Appears in collection : 2025 - T2 - WS1 - Higher rank geometric structures, Higgs bundles and physics

This talk will present a survey on the analytic Langlands correspondence proposed by Etingof, Frenkel and Kazhdan, strengthening and generalizing an earlier proposal of myself. Whenever it is established, this correspondence amounts to a classification of the eigenstates of the quantised Hitchin Hamiltonians in terms of holomorphic connections called opers with real holonomy. The Separation of Variables (SOV) method known from the theory of integrable systems offers a promising proof strategy. If time permits, I will outline some results exhibiting the relation between the SOV method and the Hecke correspondences, revealing the geometry underlying the success of the SOV method in the case of Hitchin systems.