrete examples, hands-on constructions, and viewpoints that make advanced techniques accessible across neighboring areas of research.The lectures provide deep and coherent coverage of the subject. They explore the geometry of complex Lagrangians and holomorphic symplectic manifolds, spectral and quantum curves through enumerative and analytic approaches, asymptotic and metric aspects of Higgs-bundle moduli spaces, and 2D TQFT and ribbon-graph methods for quantization and recursion. Together, these chapters form a foundational companion to the related research volume and a practical guide for advanced study, seminars, and ongoing research in this rapidly evolving field.