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We refine the regularity of noncollapsed limits of Einstein 5-manifolds. In particular, we prove uniqueness of tangent cones along the full top stratum of singular set and show that the entire singular set is contained in a countable union of bi-Lipschitz curves and points. Moreover, we establish that thesingular curve carries a real-analytic Einstein orbifold structure and is a geodesic in the limit space. The proofs rely on new 4-dimensional gap theorems for spherical and hyperbolic Einstein orbifolds. This is joint work with Tristan Ozuch.

Given a smooth projective curve X and a reductive group G, the geometric Langlands equivalence proved by Gaitsgory, Raskin et al. (roughly) gives an equivalence between sheaves on the stack Bun_G(X) of G-bundles on X (automorphic side) and quasi-coherent sheaves on the stack of G^-local systems on X (spectral side). To compute the image of an object under the geometric Langlands equivalence, one usually bootstraps from the Whittaker model. This method fails for the constant sheaf on Bun_G(X), which is “maximally singular.” Still, we will compute its image under the equivalence, confirming a conjecture of V. Lafforgue. As a consequence, when X is over F_q we find a spectral description for the constant function on Bun_G(X)(F_q