Harmonic maps to buildings and singular perturbation theory

摘要: The notion of a (uni)versal building associated with point in the Hitchin base is introduced. universal equipped harmonic map from cover given Riemann surface that initial among maps which induce cameral surface. versal obtained by relaxing uniqueness condition definition an object. In rank one case, leaf space quadratic differential defining base. main conjectures this paper are: (1) always exists; (2) asymptotics Riemann–Hilbert correspondence and non-abelian Hodge are controlled to any building; (3) spectral networks arise as inverse images singularities (4) buildings encode data 3d Calabi-Yau category whose stability conditions has connected component contains theorem establishes existence building, conjecture (3), well part (2), case two example introduced seminal work Berk–Nevins–Roberts on higher order Stokes phenomena. It also shown certain constructed asymptotic cone symmetric space.