We make rigorous (a generalization of) the formalism of Gaiotto, Moore, and Neitzke for constructing hyperkahler manifolds near semi-flat limits. In particular, we account for the effects of complicated (e.g., densely wall crossing) BPS spectra and singular fibers. This provides a general framework for proving results about Gromov-Hausdorff collapse of hyperkahler manifolds to semi-flat limits and completes the Strominger-Yau-Zaslow conception of mirror symmetry for hyperkahler manifolds at the level of hyperkahler geometry (as opposed to only constructing one complex structure). We characterize the dependence of the periods of the three canonical Kahler forms on the natural parameters of the construction, and in particular prove that for non-compact manifolds this dependence is affine- linear. Specializing to the case of moduli spaces of weakly parabolic SU(2) Higgs bundles on a sphere with four punctures, we prove that this construction yields all such manifolds which are sufficiently close to the semi-flat limit. This talk is based on joint work with Arnav Tripathy and Max Zimet.

Hitchin's equations are a system of gauge theoretic equations on a Riemann surface that are of interest in many areas including representation theory, Teichm\"uller theory, and the geometric Langlands correspondence. The Hitchin moduli space carries a natural hyperk\"ahler metric.  An intricate conjectural description of its asymptotic structure appears in the work of Gaiotto-Moore-Neitzke and there has been a lot of progress on this recently.  I will discuss some recent results using tools coming out of geometric analysis which are well-suited for verifying these extremely delicate conjectures. This strategy often stretches the limits of what can currently be done via geometric analysis, and simultaneously leads to new insights into these conjectures.