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Delve into the intricacies of quantum geometry and Fuchsian systems in this comprehensive lecture from the TRSalento school held in September 2022. Explore advanced mathematical concepts as presented by Raphaël Belliard over the course of nearly two hours. Gain insights into the complex interplay between quantum mechanics and geometric structures, with a specific focus on Fuchsian systems. Access additional resources and materials related to this in-depth exploration by visiting the official school website at https://sites.google.com/view/tr-salento-2022/.