Singular Lagrangian varieties and related normal form problems in the semi-classical analysis of quantised integrable systems

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The development of resurgent analysis has provided a new look on the perturbation series arising in quantum mechanics. These studies have in turn raised new attention on the theory of singular Lagrangian varieties in connection with the singular Bohr-Sommerfeld rules. For instance, problems such as the convergence of the series in the normal forms turned out to be of particular interest. The methods coming from algebraic geometry and homological algebra provide powerful tools for studying universal models of quantised integrable systems. The objective of this research program is to fullfill this study and to explore the universal modells by sheaf theoretical methods. In particular, we wish to find interactions between semi-classical analysis and standard concepts of singularity theory such as the Gauss-Manin connections, or mixed Hodge structure.