Explore complex Calabi-Yau threefolds with boundary in this 56-minute lecture by Simon Donaldson from Imperial College London, presented at the University of Miami. Delve into joint work with Fabian Lehmann, examining 3-dimensional complex manifolds with nowhere-vanishing holomorphic 3-forms. Learn about the extension of Hitchin's local Torelli Theorem to manifolds with boundary, focusing on the new data of closed 3-forms on 5-dimensional boundaries. Discover the application of sub-elliptic and Nash-Moser theory in proving results for pseudoconvex boundaries, with discussions on potential finite-dimensional obstruction spaces. Gain insights into related boundary value problems for G2 structures in seven dimensions and explore fascinating connections to classical affine differential geometry in real dimension three.